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<Sk^-

Conceptual

C^L
sct

A High School Physics Program

written ana illustrated by

AOL ^4(=lf("

Addison-Wesley Publishing Company, Inc.

Mcnlo Park, California- Reading, Massachusetts* Don Mills, Ontario
Wokingham, England Amsterdam Sydney Singapore
Tokyo Madrid Bogota Santiago San Juan

'

Pilot Teachers

Consultants

Marshall Ellenstein
Ridgewood High School
Norridge, Illinois

Clarence Bakken

Paul Robinson
Edison Computech High School
Fresno, California

Art

Nathan A. Untennan
Glenbrook North High School

Sheron Snyder
Mason High School
Mason, Michigan

Northbrook,

Palo Alto High School

Palo Alto, California

Illinois

Farmer

Gunn High School

Palo Alto, California

Nancy T. Watson

Charles A. Spiegel

Burris Laboratory School

Muncie, Indiana

California State University,

Dominguez

Developmental editor: Andrea G. Julian

Text and cover designer: Emily Kissin

Production designer: Rodelinde Albrecht

Photo editor: Barbara B. Hodder
Indexer: Suzanne Aboulfadl

Cover Photograph:

Ben Rose/The Image Bank

1987 by Addison-Wesley Publishing Company, Inc.

No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by
any means, electronic, mechanical, photocopying, recording, or
otherwise, without the prior written permission of the publisher.
Printed in the United States of America. Published simultaneously

Copyright

All rights reserved.

in

Canada.

ISBN D-EDl-2D7ca-l
13 14 15 16 17 18 19 20

Vh

94 93 92 91

Dominguez

Hills, California.

Hills

Acknowledgments
wish to especially thank my two friends Marshall Ellenstein
and Paul Robinson, not only for their helpful critiques and for
I

piloting the

first

draft in their classes, but for urging

me to write

book in the first place. Special thanks are also due Charlie
Spiegel, whose many suggestions have been incorporated in
this

nearly every chapter. I am thankful to the following teachers for

critiquing sample chapters of the pilot manuscript: Hal Eastin,
Cortez High School, Phoenix, AZ; Donald G. Gielow, Sequoia
High School, Redwood City, CA; Robin Gregg, Menlo-Atherton
High School, Atherton, CA; Willa D. Ramsey, Gompers Secondary
School Center, San Diego, CA; and Nathan A. Unterman,
Glenbrook North High School, Northbrook, IL. I thank my son

James (Figure

14-24) for cutting the stencils for the artwork

and

my nephew Robert Baruffaldi (Figure 20-9) for his suggestions. Finally, I am most grateful to my assistant Helen Yan (Figures 20-8,
27-20) for helping through every stage of the book, from
1
planning and typing all manuscript to hand-lettering the final
artwork.
23-1

San Francisco

Paul G. Hewitt

in

Contents

To the Student

xi

Newton's Second Law of

Motion Force and

37

Acceleration

About Science

4.1
1.1
1

.2

The Basic Science Physics

Mathematics The Language

4.2

Mass

4.3

Newton's Second

4.4

Statics

4.5

Friction

4.6

4.7

Applying Force Pressure

Free Fall Explained

4.8

Falling

of Science
1 .3

1.4
1

.5

The
The

Scientific

Method

Scientific Attitude

Scientific

Hypotheses Must Be

Testable
1

.6

.7

Unit

Science and Technology

In Perspective

10

2.3

Velocity

13

2.4

Acceleration

14

2.5

Free Fall: How Fast

Free Fall: How Far
Air Resistance and Falling
Objects

17

Is

Relative

How Fast, How Far, How

Quick How Fast Changes

19

21

25

3.1

Aristotle

on Motion
Copernicus and the Moving
Earth
Galileo on Motion
Newton's Law of Inertia
Mass A Measure of Inertia
The Moving Earth Again

25
26

3.5
3.6

Newton's Third

5.3

Identifying Action

5.4

5.5

5.6
5.7

6.1

Newton's First Law of

Motion Inertia

3.4

Interactions Produce Forces

5.2

21

ES

3.3

5.1

10
11

^^^

3.2

and Air Resistance

Motion
Speed

2.8

42
44
46
48

2.2

2.7

41

Newton's Third Law of

Motion Action and
Reaction

Motion

2.6

Law

37
38
39

Mechanics

2.1

Force Causes Acceleration

Resists Acceleration

6.2

6.3

Law
and

54

54
55
56

Reaction
Action and Reaction on
Bodies of Different Masses
Why Action and Reaction
Forces Don't Cancel
The Horse-Cart Problem
Action Equals Reaction

61

Vectors

65

Vector and Scalar Quantities

Vector Representation of
Force
Vector Representation of

65
66

57
58

59

67

Velocity
6.4
6.5

Geometric Addition of Vectors

Equilibrium

26
28

6.7

Components
Components

31

6.8

Projectile

33

6.9

Upwardly Moving

6.6

of Vectors

of Weight
Motion
Projectiles

68
70
73
74
75
78

.entum

M(
7.1

Gravitational Interactions

146

11.2

Acceleration Due to Gravity

Gravitational Fields

146
148

85

omentum

7.2

Impulse Equals Change

7.3

Bouncing

7.4

Conservation of

7.5

Collisions

7.6

Momentum

in

85
86

Momentum
Momentum

11.1

11.3

Weight and Weightlessness

151

90

11.4

Ocean Tides

153

91

11.5

Tides in the Earth and

157

Atmosphere

91

Vectors

97

11.6

Black Holes

Motion

158

Energy

101

8.1

Work

101

12.1

Earth Satellites

162

8.2

Power

103

12.2

165

8.3

Mechanical Energy
Potential Energy
Kinetic Energy
Conservation of Energy
Machines

103

12.3

Circular Orbits
Elliptical Orbits

104

12.4

Energy Conservation and

Satellite Motion
Escape Speed

168

Circular Motion

175

Rotations and Revolutions

Rotational Speed
Centripetal Force
Centripetal and Centrifugal
Forces
Centrifugal Force in a
Rotating Reference Frame
Simulated Gravity

176
176
178
179

8.4

8.8

Efficiency

105
107
109
112

8.9

Energy

115

8.5

8.6
8.7

D
9.1

9.2

9.3

9.4

for Life

Center of Gravity-

119

Center of Gravity
Center of Mass
Locating the Center of Gravity
Toppling

119

13.5

13.6

Stability

ED

Universal Gravitation

134

10.1

The
The
The
The

134
135
137
138

10.4

10.5

10.6

VI

Falling

Moon

Falling Earth

Law

of Universal

Gravitation
Gravity and Distance: The
Inverse-Square Law
Universal Gravitation

13.4

122

Center of Gravity of People

Falling Apple

13.3

121

9.6

10.3

D
13.2

9.5

10.2

12.5

13.1

123
125
129

iPi

Satellite

139

D
14.1

14.2
14.3

14.4
14.5

14.6
14.7

141

162

166

170

181

182

Rotational Mechanics

187

Torque
Balanced Torques
Torque and Center of Gravity

187
189

Rotational Inertia
Rotational Inertia and

192

Gymnastics
Angular Momentum
Conservation of Angular

Momentum

191

195
198

199


Special Relativity Space
la and
Time
15.1

15.2
15.3

Spacetime
Motion Is Relative
The Speed of Light
Constant

Is

15.4

The

15.5

The Second Postulate of

15.6

Time Dilation
The Twin Trip
Space and Time Travel

First Postulate of Special

204

205
206
207

209

Relativity

210

15.8

Solids

251

18.1

Crystal Structure

251

18.2

Density

18.3

Elasticity

18.4

Compression and Stretching

18.5

Scaling

253
255
257
259

Liquids

266
266
270

19.1

Pressure in a Liquid

211

19.2

Buoyancy

214
220

19.3

Archimedes' Principle
The Effect of Density on

Special Relativity
15.7

Uii

19.4

271

273

Submerged Objects
Biography: Albert Einstein

to

224

Special Relativity
Length, Mass, and Energy

226

16.1

Length Contraction

16.2

The Increase of Mass with

Speed
The Mass-Energy Equivalence
The Correspondence Principle

226
228

16.3

16.4

Unit II
Properties of Matter

The Atomic Nature of

Elements

17.2

17.4

Atoms Are Ageless

Atoms Are Small
Evidence for Atoms

17.5

Molecules

17.6

Compounds

17.7

The Atomic Nucleus

17.8

Electrons in the Atom

The States of Matter

17.3

17.9

Flotation
Pascal's Principle

275
277

Gases

283

20.1

The Atmosphere

20.2

20.4

Atmospheric Pressure
The Simple Barometer
The Aneroid Barometer

20.5

Boyle's

20.6

Buoyancy

20.7

Bernoulli's Principle

20.8

Applications of Bernoulli's

283
285
287
289
290
292
293
294

20.3

230
232

237

Law
of Air

Principle

238

Matter

17.1

19.5

19.6

Unit IH

299

Heat
238
239
240
241
242
243
243
245
248

D
21.1

21.2
21.3

21.4
21.5

21.6
21.7

Temperature and Heat

300

Temperature
Heat
Thermal Equilibrium
Internal Energy
Quantity of Heat
Specific Heat
The High Specific Heat of
Water

300
302
303
304
304
306
307

vu

Thermal Expansion
22.1

22.2
22.3

22.4
22.5

Lxpansion
Expansion
Expansion
Expansion
of Gases

311

of Solids

311

of Liquids

314
315
318

Water
and Compression

of

Why Warm

Air Rises

25.10

Bow Waves

25.11

ShockWaves

363
364

Sound

369

The Origin of Sound

The Nature of Sound in Air
Media That Transmit Sound
The Speed of Sound

26.8

Interference

26.9

Beats

369
370
372
372
373
374
374
376
377

Light

382
382
383
386
387

26.1

320

26.2
26.3

26.4

Transmission of Heat

323

23.6

Radiation
Absorption of Radiant Energy
Emission of Radiant Energy
Newton's Law of Cooling

323
325
327
328
329
330

23.7

The Greenhouse

331

23.1

23.2
23.3

23.4
23.5

Conduction
Convection

Effect

26.6

Forced Vibration
Natural Frequency

26.7

Resonance

26.5

27.5

Early Concepts of Light

of Light
Electromagnetic Waves
Light and Transparent
Materials
Opaque Materials

27.6

Shadows

27.7

Polarization

390
390
392

Color

399

27.1

27.2

Change of State

335

E\aporation
Condensation
Evaporation and
Condensation Rates

335
337
338

24.4

Boiling

24.5

Freezing
Boiling and Freezing at the

339
340

24.1

24.2

24.3

24.6
24.7

Regelation

341

24.8

Energy and Changes of State

342

Unit IV

349

Sound and Light

Vibrations
25.1

25.2
25.3

25.4

and Waves

350

Wave Motion
Wave Speed

353
354
356
356
357
359

25.6
25.7

Interference

25.8

Standing \ ves
The Doppler Effect

25.9

VHl

350

Vibration of a Pendulum
Wave Description

Transverse Waves
Longitudinal Waxes

25.5

27.3

27.4

341

Same Time

351

361

The Speed

The Color Spectrum

Color by Reflection
28.3
Color bv Transmission
28.4
Sunlight
28.5
Mixing Colored Light
Complementary Colors
28.6
28.7
Mixing Colored Pigments
28.8
Why the Sky Is Blue
28.9 Why Sunsets Are Red
28.10 Why Water Is Greenish Blue
28.11 The Atomic Color Code
Atomic Spectra
28.1

28.2

Reflection

and Refraction

399
400
402
403
404
406
407
409
41
41

413

419
419
420

29.1

Reflection

29.2

The Law

29.3

Mirrors

421

29.4

Diffuse Reflection

423

of Reflection

29.5

Sound

Reflection of

Refraction
29.7
Refraction of Sound
29.8
Refraction of Light
29.9
Atmospheric Refraction
29.10 Dispersion in a Prism
29.6

29.11

The Rainbow

29.12 Total Internal Reflection

424
425
426
428
429
432
432
434

Lenses

440
440

30.3

Converging and Diverging

Lenses
Image Formation by a Lens
Constructing Images Through

30.4

Ray Diagrams
Image Formation Summarized

30.1

30.2

30.5

Some Common

447

32.6
32.7

Bfii

Electric Field Lines

33.3

Electric Shielding

33.4

Electric Potential

33.5

Electric Potential

33.6

The Van de Graaff Generator

502
504

EZ]

Electric

Current

509

34.3

451

34.4

Voltage Sources
Electrical Resistance

Defects of Lenses

452

34.5

34.6

and

457

Interference
Huygens' Principle

31.2

Diffraction

Interference

.3

31.4

Energy

Defects in Vision

Diffraction

Young's Interference

34.7

501

509
510
5

Ohm's Law
Ohm's Law and Electric Shock
Direct Current and Alternating

512
513
514
517

Current
34.8

457
460
462
463

495
497
499

Electric Fields

Electric Current

Some
Some

495

33.2

Flow of Charge

The Eye

488
490

33.1

34.2

30.7

and

487

Potential

34.1

30.6

31.1

Electric Fields

448
450

Optical

Instruments

30.8

442
444

Charging by Friction and

Contact
Charging by Induction
Charge Polarization

32.5

The Speed of Electrons

in a

518

Circuit

34.9

The Source of Electrons

in a

519

Circuit

34.10 Electric Power

520

Experiment
466

31.5

Single-Color Interference

31.6

from Thin Films

Iridescence from Thin Films

31.7

Laser Light

467
469

31.8

The Hologram

471

35.3

Series Circuits

35.4

Parallel Circuits

35.5

Schematic Diagrams
Combining Resistors

UnitV
Electricity

477

Electric Circuits

525

35.1

A Battery and a Bulb

35.2

Electric Circuits

525
526
527
529

35.6

Compound

and Magnetism
35.7

531
in a

532

Circuit

Parallel Circuits

and

535

Overloading
32.1

32.2
32.3

32.4

Electrostatics

478

Electrical Forces and Charges

Conservation of Charge

479

Coulomb's Law
Conductors and Insulators

482
486

481

i\

ctism
36

jo3

538

Magnetic Poles

Magnetic Fields
The Nature of a Magnetic

539
540

39.1

541

39.2
39.3

Field
36.4
36.5

36.6
36.7
36.8

36.9

Magnetic Domains
Electric Currents

and

Magnetic Fields
Magnetic Forces on Moving
Charged Particles
Magnetic Forces on CurrentCarrying Wires
Meters to Motors
The Earth's Magnetic Field

542
544

39.4
39.5

39.6

585

The Atomic Nucleus

Radioactive Decav

585
586
588
589

Radiation Penetrating Power

Radioactive Isotopes
Radioactive Half Life
Natural Transmutation of

39.7

547
39.8

39.9

Transmutation of
Elements
Carbon Dating
Uranium Dating
Artificial

39.10 Radioactive Tracers

39.11 Radiation

Electromagnetic
Induction

554

Electromagnetic Induction
Faraday's Law
Generators and Alternating
Current

554
556
557

37.4

Motor and Generator

Comparison

560

37.5

Transformers
Power Transmission
Induction of Electric and
Magnetic Fields
Electromagnetic Waves

560
564
566

37.1

37.2
37.3

37.6
37.7
37.8

Unit VI

598
600
600
601

40.1

Nuclear Fission

40.2

The Nuclear Fission Reactor

Plutonium
The Breeder Reactor
Mass-Energy Equivalence

606
609
610
612
613
616
618

40.4
40.5

40.6

Nuclear Fusion

40.7

Controlling Nuclear Fusion

Appendixes
A
B

Units of Measurement
Working with Units in Physics

Vector Applications
Exponential Growth and

Atomic and Nuclear Physics

and You

597

606

40.3

571

594

Nuclear Fission and

Fusion

IJ

567

591

Elements

546

548
550

The Atomic Nucleus and

623
625
628
632

Doubling Time

The Atom and the

572

Quantum

38.3

Models
Light Quanta
The Photoelectric Effect

38.4

Waves

38.1

38.2

38.5

as Particles
Particles as Waves

38.7

Electron
Relative

38.8

Quantum

38.6

V.

Si.

Pi

of

Atoms

572
573
574
575
576
577
579
582

Glossary

635

Index

649

Photo Credits

660

To the Student
you can't enjoy a game unless you first know the rules.
Whether it's a ball game, computer game, or simply a
party game if you don't know the rules, the game
you aaiss out. the
is pointless and of little value.

same applies to nature and its rules what physics is

about. Nature means more to people who understand its
rules than to those who don't. To help you learn about
and comprehend nature's rules. physics is treated
Conceptually rather than mathematically in this book,
physics concepts are in english, with equations as
"guides to thinking" rather than recipes for algebraic
problem solving. this means you'll really be able to
comprehend the physics you study. then if you take a
follow-up course in physics, you'll get algebraic problem
solving. so comprehension now, and if computation
follows later, it will be with understanding .?
Enjoy your physics?

XI

About Science

Advances

in science

have brought many changes to the world.

were unfamiliar with television, jet air-

Fifty years ago, people

and how to prevent polio and simple tooth decay. Five

hundred years ago, the earth was considered to be unmoving
and the center of the universe. No one knew what makes the
stars shine; yet today, we are preparing to travel to them
with
the same energy that makes them shine.
Science is not something new. It goes back before recorded
history, when people first discovered regularities and relationships in nature. One regularity was the appearance of the star
patterns in the night sky. Another was the weather patterns during the year when the rainy season started or the days grew
longer. People learned to make predictions based on these regularities, and to make connections between things that at first
seemed to have no relationship. More and more they learned
about the workings of nature. That body of knowledge, growing
all the time, is part of science. The greater part of science is the
methods used to produce that body of knowledge. Science is an
activity
a human activity as well as a body of knowledge.
planes,

.1

The Basic Science

Physics

Science is the present-day equivalent of what used to be called

natural philosophy. Natural philosophy was the study of unanswered questions about nature. As the answers were found,
they became part of what is now called science.
The study of science today branches into the study of living
things and nonliving things: the life sciences and the physical

About Science

The life sciences branch into areas such as biology, zooland botany. The physical sciences branch into areas such as
geology, astronomy, chemistry, and physics.
Physics is more than a part of the physical sciences. It is the
most basic of all the sciences. It's about the nature of basic things
such as motion, forces, energy, matter, heat, sound, light, and the
insides of atoms. Chemistry is about how matter is put together,
how atoms combine to form molecules, and how the molecules
combine to make up the many kinds of matter around us. Biology is more complex still and involves matter that is alive. So
underneath biology is chemistry, and underneath chemistry is
physics. The ideas of physics reach up to these more complicated sciences. That's why physics is the most basic science. You
can understand science in general much better if you first have
some understanding of physics.
sciences.
ogy,

1.2

Mathematics

The Language of Science

Science
it

made its greatest headway in the sixteenth century, when

was found

matically.

that nature can be analyzed

When

and described mathe-

the ideas of science are expressed in mathe-

matical terms, they are unambiguous. They don't have the

"double meanings" that so often confuse the discussion of ideas
expressed in common language. When the findings in nature are
expressed mathematically, they are easier to verify or disprove
by experiment." The methods of mathematics and experimentation led to the

1.3

The

Scientific

enormous success

of science.

Method

Italian physicist Galileo Galilei (1564-1642) and the Engphilosopher Francis Bacon (1561-1626) are usually credited
as being the principal founders of the scientific method
method that is extremely effective in gaining, organizing, and
applying new knowledge. This method is essentially as follows:

The
lish

Although mathematics

is

very important to scientific mastery, it will not be

book focuses instead upon what should

the focus of attention in this book. This

and concepts of physics in English. When you

word descriptions that help you to visualize
ideas and concepts, with only secondary emphasis on mathematical descriptions, and when you postpone to a follow-up course the practice of algebraic
problem solving (which often tends to obscure the physics), you gain a better
comprehension of the conceptual foundation of physics.

come

first:

the basic ideas

learn physics primarily through

The

1.4

Scientific Attitude

Galileo (left) and Francis Bacon (right) have been credited as the
founders of the scientific method.

Fig. 1-1

1.

Recognize a problem.

2.

Make an educated guess

3.

4.
5.

a hypothesis about the answer.

Predict the consequences of the hypothesis.

Perform experiments to test predictions.

Formulate the simplest general rule that organizes the
three main ingredients: hypothesis, prediction, experimental outcome.

Although this cookbook method has a certain appeal, it has

not always been the key to the discoveries and advances in science. In many cases, trial and error, experimentation without
guessing, or just plain accidental discovery accounts for much of
the progress in science. The success of science has more to do

with an attitude common to scientists than with a particular

method. This attitude is one of inquiry, experimentation, and
humility before the facts.

The

Scientific Attitude

In science, a fact is generally a close agreement by competent

observers of a series of observations of the same phenomena. A
scientific hypothesis, on the other hand, is an educated guess
that is only presumed to be factual until proven so by experiment. When hypotheses have been tested over and over again
and have not been contradicted, they may become known as
laws or principles.
If a scientist believes a certain hypothesis, law, or principle
is true, but finds contradicting evidence, then in the scientific
spirit, the hypothesis, law, or principle must be changed or
abandoned. In the scientific spirit, the idea must be changed or
abandoned in spite of the reputation of the person advocating it.

About Science

As an example, the greatly respected Greek philosopher Aristotle (384-322 B.C.) claimed that falling objects fall at a speed
proportional to their weight. This false idea was held to be true
for more than 2000 years because of Aristotle's compelling
authority. In the scientific spirit, however, a single verifiable
experiment to the contrary outweighs any authority, regardless
of reputation or the number of followers or advocates. In modern science, argument by appeal to authority is of no value
whatever.
Scientists must accept their findings and other experimental
evidence even when they would like them to be different. They

between what they see and what they

most people, have a vast capacity
for fooling themselves.
People have always tended to adopt
general rules, beliefs, creeds, ideas, and hypotheses without
thoroughly questioning their validity and to retain them long
after they have been shown to be meaningless, false, or at least
questionable. The most widespread assumptions are often the
least questioned. Most often, when an idea is adopted, particular attention is given to cases that seem to support it, while cases

must

strive to distinguish

wish

to see, for scientists, like

that

seem

to refute

Scientists use the

it

are distorted, belittled, or ignored.

theory in a different way from its usage

word

everyday speech. In everyday speech a theory is no different

from a hypothesis a supposition that has not been verified. A
scientific theory, on the other hand, is a synthesis of a large body
of information that encompasses well-tested and verified hypotheses about certain aspects of the natural world. Physicists,
for example, speak of the theory of the atom; biologists have the
in

cell theorv.

The theories

undergo change. Scievolve as they go through stages of redefinition

and refinement. During the last hundred years, the theory of the
atom has been refined, as new evidence was gathered. Similarly,
biologists have refined the cell theory.
The refinement of theories is a strength of science, not a weakness. Many people feel that it is a sign of weakness to "change
your mind." Yet competent scientists must be experts at changing their minds. Thev change their minds, however, only when
confronted with solid experimental evidence to the contrary or
when a conceptually simpler hypothesis forces them to a new
point of view. More important than defending beliefs is improving them. Better hypotheses are made by those who are honest
of science are not fixed, but

entific theories

in the face of fact.

In

your education it is not enough to be aware that other people mav

but mainlv to be aware of \our own tendency to fool yourself.

fool vou,

try to

Scientific

1.5

Hypotheses Must Be Testable

Scientific

1.5

Hypotheses Must Be Testable

Before a hypothesis can be classified as scientific, it must conform to a cardinal rule. The rule is that the hypothesis must be
testable. It is more important that there be a means of proving it
wrong than that there be a means of proving it correct. At first
thought this may seem strange, for we think of scientific hypotheses in terms of whether they are true or not. For most things we
wonder about, we concern ourselves with ways of finding out
whether they are true. Scientific hypotheses are different. In
fact, if you want to distinguish whether a hypothesis is scientific
or not, look to see if there is a test for proving it wrong. If there is
no test for its possible wrongness, then it is not scientific.
Consider the hypothesis "Intelligent life exists on other planets somewhere in the universe." This hypothesis is not scientific.
Reasonable or not, it is speculation. Although it can be proved
correct by the verification of a single instance of intelligent life
existing elsewhere in the universe, there is no way to prove the
hypothesis wrong if no life is ever found. If we searched the far
reaches of the universe for eons and found no life, we would not
prove that it doesn't exist "around the next corner." A hypothesis
that is capable of being proved right but not capable of being
proved wrong is not a scientific hypothesis. Many such statements are quite reasonable and useful, but they lie outside the

domain

of science.

Question

Which of these is a scientific hypothesis?

a. Atoms are the smallest particles of matter

that exist.

b.

The universe is surrounded by a second universe, the

istence of which cannot be detected by scientists.

c.

Albert Einstein

is

ex-

the greatest physicist of the twentieth

centurv.

Answer
Only a

is

scientific,

because there

is

a test for

not only capable of being proved wrong, but

Statement b has no

test for possible

it

its

wrongness. The statement is

has been proved wrong.

in fact

wrongness and

is

therefore unscientific

Some

pseudoscientists and other pretenders of knowledge will not even consider

a test for the possible wrongness of their statements. Statement c is an assertion,

which has no test for possible wrongness. If Einstein was not the greatest ph\ sihow would we know? It is important to note that because the name Einstein
is generally held in high esteem, it is a favorite ol pseudoscientists. So we should

cist,

not be surprised that the

is

name ol Einstein, like

who uish to bring

cited often bv charlatans

points of view.

that of various religious figures,

respect to themselves

and

their

l.(

About Science

Science and Technology

Science and technology are different from each other. Science is
method of answering theoretical questions; technology is a
method of solving practical problems. Science has to do with
discovering facts and relationships between observable phenomena in nature, and with establishing theories that organize
and make sense of these facts and relationships. Technology has
to do with tools, techniques, and procedures for putting the find-

ings of science to use.

Another difference between science and technology has to do

with its effect on human lives. Science excludes the human factor. Scientists who seek to comprehend the workings of nature
cannot be influenced by their own or other people's likes or dislikes, or to popular ideas about what is correct. What scientists
discover may shock or anger people
as did Darwin's theory of
evolution. If a scientific finding or theory is unpleasant, we have
the option of ignoring it. Technology, on the other hand, can
hardly be ignored once it is developed. We do not have the option of refusing to breathe polluted air; we do not have the option of refusing to hear the sonic boom of a supersonic jetliner
overhead; we do not have the option of living in a nonnuclear
age. Unlike science, advances in technology must be measured
in terms of the human factor.
We are all familiar with the abuses of technology. Many people blame technology itself for the widespread pollution and resource depletion and even social decay in general so much so
that the promise of technology is obscured. That promise is a
cleaner and healthier world. It is much wiser to combat the dangers of technology with knowledge than ignorance. Wise applications of science and technology can lead to a better world.

Fig. 1-2

Science comple-

ments technology.

1.7

In Perspective
More than 2000 years ago enormous human effort went into the
construction of great pyramids in Egypt and in other parts of
the world. It was only a few centuries ago that the most talented
skilled artists, architects, and artisans of the world directed their genius and effort to the building of the great stone
and marble structures the cathedrals, synagogues, temples,
and mosques. Some of these architectural structures took more

and most

than a century to build, which means that nobody witnessed

both the beginning and the end of construction. Even the archi-

1.7

In Perspective

tects

and early builders who

lived to ripe old ages never

saw the

finished results of their labors. Entire lifetimes were spent in the

of construction that must have seemed without beginning or end. This enormous focus of human energy was inspired
by a vision that went beyond world concerns a vision of the
cosmos. To the people of that time, the structures they erected
were their "spaceships of faith," firmly anchored but pointing to
the cosmos.
Today the efforts of many of our most skilled scientists, engineers, artists, and artisans are directed to building the spaceships that already orbit the earth, and others that will voyage
beyond. The time required to build these spaceships is extremely
brief compared to the time spent building the stone and marble
structures of the past. Many people working on today's spaceships were alive before Charles Lindbergh made the first solo
airplane flight across the Atlantic Ocean. Where will younger
lives lead in a comparable time?
We seem to be at the dawn of a major change in human growth,
not unlike the stage of a chicken embryo before it fully matures.
When the chicken embryo exhausts the last of its inner-egg resources and before it pokes its way out of its shell, it may feel it is
at its last moments. But what seems like its end is really only its
beginning. Are we like the hatching chicks, ready to poke through
to a whole new range of possibilities? Are our spacefaring efforts

shadows

the early signs of a

new human

era?

The earth is our cradle and has served us well. But cradles,
however comfortable, are one day outgrown. With inspiration
that in many ways is similar to the inspiration of those who
built the early cathedrals, synagogues, temples, and mosques,
we aim for the cosmos.
We live at an exciting time!

Fig. 1-3

A NASA conception of a spaceship of the future. New discoveries await

who will venture beyond our solar system.

the people

About Science

chapter Review

Concept Summary

2.

Science is an activity as well as a body of

knowledge.
Physics is the most basic of all the sciences.
The use of mathematics helps make ideas
in science

The

scientific

is

3.

What

4.

Is

to science?

the usage of mathematics minimized

book?

is

(1.2)

the scientific

method?

(1.3)

a scientific fact something that

and unchanging? Explain. ( .4)

is

absolute

a procedure for answer-

ing questions about the world by testing edu-

cated guesses, or hypotheses, and formulating

general rules.
Hypotheses
in science must be testable;
they are changed or abandoned if they are
contradicted by experimental evidence.

mathematics important

is
is

in this

unambiguous.

method

Why
Why

5.

Distinguish between a hypothesis and a theory. (1.4)

6.

Theories in science undergo change. Is this a

strength or a weakness of science? Explain.
(1.4)

A theory

is

a body of knowledge and well-tested

hypotheses about some aspect of the natural

world.
Theories are modified as new evidence is

7.

proving
8.

wrong?

( 1

.5)

Distinguish between science and technologv.

Think and Explain

Important Terms

1.

fact (1.4)

Why
ing"

(1.3)

does science tend to be a "self-correctof knowing about things?

way

(1.4)
2.

principle (1.4)
scientific

method

(1.3)

tific

3.

is

ences?

physics the most basic of the

(1.1)

is

likely

a.

misunderstanding of
"But that's only a scien-

the

says,

theory"?

Make an argument

for bringing to a halt

the advances of technology.

Review Questions

Why

What

someone who

theory (1.4)

1.

it

mean to say that if a hypothesis

then there must be a means of

(1.6)

Science deals with theoretical questions, while

technology deals with practical problems.

law

it

is scientific

gathered.

hypothesis

What does

b.
scic.

Make an argument

that advances
nology should continue.
Contrast your two arguments.

in tech-

Mechanics

THERE'S A LOT OF PHYSICS IN PLAYING TUGO'WAR.

FOR EXAMPLE, ISN'T THE RIGHT END OF THE ROPE BEING
PULLED AS HARD AS PULL THIS LEFT END'? ISN'T IT
STILL BEING PULLED IF IT'S TIED TO A TREE INSTEAD OF
IN MY FRIEND'S HANDS?
DON'T WIN IF PUSH HARDER
ON THE GROUND RATHER THAN PULL HARDER ON THE ROPE?
AND DOESN'T IT TAKE ENER6Y TO EXERT THIS FORCE AND
impart MOTION 1 These sample ideas are from the
FOUNDATION OF PHYSICS --MECHANICS
WHAT THE NEXT
15 CHAPTERS ARE ABOUT.
READ, THINK, AND ENJOY
I

J?

\Ts^r
*:*.

ff^.

Motion

CA_

Motion

around us. We see

on the highway, in

^Y/a^:

everyday activity of peosway in the wind, and

with patience, we see it in the nightime stars. There is motion at
the microscopic level that we cannot see directly: jostling atoms
make heat or even sound; flowing electrons make electricity;
and vibrating electrons produce light. Motion is everywhere.
Motion is easy to recognize but harder to describe. Even the
Greek scientists of 2000 years ago, who had a very good understanding of many of the ideas of physics we study today, had great
difficulty in describing motion. They failed because they did not
understand the idea of rate. A rate is a quantity divided by time.
It tells how fast something happens, or how much something
changes in a certain amount of time. In this chapter you will
learn how motion is described by the rates known as speed,
velocity, and acceleration. It would be very nice if this chapter
helps you to master these concepts, but it will be enough for you
to become familiar with them, and to be able to distinguish
among them. The next few chapters will sharpen your understanding of these concepts.
is all

ple, of cars

2.1

Motion

it

in the

trees that

Is Relative
Everything moves. Even things that appear to be at rest move.
They move with respect to, or relative to, the sun and stars.
A book that is at rest, relative to the table it lies on, is moving at
about 30 kilometers per second relative to the sun. And it moves

10

Speed

2.2

11

even faster relative to the center of our galaxy. When we discuss

the motion of something, we describe its motion relative to something else. When we say that a space shuttle moves at 8 kilometers per second, we mean relative to the earth below. When we
say a racing car in the Indy 500 reaches a speed of 300 kilometers
per hour, of course we mean relative to the track. Unless stated
otherwise, when we discuss the speeds of things in our environment, we mean with respect to the surface of the earth. Motion
is

relative.

2.2
A moving

Speed
object travels a certain distance in a given time.

car,

example, travels so many kilometers in an hour. Speed is a

measure of how fast something is moving. It is the rate at which
distance is covered. Remember, the word rate is a clue that something is being divided by time. Speed is always measured in terms
of a unit of distance divided by a unit of time. Speed is defined
as the distance covered per unit of time. The word per means
for

"dividedby."

Fig. 2-1
A cheetah is the fastest land animal over distances
and can achieve peak speeds of 100 km/h.

Any combination

of distance

and time units

less

is

than 500 meters

legitimate for

miles per hour (mi/h); kilometers per hour (km/h); centimeters per day (the speed of a sick snail?); lightyears per century whatever
useful and convenient. The slash symbol

speed

is

(/) is

read as "per." Throughout this book we'll primarily use meters

per second (m/s). Table 2-1 shows some comparative speeds in
different units.

Motion

12

Table 2-1

Approximate Speeds

20km/h =
40 km/h =
60km/h =
80 km/h =
100 km/h =
120 km/h =

in Different Units

12 mi/h = 6 m/s
25 mi/h = 11 m/s
37 mi/h = 17 m/s
50 mi/h = 22 m/s
62 mi/h = 28 m/s
75 mi/h = 33 m/s

Instantaneous speed

Fig. 2-2

The speedometer

tor

A car dues not always move at the same speed. A car may travel
down a street at 50 km/h, slow to km/h at a red light, and
speed up to only 30 km/h because of traffic. You can tell the
speed of the car at any instant by looking at the car's speedometer The speed at any instant is called the instantaneous
speed. A car traveling at 50 km/h may go at that speed for only
one minute. If it continued at that speed for a full hour, it would
cover 50 km in that hour. If it continued at that speed for only
half an hour, it would cover only half that distance: 25 km. If it
continued for only one minute, it would cover less than
km.
1

a North American car gives

readings

ot

speed

both mi h and

in

Odometers

instantaneous
for the

km h.
US market

gi\e readings in mi; those for

the

Canadian market give

in km.

readings

Average speed
In

planning a

long

it

trip

by

car, the driver often

wants to know how

The car will cer-

will take to cover a certain distance.

tainly not travel at the same speed all during the trip. All the
driver cares about is the average speed for the trip as a whole.
The average speed is defined as follows:
,

average speed =

total

distance covered
time interval
:

Average speed can be calculated rather easily. For example, if

hour, we say
drive a distance of 60 kilometers in a time of
our average speed is 60 kilometers per hour (60 km/h). Or if we
travel 240 kilometers in 4 hours, we find:

we

total distance covered

average speed =
time interval
,

km
4h

240

60 km/h

Note that when a distance in kilometers (km) is divided by a time

in hours (h), the answer is in kilometers per hour (km/h).
Since average speed is the whole distance covered divided by
the total time of travel, it does not indicate the different speeds
and variations that may have taken place during shorter time
intervals. In practice, we experience a variety of speeds on most
trips, so the average speed is often quite different from the speed
at any instant, the instantaneous speed. Whether we talk about
average speed or instantaneous speed, we are talking about the
rates at which distance is traveled.

13

Velocity

2.3

Questions

With the speedometer on the dashboard of every car is

an odometer, which records the distance traveled. If

a.

the initial reading

trip

b.

is

and the reading

set at zero at the

is

35

beginning of a

km one-half hour later, what

has been your average speed?

Would it be possible to attain this average speed and
never exceed a reading of 70 km/h on the speedometer?

a cheetah can maintain a constant speed of 25 m/s, it

meters every second. At this rate, how far
will it travel in 10 seconds? In 1 minute?

2. If

will cover 25

Velocity

2.3

we can use the words speed and velocity

interchangeably. In physics, we make a distinction between the
two. Very simply, the difference is that velocity is speed in a given
In everyday language,

ing

we

When we

say a car travels at 60 km/h, we are specifysay a car moves at 60 km/h to the north,
are specifying its velocity. Speed is a description of how fast;

direction.
its

speed. But

if

we

Answers
(Are you reading this before you have formulated a reasoned answer in your

mind?

do you also exercise your body by watching others do push-ups?

When you encounter the many questions as above
throughout this book, think before you read the footnoted answers. You'll not
If so,

Exercise your thinking!

only learn more, you'll enjoy learning more.)

la.

b.

Average speed =

total distance

covered

35

time interval

km =

0.5 h

70 km/h

No, not if the trip started from rest and ended at rest, for any intervals with
an instantaneous speed less than 70 km/h would have to be compensated
with instantaneous speeds greater than 70 km/h to yield an average of
70 km/h. In practice, average speeds are usually appreciably less than peak
instantaneous speeds.

2.

In 10 s the cheetah will cover 250

m, and

in

1500 m, more than fifteen football fields! If

the time of travel, then the distance covered

minute

we know

(or 60

s) it

will cover

the average speed

and

is

distance = average speed x time interval

distance = (25 m/s) x (10 s) = 250
distance = (25 m/s) x (60

little

s)

thought will show that this relationship

average speed =

m
m

= 1500
is

simply a rearrangement

distance

time interval

ol

14

velocity

is

how

fast

and

what

in

direction."

We

Motion

will see in the

next section that there are good reasons for the distinction be-

tween speed and

velocity.

Question

The speedometer of a car moving northward reads

It passes another car that travels southward at
Do both cars have the samt speed? Do they have

60 km/h.
60 km/h.
the

Fig. 2-3

The car on the

cular track

may have

cir-

a con-

stant speed, but not a

constant velocity because

direction of motion
ing every instant.

is

its

chang-

same

velocity?

Constant Velocity
From the definition of velocity

it

follows that to have a constant

and constant direction.

Constant speed means that the motion remains at the same
speed the object does not move faster or more slowly. Constant
velocity requires both constant speed

direction

means

that the motion

is

in a straight line

the ob-

path does not curve at all. Motion at constant velocity

motion in a straight line at constant speed.

ject's

is

Changing Velocity
speed or the direction (or both) is changing, then the
considered to be changing. Constant speed and constant velocity are not the same. A body may move at constant
speed along a curved path, for example, but it does not move
with constant velocity because its direction is changing every
If

either the

velocity

is

instant.

In a car, there are three controls that are used to change the
velocity.

One

The second
third

2.4

is

is

is

it is used to increase the speed.

used to decrease the speed. The
is used to change the direction.

the gas pedal;

the brake;

is

it

the steering wheel;

it

Acceleration
We

can change the state of motion of an object by changing its

its direction of motion, or by changing both.
Any of these changes is a change in velocity. Sometimes we are
speed, by changing

Answer
Both cars have the same speed, but they have opposite
are

moving

in

Directional quantities are called vectors; velocity

quantities are called scalars; speed

be covered

in

velocities because they

opposite directions.

Chapter

6.

is

is

a vector. Nondirectional

a scalar. Vector and scalar quantities will

Acceleration

2.4

15

how fast the velocity is changing. A driver on a

two-lane road who wants to pass another car would like to be
able to speed up and pass in the shortest possible time. The rate
at which the velocity is changing is called the acceleration. Because acceleration is a rate, it is a measure of how fast the velocity is changing per unit of time:
interested in

acceleration

We

are

all

2 of velocity
= change
r-
r^

time interval

familiar with acceleration in an automobile. The

driver depresses the gas pedal, appropriately called the acceler-

'i

The passengers then experience acceleration, or "pickup"

sometimes called, as they tend to lurch toward the rear of
the car. The key idea that defines acceleration is change. Whenever we change our state of motion, we are accelerating. A car
that can accelerate well has the ability to change its velocity
rapidly. A car that can go from zero to 60 km/h in 5 seconds has
a greater acceleration than another that can go from zero to
80 km/h in 10 seconds. So having a good acceleration is being
"quick to change" and not necessarily fast.
In physics, the term acceleration applies to decreases as well
as increases in speed. The brakes of a car can produce large reator.

as

it is

tarding accelerations; that is, they can produce a large decrease

per second in the speed. This is often called deceleration, or negative acceleration. We experience deceleration when the driver of
a bus or car slams on the brakes and we tend to lurch forward.

Fig. 2-4

A car

is

accelerating whenever there

is

a change in

its

state of motion.

Acceleration applies to changes in direction as well as changes

you ride around a curve at a constant speed of 50 km/h,

in speed. If

you

acceleration as you tend to lurch toward

You may round the curve at constant
speed, but your velocity is not constant because your direction is
changing every instant. Your state of motion is changing; you
are accelerating. Now you can see why it is important to distinfeel the effects of

the outside of the curve.

guish between speed and velocity, and why acceleration is defined as the rate of change of velocity, rather than speed. Acceleration, like velocity, is directional. If we change either speed or
direction, or both,

we change

velocity

and we accelerate.

16

most

In

of this

along a straight

book we

line.

common

will be

When

Motion

concerned only with motion

motion is being consid-

straight-line

and velocity interchangeably.

not changing, acceleration may be expressed as the rate at which speed changes.

ered,

it

When

is

to use speed

the direction

is

change

in speed
time interval

acceleration (along a straight line)

Speed and velocity are measured in units of distance per time.

The units of acceleration are a bit more complicated. Since
acceleration is the change in velocity or speed per time interval,
its units are those of speed per time. If we speed up without
change in direction from zero to 10 km/h in
second, our
change in speed is 10 km/h in a time interval of 1 s. Our acceler1

ation (along a straight line)

change
acceleration =

is

speed
time interval

km/h

10

in

10km/h-s

The acceleration is 10 km/hs (read as 10 kilometers per hour

second). Note that a unit for time enters twice: once for the unit
of speed and again for the interval of time in which the speed is
changing. If you understand this, you can answer the following
questions. If you don't, maybe the answers to the questions will
be of help.

Questions
1.

Suppose a car moving

in a straight line steadily in-

speed each second, first from 35 to 40 km/h,

then from 40 to 45 km/h, then from 45 to 50 km/h. What

creases
is its

2.

its

acceleration?

In 5 seconds a car

moving

in a straight line increases its

speed from 50 km/h to 65 km/h while a truck goes from

rest to 15 km/h in a straight line. Which undergoes the
greater acceleration? What is the acceleration of each
vehicle?
Answers
1

We see
eration

2.

that the speed increases by 5

is

therefore 5

km/h during each

km/hs during each

1-s interval.

The

accel-

interval.

their speeds by 15 km/hr during the same

time interval, so their accelerations are the same. If you realized this without

The car and truck both increase

first

calculating the accelerations, you're thinking conceptually. The accelera-

tion of

each vehicle

is:

acceleration

_ change in speed _
time interval

km/h

km/hs

Although the speeds involved are quite different, the rates of change of speed
are the same. Hence the accelerations are equal.

Free Fall:

2.5

How Fast

Free

2.5

Fall:

Drop a stone and it

it starts from a

know
know

17

How Fast
Does

falls.

accelerate while falling?

it

and gains speed as

rest position,

it

falls.

We
We

this because it would be safe to catch if it fell a meter or

two, but not from the top of a tall building. Thus, the stone must
gain more speed during the time it drops from a building than

during the shorter time it takes to drop a meter. This gain in

speed indicates that the stone does accelerate as it falls.
Gravity causes the stone to fall downward once it is dropped.
In real

life,

ject. Let's

air resistance affects the acceleration of a falling ob-

imagine that there

is

no

air resistance,

and gravity

is

the only thing that affects a falling object. Such an object would
then be in free fall. Table 2-2 shows the instantaneous speed at

the end of each second of

from

rest.

fall

The elapsed time

of a freely-falling object
is

passed, since the beginning of the

Table 2-2

dropped

the time that has elapsed, or

Fig. 2-5

fall.

Free-Fall Speeds of Object

Dropped from Rest

riicipseG

(meters/second)

(seconds)

10

20
30
40
50

4
5

10/

Note in Table 2-2 the way the speed changes. During each second of fall the instantaneous speed of the object increases by an
additional 10 meters per second. This speed gain per second
the acceleration:

acceleration

change

in speed
time interval

Note that when the change

10 m/s
1

is

10 m/s J

speed is in m/s and the time inter2

in m/s (read "meters per second
squared"). The unit of time, the second, enters twice once for
the unit of speed, and again for the time interval during which
the speed changes.
val

is

in

s,

the acceleration

in

is

If

a falling rock

were somehow equipped with

a speedometer, in each succeeding second of fall its reading would increase by 10 m/s.
Table 2-2 shows the speeds we
would read at various seconds of fall.

18

The acceleration

Motion

of an object falling

under conditions where

about 10 meters per second squared
(10 m/s'). For free fall, it is customary to use the letter g to represent the acceleration (because in free fall, the acceleration is due
to gravity). Although g varies slightly in different parts of the
world, its average value is nearly 10 m/s More accurately, it
2
is 9.8 m/s
but it is easier to see the ideas involved when it is
rounded off to 10 m/s-. Where accuracy is important, the value
of 9.8 m/s-' should be used for the acceleration during free fall.
Note in Table 2-2 that the instantaneous speed of an object
falling from rest is equal to the acceleration multiplied by the
amount of time it falls.
air resistance

negligible

is

is

instantaneous speed = acceleration x elapsed time

SECONDS ;iT'0

2S

The instantaneous speed v of an object falling from rest after an

elapsed time t can be expressed in shorthand notation:
v

9 4s
u--IO"%

The

9 5s

gt

symbolizes both speed and velocity. Take a moment

equation with Table 2-2. You will see that whenever
the acceleration g 10 m/s 2 is multiplied by the elapsed time in
seconds, you have the instantaneous speed in meters per second.
to

IS

letter v

check

this

ir =

20"*$

<

Question

What would the speedometer reading on the falling rock

shown in Figure 2-5 be 4.5 seconds after it drops from rest?

How
u- =

s after

it is

dropped? 100

s?

30 "Vs

0s.

about 8

So

far,

we have been

ward under

looking at objects moving straight downNow, when an object is thrown upward, it

move upward for a while. Then it comes back down.

gravity.

continues to
At the highest point,

from upward

Then

it

starts

when it is changing its direction of motion

downward, its instantaneous speed is zero.
downward just as if it had been dropped from rest
to

at that height.

Answer

Ts
U = 40
Fig. 2-6

The

rate at

"/$

which

the speed changes each sec-

same

ond

is

the

ball

is

going upward

downward.

wh<. her the

The speedometer readings would be 45 m/s, 80 m/s, and 1000 m/s, respectively. You can reason this from Table 2-2 or use the equation v = gt, where g is
replaced by 10 m/s 2
.

This relationship follows from the definition of acceleration when the acceleration is g and the initial speed is zero. If the object is initially moving
downward at speed v, the speed v after any elapsed time t is v = v + gt. This
book will not be concerned with such added complications. You can learn a lot

from even the most simple cases!

2.6

How Far

Free Fall:

19

What about the upward part of the path? During the upward
motion, the object slows from its initial upward velocity to zero
velocity. We know it is accelerating because its velocity is changing
its speed is decreasing. The acceleration during the upward motion is the same as during the downward motion: g =
10 m/s 2 The instantaneous speed at each point in the path is the
same whether the object is moving upward or downward (see
Figure 2-6). The velocities are different, of course, because they
are in different directions. During each second, the speed or the
velocity changes by 10 m/s. The acceleration is 10 m/s 2 the whole
time, whether the object is moving upward or downward.

2.6

Free

Fall:

How Far

How fast a falling object moves is entirely different from how far
it

moves. To understand

the

first

this,

second, the object

is

return to Table 2-2. At the end of

moving downward with an

instan-

taneous speed of 10 m/s. Does this mean it falls a distance of 10

during the first second? No. If it fell 10
the first second, it
would have to have had an average speed of 10 m/s. But we know
the speed started at zero and increased to 10 m/s only at the end
of a full second. How do we find average speed for an object
moving in a straight line with constant acceleration, as it is doing here? We find it the same way we find the average of any two
numbers: add them and divide by 2. So if we add the initial
speed, zero in this case, and the final speed of 10 m/s, and then
divide by 2, we get 5 m/s. During the first second, the object has
an average speed of 5 m/s. It falls a distance of 5 m. To check
your understanding of this, carefully consider the following
check question before going further.

Question
During the span of the second time interval in Table 2-2,
and ends at 20 m/s. What is the
average speed of the object during this 1 -second interval?

the object begins at 10 m/s

What

is its

acceleration?

Answer
The average speed
beginning speed +

The acceleration

will be
final

speed _ (10 m/s) + (20 m/s) _ 30 m/s

15 m/s

will be

change

in speed
time interval

(20 m/s) - (10 m/s) _

1

10 m/s
1

10 m/s 2

20

Motion

Table 2-3 shows the total distance moved by a freely falling

it has fallen
5 m. At the end of 2 s it has dropped a total distance of 20 m. At
the end of 3 s it has dropped 45 m altogether. These distances
form a mathematical pattern: at the end of time t, the object has
l
fallen a distance d of \gt * Try using g = 10 mis 1 to calculate the
distance fallen for some of the times shown in Table 2-3.

object dropped from rest. At the end of one second

Table 2-3

Dropped from Rest

Free-Fall Distances of Object

Elapsed Time (seconds)

Distance Fall*

Fig. 2-7

ing rock

Pretend that a
is

20
45
80

fall-

somehow

125

equipped with an odometer.

The readings ot distance
fallen increase with time and
are

shown

in

Table 2-3.

fr

Question

An apple drops from a tree and hits the ground in one

second. What is its speed upon striking the ground? What is
its average speed during the one second? How high above
ground was

when

it

it first

dropped?

Answer
Using 10
speed v

m
=

gt

for g

=
,

we

(10
.

average speed v

find

ms

10 m/s

+
beginning
srv

final v

J
)

(1 s)

(0

(The bar over the symbol v denotes average speed

m s)

- (10 m/s)

m/s

v.)

distance d = average speed x time interval = (5 m/s) x

(1 s)

or equivalently,

distance d = igt J =

(i)

x (10

m's-')

(1 s)

Notice that the distance can be found by either of these equivalent relationships.
'

distance = average speed x time interval

=
=

beginning speed +
^

+ hgt
-x

-igf

final

speed

x t,me

How Fast, How Far, How Quick How Fast Changes

2.8

21

Air Resistance and Falling Objects

2.7

Drop a feather and a coin and you'll notice that the coin reaches
the floor way ahead of the feather. Air resistance is responsible
for these different accelerations. This fact can be shown quite
nicely with a closed glass tube connected to a vacuum pump.
The feather and coin are placed inside. When air is inside the
tube and it is inverted, the coin falls much more rapidly than the
feather. The feather flutters through the air. But if the air is removed with a vacuum pump and the tube is quickly inverted,
the feather and coin fall side by side at acceleration g (Figure
2-8). Air resistance noticeably alters the motion of falling pieces
of paper or feathers. But air resistance less noticeably affects the
motion of more compact objects like stones and baseballs. Most
objects falling in air can be considered to be falling freely. (Air
resistance will be covered in

w
Much

detail in

Chapter

4.)

of the confusion that arrives in analyzing the

far."

comes about from mixing up "how

When we wish

to specify

how

fast

motion of
fast"

something

with
freely

a rate (the rate at which distance is covered). Acceleranot velocity, nor is it even a change in velocity; acceleration is the rate at which velocity itself changes.
Please be patient with yourself if you find that you require a
few hours to achieve a clear understanding of motion. It took
people nearly 2000 years, from the time of Aristotle to Galileo, to
achieve as much!
tion

is

Fig. 2-8

feather

accelerate equally
is

falls from rest after a certain elapsed time, we are talking about
speed or velocity. The appropriate equation is v = gt. When we
wish to specify how far that object has fallen, we are talking
about distance. The appropriate equation is d \gt 2 Velocity or
speed (how fast) and distance (how far) are entirely different
from each other.
The most confusing concept, and one of the most difficult encountered in this book, is "how quickly does speed or velocity
change": acceleration. What makes acceleration so complex is
that it is a rate of a rate. It is often confused with velocity, which
is itself

00

no

air

and a coin

when

around them.

How Fast, How Far, How Quick How Fast Changes

falling objects

"how

more

&

there

22

Motion

Chapter Review

Concept Summary

instant.

Note: Each chapter in this book concludes with

Review Questions and Think and Explain exercises. The Review Questions are designed to help you fix ideas and catch the essentials of the chapter material. You'll notice that
answers to the questions can be found within the
chapter. Think and Explain exercises stress
thinking rather than mere recall of information
and call for an understanding of the definitions,
principles, and relationships of the chapter ma-

Average speed is the total distance covered

divided by the time interval.

Think and Explain exercises

Motion

is

a set of

described relative to something.

Speed is a measure of how fast something is

moving.
Speed is the rate at which distance is covered; it is measured in units of distance divided by time.
Instantaneous
speed is the speed at any

many

terial. In

cases the intention of particular

is

to help

you apply

the ideas of physics to familiar situations.

Velocity

is

speed together with the direction of

The

travel.

velocity is constant only when the

speed and the direction are both constant.

The

Acceleration

is

the rate at which the velocity

Activities,

which are

at the

end of most

chapters, are pre-lab activities that can be done

in or out of class, or simple home projects to do

on your own. Their purpose

is

to

encourage you

to discover the joy of doing physics.

is

changing.

In physics, an object

considered to be acspeed is increasing,

when its speed is decreasing, and/or when
the direction is changing.
Acceleration is measured in units of speed
divided by time.
celerating

when

Review Questions

is

its

What

object in free

fall is falling

under the

meant bv saving

that motion is relaeveryday motion, what is motion

usually relative to? (2.1)

2.

An

is

tive? For

influ-

Speed

is

the rate at which

what happens?

\J>'(2.2)

ence of gravity alone, where air resistance does

not affect its motion.
An object in free fall has a constant acceleration of about 10 m/s 2

3.

You walk across the room at 1 kilometer per

/second. Express this speed in abbreviated
units, or symbols. (2.2)

\.

Important Terms

What

is

the difference between instantane-

ous speed and average speed?

(2.2)

acceleration (2.4)

\~5

average speed (2.2)

elapsed time (2.5)

Does the speedometer of a car read instantaneous speed or average speed? (2.2)

free fall (2.5

instantaneous speed

(2.2)

'

6.

What

is

the difference between speed and

velocity? (2.3)

rate (2.1)
relative (2.1)

speed

(2.2)

velocity (2.3)

the speedometer in a car reads a constant

speed of 40 km/h, can you say that the car

7. If

23

Chapter Review

Why

has a constant velocity?

or

why

ow far will a freely-falling object fall from

a position of rest in five seconds? Six sec-

not?

(2.3)

onds?

What two
in speed?

controls on a car enable a change

Name another control that enablesv

a change in velocity. (2.3)

rf$jQ

How

\v\

(2.6)

Ylf}$r>
/ ^2^

quantity describes how quickly you

change how fast you're traveling, or how
quickly you change your direction? (2.4)

pens?

What

(2.4)

is

^W

is

the rate at which

an object move in one second

average speed during that second is
m/s? (2.6)
far will

<5vn

(3

"How

22

have

far will a freely-falling object

from a position of rest when

stantaneous speed is 10 m/s? (2.6)
fallen

its in-

what hap-

^W^A^
constant speed of

#^

100km/h?(2.4)
is

^l\Ss
D^2)

<

/ 23. Does air resistance'iricrease

ice lncrea
or decrease the

V y

acceleration of a falling object? (2.7)

the acceleration of a car that travels

in a straight line at

What

its

What

Acceleration

What

Y24I.

is

the appropriate equation for

how

something freely falls from a position of

rest? For how far that object falls? (2.8)
fast

L--

t-,
r

the acceleration of a car that, while

moving in a straight line, increases its speed

from zero to 100 km/h in lOi seconds? (2.4)

By how much does the speed -of a vehicle

moving in a straight line change each second when it is accelerating at 2 km/h-s? At
4 km/h-s? At 10 km/h-s?

Why

25.

is it

said that acceleration

is

a rate of a

rate? (2.8)

Activities

By any method you

1.

(2.4)

choose, determine your

average speed of walking.

Under what conditions can acceleration be

defined as the rate at which the speed i&
changing?

(2.4)

wW^-^" (JU

Why

does the unit of time enter twice in the

unit of acceleration? (2.4) fyMJls J5]("W^'

How much

gain in speed each second does

freely falling object acquire? (2.5)

For a freely-falling object dropped from

rest, what is its instantaneous speed at the
end of the fifth second of fall? The sixth sec-

ond?

this with your friends. Hold a dollar

,$TlTry
so that the midpoint hangs between

bill

friend's fingers (Figure A). Challenge

friend to catch
A /\jnf

gers shut

gt

),

which

For a freely-falling object dropped from

rest, what is its acceleration at the end of the
fifth second of fall? The sixth second? At the
end of any elapsed time t? (2.5) pr\
For an object thrown straight upward, what
is its instantaneous speed at the top of its
path? Its acceleration? (Why are your answers different?) (2.5)
. / n
_^
i

by snapping his or her

release

it.

The

bill

fin-

won't

be caught! Explanation: It takes at least Vi

econd for the necessary impulses to travel
from the eye to the brain to the fingers. But
in only Vs second, the bill falls 8 cm (from d

(2.5)

VD

it

when you

your

u5

is

half the length of the

bill.

4.

24

3.

You can compare your reaction time with

f.

by catching a ruler that is

dropped between your fingers. Let your
that of a friend

friend hold the ruler as

Snap your

shown

in

fingers shut as soon as

ruler released.

The number

Figure B.

you see the

b.

of centimeters

that pass through your fingers depends on

your reaction time. You can find your reac1
tion time n seco nds by solving d = igt for
time: t = Vldlg = 0.045 Vd, where d is in

Motion

a freely-falling rock were equipped

with a speedometer, by how much would
its speed readings increase with each
second of fall?
Suppose the freely-falling rock were
drop'ped near the surface of a planet
where g = 20 m/s 2 By how much would
its speed readings change each second?
If

a freely-falling rock were equipped with

an odometer, would the readings for distance fallen each second be the same, increase with time, or decrease with time?
If

centimeters.

a.

When
ward

ball

is

thrown straight up-

absence of air resistance, by

does the speed decrease each

in the

how much
second?
b.

After

turn

Fig.

c.

it reaches the top and begins its

downward, by how much does

speed increase each second?

How much time is required

compared

Think and Explain

(D Why is

it

(JJ

that an object can accelerate while

traveling at constant speed, but not at constant velocity?

Light travels in a straight line at a constant speed of 300 000 km/s.

^rv
i--\
acceleration?

Which has
moving in

What

is

its

when

car that increases its speed from 50 to 60 km/h, or a

bicycle that goes from zero to 10 km/h in the
same time? Defend vour answer.
a straight line

in

its

going up

coming down?

Table 2-2 shows that the instantaneous speed

of an object dropped from rest is 10 m/s
after 1 s of fall. Table 2-3 shows that the object has fallen only 5 m during this time.
Your friend says this is incorrect, because
distance traveled equals speed times time,
so the object should fall 10 m. What do you
say?
a.

YV
the greater acceleration

to

re-

What

is

the instantaneous speed of a

freely-falling object

10 s after

it

is

re-

leased from a rest position?

b.

What

is

its

average speed during this

time?
c.

How

far will

it

travel in this time?


Newton's

First

Law of Motion

Inertia

you saw a boulder in the middle of a flat field suddenly begin

moving across the ground, you'd look for the reason for its motion. You might look to see if somebody was pulling it with a

If

it with a stick or something. You'd reason that

something was the cause of motion. Nowadays we don't believe
that such things happen without cause. In general we would say
the cause of the boulder's motion was a force of some kind. We
know that something forces the boulder to move.

rope, or pushing

3.1

Aristotle

Do boulders
without cause?
Fig. 3-1

move

on Motion

The idea that a force causes motion goes back to the fourth century B.C., when the Greeks were developing ideas of science. The
foremost Greek scientist was Aristotle, who studied motion and
divided it into two kinds: natural motion and violent motion.
Natural motion on earth was thought to be either straight up
or straight down, such as the falling of a boulder toward the
ground or the rising of a puff of smoke in air. Objects would seek
their natural resting places: boulders on the ground, and smoke
high in the air like the clouds. It was natural for heavy things to
fall and very light things to rise. Aristotle proclaimed that, for
the heavens, circular motion was natural, as it was without beginning or end. So the planets and stars moved in perfect circles
about the earth. Since their motions were natural, they were not
caused by forces.
Violent motion, on the other hand, was imposed motion. It
was the result of forces which pushed or pulled. A cart moved
because it was pulled by a horse; a tug-of-war was won by pulling on a rope; a ship was pushed by the force of the wind. The
25

26

Newton's

First

Law of Motion

important thing about violent motion was that it had an exterwas imparted to objects. Objects in
their resting places could not move by themselves, but were

nal cause; violent motion

pushed or pulled.

was commonly thought for nearly 2000 years that if an obwas moving "against its nature," then a force of some kind
was responsible. Such motion was possible only because of an
outside force; if there were no force, there would be no motion.
So the proper state of objects was one of rest, if they were not
pushed or pulled. Since it was evident to most thinkers up to the
sixteenth century that the earth must be in its proper place, and
that a force large enough to move it was unthinkable, it seemed
It

ject

clear that the earth did not move.

3.2

Copernicus and the Moving Earth

It was in this climate that the astronomer Nicolaus Copernicus
(1473-1543) formulated his theory of the moving earth. Copernicus reasoned from his astronomical observations that the
earth traveled around the sun. This idea was extremely controversial in his time, and he worked on his ideas secretly to escape
persecution. In the last days of his life, at the urging of close
friends, he sent his ideas to the printer. The first copy of his
work, De Revolutionibus, reached him on the dav he died, May

24, 1543.

3.3

Galileo

on Motion
It

was

Galileo, the foremost scientist of the sixteenth century,

who was

the first to show that Copernicus's idea of a moving

was reasonable. Galileo did this by demolishing the notion
a force was necessary to keep an object moving.

earth
that

any push or pull. Friction is the name given to the

between materials that are moving past each
other. Friction arises from the irregularities in the surfaces of
sliding objects. Even very smooth surfaces have microscopic irregularities that act as obstructions to motion. If friction were
absent, a moving object would need no force whatever for its

force

is

force that acts

continued motion.

showed that only when friction is present, as it usua force necessary to keep an object moving. He tested
his idea with inclined planes
flat surfaces that are raised at
Galileo

ally

is, is

Galileo on Motion

3.3

27

one end. He noted that balls rolling down inclined planes pick
up speed (Figure 3-2 left). They rolled to some degree in the direction of the earth's gravity. Balls rolling up inclined planes slowed
down (Figure 3-2 center). They rolled in a direction that opposed
gravity. What about balls rolling on a level surface, where they
would neither roll with nor against gravity (Figure 3-2 right)?
He found that for smooth horizontal planes, balls rolled without
changing speed. He stated that if friction were entirely absent, a
horizontally-moving ball would move forever. No push or pull
would be required to keep it moving, once it was set in motion.
SLOPE UPWARD
SPEED DECREASE 5y

SLOPE DOWNWARD
SPEED INCREASES

NO SLOPE
DOES SPEED CHANGE ?

&

Fig. 3-2
its

When

(Left)

speed. (Right)
gravity.

Does

When

its

moves with the earth's gravity and

it moves against gravity and loses
plane, it moves neither with nor against

the ball rolls down,

speed increases. (Center)

it

rolls

When
on a

it

it

rolls up,

level

speed change?

Galileo's conclusion

was supported by another

line of rea-

He placed two of his inclined planes facing each other, as

in Figure 3-3. He found that a ball rolling down one plane would
soning.

up the other to nearly the same height. The smoother the

more nearly equal were the initial and final heights.
He found that the ball tended to attain the same height even
when the second plane was longer and inclined at a smaller
angle. In rolling to the same height, the ball had to roll farther.
roll

planes, the

Additional reductions of angle for the upward plane gave the

same results. Always the ball went farther as it tended to reach
the same height.

FROM

HERE...

Fig. 3-3

TO HERE

(Left)

FROM

ball that rolls

(Center) As the angle of the

greater distance to reach

horizontal?

TO HERE

HERE...

down an

incline will roll

up

HERE...

to its initial height.

reduced, the ball must roll a

height. (Right) How far will it roll along the

upward

its initial

FROM

incline

is

28

What

Newton's

Law of Motion

First

was reduced
plane was perfectly horizontal? How far
would the ball roll? He realized that only friction would keep it
from rolling forever. It was not the nature of the ball to come to
rest, as Aristotle had claimed. In the absence of friction, the moving ball would naturally keep moving. Galileo said that every
material object has a resistance to change in its state of motion.
if

the angle of incline of the second plane

to zero, so that the

He

called this resistance inertia.

Galileo's concept of inertia discredited the Aristotelian theory

It would be seen that although a force (gravity) is necessary to hold the earth in orbit around the sun, no force was

of motion.

required to keep the earth

in

motion. There

is

no

friction in the

empty space ot the solar system, and the earth therefore coasts
around and around the sun without loss in speed. The way was
open for Isaac Newton (1642-1727) to synthesize a new vision of
the universe.

Question

ball

is

rolled across the top of a pool table

rolls to a stop.

ior?

pret

3.4

How would

How would Galileo

and slowly

Aristotle interpret this behav-

interpret it?

How would you inter-

it?

Newton's Law of Inertia

Within a year of Galileo's death, Isaac Newton was born. In 1665,
at the age of 23, Newton developed his famous laws of motion.

They replaced the Aristotelian ideas that had dominated the

thinking of the best minds for nearly 2000 years. This chapter
covers the first of Newton's three laws of motion. The other two
are covered in the next two chapters.
Newton's first law, usually called the law of inertia, is a restatement of Galileo's idea.
in its state of rest, or of motion
constant speed, unless it is compelled to change that state by forces exerted upon it.

Every body continues

Fig. 3-4
to

remain

Objects at rest tend

at rest.

in a straight line at

Answer
would likely say that the ball comes to a stop because it seeks its
proper state, one of rest. Galileo would likely say that once in motion the ball
would continue in motion; what prevents continued motion is not its nature or
its proper rest state, but the friction between the table and the ball. Only you can
Aristotle

answer the

last

question!

3.4

Newton's

Law of Inertia

29

Simply put, things tend to keep on doing what they're already

doing. Dishes on a table top, for example, are in a state of rest.

They tend to remain at rest, as is evidenced if you snap a table

cloth from beneath them. (Try this at first with some unbreakable dishes! If you do it properly, you'll find the brief and small
force of friction between the dishes and the fast-moving tablecloth is not significant enough to appreciably move the dishes.)
If an object is in a state of rest, it tends to remain at rest. Only a
force will change that state.
Now consider an object in motion. If you slide a hockey puck
along the surface of a city street, the puck quite soon comes to
rest. If you slide it along ice, it slides for a longer distance. This
is because the friction force is very small. If you slide it along an
air table where friction is practically absent, it slides with no ap-

Fig. 3-5

surface.

An

air tabic. Blasts of air

from many tiny holes provide a

friction-free

30

Newton's

parent loss

in

We

speed.

ing object tends to

own

will

move

^M

'

forever.

It

vacuum
move by

will

virtue of

its

of inertia provides a completely different

Whereas the ancients thought forces

were responsible for motion, we now know that objects will continue to move by themselves. Forces are needed to overcome any
friction that may be present and to set objects in motion initially. Once an object is moving in a force-free environment, it
will move in a straight line indefinitely. The next chapter will
show that forces are needed to accelerate objects but not to maintain motion if there is no friction.

41

1B

mov-

Toss an
of outer space,

in a straight line indefinitely.

inertia.

So we see the law

way

Law of Motion

see that in the absence of forces, a

move

object from a space station located in the

and the object

First

'

of viewing motion.

mBI^SL

Questions
1

^IfrfHH
V
^^y

suddenly the force of gravity of the sun stopped acting

on the planets, in what kind of path would the planets
If

move?
2.

Would
sists

it

be correct to say that the reason an object rein its state of motion is because

change and persists

of inertia?

1
Fig. 3-6

The spacecraft

launched in the late 1970s on

the Pioneer and Voyager missions have gone pas: the orbit
of Saturn (show
and are
still in motion. L
for the
,

gravitational effect- of stars

and planets in the

their motion will c
without change.

erse,

iue

2.

Answers
The planets, like any
acted upon them.

objects,

would move

In a strict sense, no. Scientists don't

know

in a straight-line

path

if

no forces

the reason that objects exhibit this

property. Nevertheless, the property of behaving in this predictable

We

way

is

understand many things and have labels and names for

these things. There are also many things we do not understand, and we have
labels and names for these things as well. Education consists not so much in
acquiring new names and labels but in learning what is understood and what
called inertia.

is

not.

a
Mass

3.5

A Measure of Inertia

Mass

3.5

Kick an empty

and
and

31

A Measure of Inertia

tin

can and

move

it

moves. Kick a can

filled

with sand,

much. Kick a tin can filled with solid lead,

you'll hurt your foot. The lead-filled can has more inertia
than the sand-filled can, which in turn has more inertia than the
empty can. The can with the most matter has the greatest inertia. The amount of inertia an object has depends on its mass
that is, on the amount of material present in the object. The
more mass an object has, the more force it takes to change its
state of motion. Mass is a measure of the inertia of an object.
it

doesn't

as

Mass Is Not Volume

Many people confuse mass with volume. They

Fig. 3-7

think that

if

an

You can

tell

much matter is in
when you kick it.

how

the can

object has a large mass, it must have a large volume. But volume
is a measure of space and is measured in units such as cubic centi-

meters, cubic meters, or liters. Mass is measured in kilograms.

(A liter of milk, juice, or soda
anything that is mainly water
has a mass of about one kilogram.) How many kilograms of

matter are in an object, and

object, are

two

how much

different things.

space

Which has

is

taken up by that

the greater

mass

common

automobile battery? Clearly the

more difficult to set in motion is the battery. This is evidence of
the battery's greater inertia and hence greater mass. The pillow
may be bigger that is, it may have a larger volume but it has
less mass. Mass is different from volume.
feather pillow or a

Mass

Is

Fig. 3-8

The pillow has a

larger size (volume) but a

smaller mass than the battery.

Not Weight

We say something has

a lot of matter if it is heavy. That's because we are used to measuring the quantity of matter in an object by its gravitational attracMass

is

most often confused with weight.

But mass is more fundamental than weight;

mass is a measure of the actual material in a body. It depends
only on the number and kind of atoms that compose it. Weight is
a measure of the gravitational force that acts on the material,
and depends on where the object is located.
The amount of material in a particular stone is the same
whether the stone is located on the earth, on the moon, or in
outer space. Hence, its mass is the same in any of these locations. This could be shown by shaking the stone back and forth.
The same force would be required to shake the stone with the
same rhythm whether the stone was on earth, on the moon, or in
tion to the earth.

a force-free region of outer space. That's because the inertia of

the stone is solely a property of the stone and not its location.
But the weight of the stone would be very different on the earth

Fig. 3-9

finds

it

The person

shake the stone

space

weightshake it in
weighted stale on earth.

less state as
its

in

just as difficult to

it

is

in its

to

32

Newton's First

Law of Motion

and on the moon, and still different in outer space if the stone
were away from strong sources of gravitation. On the surface of
the moon the stone would have only one-sixth its weight on earth.
This is because gravity is only one-sixth as strong on the moon as
compared to on the earth. If the stone were in a gravity-free region of space, its weight would be zero. Its mass, on the other
hand, would not be zero. Mass is different from weight.
We can define mass and weight as follows:

The quantity

of matter in a body.

More

specifi-

measure of the inertia or "laziness" that a body exhibits in response to any

effort made to start it, stop it, or change in any
cally,

way
<U^\
\>

\0

no

The

Weight:

it

its

is

state of motion.

force

due

to gravity

upon a body.

Mass and weight are not the same thing, but they are proportional to each other. Objects with great masses have great
weights. Objects with small masses have small weights. In the
same location, twice as much mass weighs twice as much. Mass

and weight are proportional to each other but not equal to each
other. Mass has to do with the amount of matter in the object.
Weight has to do with how strongly that matter is attracted by
the earth's gravity.

Questions
1

Does a 2-kilogram iron block have twice as much inertia

much mass? Twice
is much volume? Twice as much weight (when weighed

as a 1-kilogram block of iron? Twice as

in the
2.

same

location)?

Does a 2-kilogram bunch of bananas have twice as much

-kilogram loaf of bread? Twice as much
inertia as a
mass? Twice as much volume? Twice as much weight
(when weighed in the same location)?
1

^
1.

Answers
The answer

is

iron atoms,

and therefore twice the amount

blocks are

yes to

made

all

of the

questions.

same

A 2-kilogram block of iron has twice as many

of matter, mass,

and weight. The

material, so the 2-kilogram block also has twice

the volume.
2.

Two

kilograms of anything has twice the inertia and twice the mass of one
kilogram of anything else. In the same location, two kilograms of anything
will weigh twice as much as one kilogram of anything (mass and weight are
proportional). So the answer to all questions is yes, except for volume. Volume
and mass are proportional only when the materials are the same, or when
they are equally compact for their mass when they have the same density.
Bananas are denser than bread enough so that two kilograms of bananas
have less volume than one kilogram of ordinary bread.

nn

The Moving Earth Again

3.6

33

Kilogram Weighs 9.8 Newtons

In the United States

it

of matter in an object

has been

by

its

common

to describe the

amount

gravitational pull to the earth

by

weight. The common, traditional unit of weight is the pound.

In most parts of the world, however, the measure of matter is
commonly expressed in mass units. The kilogram is the international, metric
SI"
unit of mass. The SI symbol for kilogram
is kg. At the earth's surface, a 1-kg bag of nails has a weight of
its

9.8

pounds.

2.2

The SI unit of force is the newton (named after guess who?).

One newton is equal to a little less than a quarter of a pound
(like the weight of a quarter-pound burger after it is cooked). The
SI symbol for newton is N (with a capital letter because it is
named after a person). A 1-kg bag of nails has a weight in metric
units of 9.8 N. Away from the earth's surface, where the force of
gravity is less, it would weigh less.

One kilogram of
weighs 2.2 pounds,
which is the same as 9.8
newtons.
Fig. 3-10

nails

Question

The text states that a 1-kg bag of nails weighs 9.8

the earth's surface. Does 1 kg of yogurt also weigh 9.8

at

N?

|
1

3.6

The Moving Earth Again

When

Copernicus announced the idea of a moving earth in the

sixteenth century, there was much arguing and debating of this
controversial idea. One of the arguments against a moving earth
was the following: Consider a bird sitting at rest at the top of a

On the ground below is a fat, juicy worm. The bird sees

worm and drops vertically below and catches it. This would

tall tree.

the

not be possible, it was argued, if the earth moved as Copernicus

suggested. If Copernicus were correct, the earth would have
to travel at a speed of 107 000 km/h to circle the sun in one
year. Convert this speed to kilometers per second and you'll get
30 km/s. Even if the bird could descend from its branch in one

Answer
kg of anything weighs 9.8 N. (We used nails in this
example because most everybody identifies with nails at least everybody who
likes to build things. But not everybody likes yogurt.)
Yes, at the earth's surface

'^a/'+ssSs'.

***&:

name, Le Systeme International d'Unites, for the

international, metric system of measurement. The short forms of the SI units

at rest for the bird to catch

are called symbols rather than abbreviations.

the

SI stands for the French

Fig. 3-11

worm?

Must the earth

bi

34

Newton's

First

Law of Motion

second, the worm would have been swept by the moving earth a
distance of 30 kilometers away. For the bird to catch the worm
under this circumstance would be an impossible task. But birds
in fact do catch worms from high tree branches, which seemed
clear evidence that the earth must be at rest.
Can you refute this argument? You can if you invoke the idea
of inertia. You see, not only is the earth moving at 30 km/s, but
so are the tree, the branch of the tree, the bird that sits on it,
the worm below, and even the air in between. All are moving at

30 km/s. A body

motion remains in motion if no unbalanced

on it. So when the bird drops from the branch,
its initial sideways motion of 30 km/s remains unchanged. It
catches the worm quite unaffected by the motion of its total
environment.
Stand next to a wall. Jump up so that your feet are no longer
in contact with the floor. Does the 30-km/s wall slam into you?
Why not? Because you are also traveling at 30 km/s before, during, and after your jump. The 30 km/s is the speed of the earth
relative to the sun
not the speed of the wall relative to you.
People three hundred years ago had difficulty with ideas like
these not only because they failed to acknowledge the concept of
inertia, but because they were not accustomed to moving in highspeed vehicles. Slow, bumpy rides in horse-drawn carriages did
not lend themselves to experiments that would reveal inertia.
Today we flip a coin in a high-speed car, bus, or plane, and we
catch the vertically-moving coin as we would if the vehicle were
at rest. We see evidence for the law of inertia when the horizontal motion of the coin before, during, and after the catch is the
same. The coin keeps up with us. The vertical force of gravity affects only the vertical motion of the coin.
Our notions of motion today are very different from those of
our ancestors. Aristotle did not recognize the idea of inertia because he failed to imagine what motion would be like without
friction. In his experience, all motion was subject to resistance,
and he made this fact central to his theory of motion. We can
only wonder how differently science might have progressed if
Aristotle had recognized friction for what it is, namely a force
like any other, which may or may not be present.
in

forces are acting

Fig. 3-12

Flip a coin in a
high-speed airplane, and it

behaves as
s

.[f

if

the plane
^eeps

were
up

with you: inertia in action.

Chapter Review

35

Chapter Review

Concept Summary

The speed of a ball increases as it rolls down

an incline, and the speed decreases as the
ball rolls up an incline. What happens to the
speed on a smooth horizontal surface? (3.3)

if it were not for friction

an object in motion would keep moving forever.

Galileo concluded that

According to Newton's first law of motion the

law of inertia every body continues in its state

of rest or of

motion

in a straight line at constant

speed unless forces cause

Inertia

is

the resistance an

in its state of

it

to

change

Galileo found that a ball rolling

its state.

object has to a

Mass is a measure of inertia.

Mass is not the same as volume.
Mass is not the same as weight.
The mass of an object depends on the
amount and type of matter in it, but does
not depend on the location of the object.
The weight of an object is the gravitational
force on it and depends on the location.

its initial

6 J Does the law of inertia pertain to moving

V/

objects, objects at rest, or both?

your answer with examples.

The law

of inertia states that

no force

is re-

you were in a spaceship and fired a cannonball into frictionless space, how much
force would have to be exerted on the ball to
keep it going? (3.4)
If

friction (3.3)

inertia (3.3)

kilogram (3.5)
law of inertia (3.4)

Does a 2-kilogram rock have twice the mass

1 -kilogram
rock? Twice the inertia?
Twice the weight (when weighed in the

mass (3.5)
newton (3.5)

of a

first

law

same

(3.4)

Review Questions

(10.

(What was the distinction

made between natural motion and
that

to

location)? (3.5)

Does a

liter

of molten lead have the

volume as a liter of apple juice? Does

the same mass? (3.5)

Aristotle
violent

it

same
have

(3.1)
1

\J>.

Support

(3.4)

quired to maintain motion. Why, then, do

you have to keep peddling your bicycle to
maintain motion? (3.4)

force (3.3)

motion?

height? (3.3)

Important Terms

Newton's

How

another.

change

motion.

down one

up enough speed to roll up

high will it roll compared to

incline will pick

Why was

Copernicus reluctant to publish

\)

Why do physics

types say that

fundamental than weight?

mass

is

more

(3.5)

his ideas? (3.2)

What

is

the effect of friction on a

How

moving ob-

an object able to maintain a

constant speed when friction acts upon it?

ject?

(3.3)

is

An elephant and a mouse would both have

the same weight
zero
in gravitation-free
space. If they were moving toward you with
the same speed, would they bump into you
with the same effect? Explain. (3.5)

[^

36

Soo

>iLAl^ijLVAft>oW^

fl

13.

What

14.

If you hold a coin above your head while in a

bus that is not moving, the coin will land at
your feet when you drop it. Where will it
land if the bus is moving in a straight line at
constant speed? Explain. (3.6)

15.

In the cabin of a jetliner that cruises at

600 km/h, a pillow drops from an overhead
rack to your lap below. Since the jetliner is
moving so fast, why doesn't the pillow slam

is

Newton's First Law of Motion

.ll

the weight of 2 kg of yogurt? (3.5)

compartment when

into the rear of the

drops? (What

is

it

Fig.

the horizontal speed of the

Fig.

pillow relative to the ground? Relative to

you inside the

jetliner?) (3.6)

6.

and Explain
Many automobile

hammer

is loose, and you

by banging it against the
top of a work bench, why is it best to hold it
with the handle down (Figure B) rather than
with the head down? Explain in terms of

If

the head of a

' wish

passengers have suffered

neck injuries when struck by cars from
behind. How does Newton's law of inertia
apply here? How do headrests help to guard

Two

against this type of injury?

one

to tighten

it

inertia.

closed containers look the same, but

packed with lead and the other with a

How could you determine
which had more mass if you and the containers were orbiting in a weightless condiis

few

Suppose you place a ball in the middle of

and then accelerate the wagon
forward. Describe the motion of the ball
relative to (a) the ground and (b) the wagon.
a wagon,

feathers.

tion in outer space?

8.

an elephant were chasing you, its enormous mass would be most threatening. But
if you zigzagged, its mass would be to your
advantage. Why?

What

is

weight

If

9.)lf

in

your actual (or most desirable)

newtons?

you are

\_y along a

sitting in a

are traveling at 100

When you compress

changes:
weight?
tity

A massive

mass,

a sponge,

which quan-

inertia,

volume, or

a.

Which

the

is

string

is

more

likely

break? Which property mass or

weight is important here?
If the string is instead snapped downward, which string is more likely
to break? Which property
mass or
weight is important this time?
to

km/h

traveling

you

too.

it

moving

relative to the road?

Relative to you?
b. If you drop the apple, does
ball

is

you hold an apple over your head, how

fast is

suspended by a string
from above, and slowly pulled by a string
from below (Figure A). Is the string ten
sion greater in the upper or the lowery
string?

If

bus that

straight, level road at 100 km/h,

it

still

have

same horizontal motion?

As the earth rotates about its axis, it takes

three hours for the United States to pass
beneath a point above the earth that is stationary relative to the sun.

What

is

wrong

with this scheme: To travel from Washington D.C. to San Francisco using very little
fuel, simply ascend in a helicopter high over
Washington D.C. and wait three hours until
San Francisco passes below?


Newton's Second Law of Motion
Force and Acceleration

Kick a football and

it moves. Its path through the air is not a

curves due to gravity. Catch the ball and it stops.
Most of the motion we see undergoes change. Most things start
up, slow down, or curve as they move. The last chapter covered
objects at rest or moving at constant velocity. There was no net
force acting on these objects. This chapter covers the more com-

straight line

it

mon cases, in which there is a change in motion that is, accelerated motion.
Recall from Chapter 2 that acceleration describes how fast
motion is changing. Specifically, it is the change in velocity per
certain time interval. In shorthand notation,
,

acceleration

Fig. 4-1
it

Kick a football and

neither remains at rest nor

moves

in a straight line.

change in velocity
-:
^
time interval
-,

This is the definition of acceleration."' This chapter focuses on

the cause of acceleration: force.

E3

Force Causes Acceleration

Consider an object at rest, such as a hockey puck on ice. Apply a

force and it moves. Since it was not moving before, it has accelerated
changed its motion. When the stick is no longer in
contact with the puck, it moves at constant velocity. Apply another force by striking it with the stick again, and the motion
changes. Again, the puck has accelerated. Forces are what produce accelerations.

/I6
*
The Greek letter A (delta) is often used as a symbol for "change in" or "difference in." In "delta" notation, a = Av/A/, where Av is the change in velocity,

and \t

is

the change in time (the time interval).

Fig. 4-2

Puck about

to be hit.

37

Newton's Second Law of Motion

38

APPLIED

FORCES

NET
FORCE

5n

*N
15

ION
ION

^H

5N

Most often, the force we apply is not the only force that acts
on an object. Other forces may act as well. The combination of
all the forces that act on an object is called the net force. It is the
net force that accelerates an object.
We see how forces combine to produce net forces in Figure
4-3. If you pull horizontally with a force of 10 N on an object
that rests on a friction-free surface, an air track for example,
then the net force acting on it is 10 N. If a friend assists you and
pulls at the same time on the same object with a force of 5 N in
the

same

forces, 15

*Q

5n

.&

When

is

the

sum

of the forces.

forces act in opposite

directions, the net force

were pulled with a single force of

On

Fig. 4-3
When more than
one force acts in the same di
rection on an object, the net

force

sum of these
(Figure 4-3 top). The object will accelerate as if it

direction, then the net force will be the

is

difference of the forces.

the

15 N. If, however, your friend

opposite direction, the net force will be the
difference of these forces, 5 N (Figure 4-3 center). The acceleration of the object would be the same as if it were instead pulled
with a single force of 5 N.

pulls with 5

We

in the

amount

of acceleration depends on the

To increase the acceleration of an object, you must increase the net force. This makes good sense.
Double the force on an object, and you will double the acceleration. If you triple the force, you'll triple the acceleration, and so
on. We say that the acceleration produced is directly proporfind that the

amount

of the net force.

tional to the net force.

We write:

acceleration

The symbol ~ stands

4.2

net force

for "is directly proportional to."

Mass Resists Acceleration

Push on an empty shopping cart. Then push equally hard on a
heavily loaded shopping cart, and you'll produce much less acceleration. This is because acceleration depends on the mass
being pushed upon. For objects of greater mass we find smaller
accelerations. Twice as much mass for the same force results in
only half the acceleration; three times the mass results in one
third the acceleration, and so forth. In other words, for a given
force the acceleration produced is inversely proportional to the
mass.

We

write:

The acceleration
produced depends on the
mass being pushed.

Fig. 4-4

acceleration

By

we mean

two values change in opposite

see that as the denominator
the whole quantity decreases. The quantity 753 is less

inversely

that the

directions. (Mathematically

increases,

mass

than the quantity

75

we

for example.)

Newton's Second

4.3

Law

39

Newton's Second Law

Newton was the first to realize that the acceleration we produce
when we move something depends not only on how hard we
push or pull (the force) but on the mass as well. He came up with
one of the most important rules of nature ever proposed, his
ond law of motion. Newton's second law states:

sec-

The acceleration produced by a net force on a body is

directly proportional to the magnitude of the net force,
in the same direction as the net force, and inversely
proportional to the mass of the body.

Or

in shorter notation,

net force

acceleration

By using
grams

mass

consistent units such as newtons (N) for force, kilo-

(kg) for

acceleration,

mass, and meters per second 'squared (m/s 2 ) for

we

get the exact equation.

acceleration
In briefest form,

where a

is

= net force
mass

acceleration,

is

net force,

and

m is

mass:
F_
a

The acceleration

From

is

equal to the net force divided by the mass.

we can see that if the net force that acts

doubled, the acceleration will be doubled. Suppose instead that the mass is doubled. Then the acceleration will
be halved. If both the net force and the mass are doubled, then
the acceleration will be unchanged.
this relationship

on an object

Fig. 4-5

is

The good acceleration of the racing car

large forces.

is

due

to its ability to

produce

40

(rk,^

"^

****

Newton's Second Law of Motion

Problem Solving

If the mass of an object is measured in kilograms (kg), and

acceleration is expressed in meters per second squared
(m/s ), then the force will be expressed in newtons (N). One
2

newton is the force needed to give a mass of one kilogram

an acceleration of one meter per second squared. We can
arrange Newton's second law to read

c*'

force

>/>

As you can

,
.

e
<^r

= mass x acceleration
2
(1 kg) x (1 m/s )

N =

see,

kg-m/s 2

(The dot between "kg" and "m/s 2 " means that the units have
been multiplied together.)
If you know two of the quantities in Newton's second
law, you can easily calculate the third. For example, how
much thrust must a 30 000-kg jet plane develop to achieve
an acceleration of 1.5 m/s 2 ? Thrust means force, so

F ma
= (30 000 kg) x (1.5 m/s
= 45 000 kg-m/s
= 45 000 N

Suppose you know the force and the mass and want to
For example, what acceleration will
be produced by a force of 2000 N on a 1 000-kg car? Using
Newton's second law we find
find the acceleration.

F_

m
If

__

2000 N _ 2000 kg-m/s

1000 kg
1000 kg

the force were 4000 N,

_

2 m/s

what would be the acceleration?

4000 N _ 4000 kg-m/s

1000 kg
1000 kg

= 4 m/s

Doubling the force on the same mass simply doubles the

acceleration.
In traditional problem-solving physics courses, the prob-

ms

are often

more complicated than

the ones given here,

on solving complicated problems,

t will instead concentrate on becoming acquainted with
e equations as guides to thinking about the basic physics
ncepts. Mathematical problems can be an objective for
low-up study in physics. Most any other physics textok will provide you with plenty of mathematical problems if you want them. But please learn the concepts first!
is

book

will not focus

41

Statics

4.4

Questions
1

If

can
2.

able to accelerate at 2 m/s 2 what acceleration

attain if it is towing another car of equal mass?

a car
it

is

What kind

of motion does an unchanging force produce

on an object of fixed mass?

4.4

Statics

How many

on your book as it lies motionless on the

weight. If that were the only force acting
on it, you'd find it accelerating. The fact that it is at rest, and not
accelerating, is evidence that another force is acting. The other
force must just balance the weight to make the net force zero.
The other force is the support force of the table (often called the
normal force*). To see that the table is pushing up on the book,
imagine an ant underneath the book. The ant would feel itself
being squashed from both sides, the top and the bottom. The
table actually pushes up on the book with the same amount of
force that the book presses down. If the book is to be at rest, the
sum of the forces acting on it must balance to zero.
Hang from a strand of rope. Your weight provides the force
which tightens the rope and sets up a tension force in it. How
much tension is in the strand? If you are not accelerating, then it
must equal your weight. The rope pulls up on you, and the gravitation of the earth pulls down on you. The equal forces cancel
because they are in opposite directions. So you hang motionless.
Suppose you hang from a bar supported by two strands of
rope, as in Figure 4-7. Then the tension in each rope is half your
weight (if the weight of the bar is neglected). The total tension
U P (2 your weight + \ your weight) then balances your downward acting weight. When you do pullups in exercising, you use
both arms. Then each arm supports half your weight. Ever try
pullups with just one arm? Why is it twice as difficult?
forces act

table? Don't say one,

its

_G 4,
f

The

net force on the

book is zero because the table
pushes up with a force that
Fig. 4-6

equals the

downward weight

of the book.

Answers
1

When
will

2.

the engine produces the

be

Motion

half. It will

at a

be

m/s

same

force on twice the mass, the acceleration

2
.

constant acceleration, in accord with Newton's second law.

This force acts at right angles to the surface; "normal to" means "at right
angles to," which is whv the force is called a normal force.

The sum ol the rope

tensions must equal your

Fig. 4-7

weight.

42

Newton's Second Law of Motion

Question

When you

step on a bathroom scale, the downward pull

and the upward support force of the floor compress a spring which is calibrated to give your weight. In
effect, the scale shows the support force. If you stand on
two scales with your weight equally divided between each
scale, what will each scale read? How about if you stand
w ith more of your weight on one foot than the other?
of gravity

Friction
Even when a

single force

is

applied to an object,

the only force affecting the motion. This

Friction

is

tion. It is

it is

usually not

because of friction.
a direction to oppose mois

a force that always acts in

in large part to irregularities in the two surfaces

due

Even very smooth surfaces are bumpy when

viewed with a microscope. When an object slides against another, it must either rise over the irregular bumps or else scrape
that are in contact.

them off. Either way requires a force.

The force of friction between two surfaces depends on the
kinds of material and how much they are pressed together. Rubber against concrete, for example, produces more friction than
steel against steel. That's

why

concrete road dividers are replac-

ing steel rails (Figure 4-8). Notice that the concrete divider

Fig. 4-8

Cross-section view

of concrete road divider,

and

road divider. Which design is best for slowing an

out-of-control, sideswiping
steel

car?

is

wider at the bottom, so that the tire of a sideswiping car rather

than the steel body makes contact with it. The greater friction
between the rubber tire and the concrete is more effective in
slowing the car than the contact between the steel body of the
car with the steel rail of the other design.
Friction is not restricted to solids sliding over one another.
Friction occurs also in liquids and gases, which are called fluids
(those which flow). Fluid friction occurs when an object moving
through a fluid pushes aside some of the fluid. Have you ever
tried running a 100-m dash through waist-deep water? The fric-

Answer
must add up to your weight. This is because the
which equals the support force of the floor, must counteract vour weight so the net force on you will be zero. If you stand equally on
each scale, each will read half your weight. If you lean more on one scale than
the other, more than half your weight w ill be read on that scale but less on the
other, so they will still add up to your weight. For example, if one scale reads
two-thirds your weight, the other scale will read one-third your weight. Get it?

The reading on both

sum

scales

of the scale readings,

43

Friction

4.5

appreciable, even at low speeds. Air resistance,

moving through air, is a very
common case of fluid friction. You don't notice it when walking
or jogging, but you'll notice it for higher speeds as in skiing downhill or skydiving.
tion of liquids

is

the friction that acts on something

When

an object may move at constant veapplied to it. In this case, the friction force
just balances the applied force. The net force is zero, so there is
no acceleration. For example, in Figure 4-9 the crate will move
at constant velocity when it is pushed just hard enough to balance the friction. The sack will fall at constant velocity when the
air resistance balances its weight.
there

is friction,

locity while a force

is

AIR RESISTANCE

PUSH

FRICTION
Fig. 4-9

Push the crate

to the right,

WEI6HT
and friction acts toward the left. The sack
upward. The direction of the force of fric-

falls

downward, and

tion

always opposes the direction of motion.

air friction acts

Questions
1

Figure 4-6 shows only two forces acting on the book: its
weight and the support force from the table. Doesn't the
force of friction act as well?

2.

Suppose a high-flying jumbo

jet

cruises at constant

when the thrust of its engines is a constant

80 000 N. What is the acceleration of the jet? What is the

velocity

force of air resistance acting

on the

jet?

Answers
1.

No, not unless the book tends to slide or does slide across the table. For example, if it is pushed toward the left by another force, then friction between
the book and table will act toward the right. Friction forces occur only when
an object tends to slide or is sliding.

2.

The acceleration is zero because the velocity is constant, which means not
changing. Since the acceleration is zero, it follows from a = Flm that the net
force is zero. This implies that the force of air resistance must just equal the
thrusting force of 80 000 N but must be in the opposite direction. So the air
resistance is 80 000 N.

44

4 Ko

Applying Force

Newton's Second Law of Motion

Pressure

a book on a table, and no matter how you place it

back, upright, or even balancing on a single corner the
force of the book on the table is the same. This can be checked by

You can put

on

its

doing this on a bathroom scale, where you'll read the same

weight in all cases. Balance the book different ways on the palm
of your hand. Although the force will be the same, you'll note
differences in the way the book presses against your palm. This
is because the area of contact is different in each case. The force
per unit of area

is

called pressure.

pressure =

where the force

is

More

precisely,

force

?
p
area of application
:

perpendicular to the surface area. In equation

form,

where P

Fig. 4-10

exerts the

and A is the area against which the force

which is measured in newtons, is different from pressure, which is measured in newtons per square meter [called a
pascal (Pa), a relatively new unit, adopted in I960].

The upright book

same force but a

greater pressure against the

because your area of contact

j-

Friction between
and the ground is the
same whether the tire is wide
or narrow. The purpose of the
Fig. 4-11

the tire

greater contact area

is

duce heating and wear.

to re-

the pressure

Many people mistakenly believe that the wider tires of dragracing vehicles produce more friction. But the larger area reduces only the pressure. The force of friction is independent of
the contact area. Wide tires produce less pressure, and narrow
tires produce more pressure. The wideness of the tires reduces
heating and wear.
You exert more pressure against the ground when you are
standing on one foot than when standing on both feet. This is

supporting surface.

|BA! p
K;_

is

acts. Force,

is less.

Stand on one

toe, like a bal-

huge. The smaller the area that supports a given force, the greater is the pressure on that surface.
You can calculate the pressure you exert on the ground when
you are standing. One way is to moisten the bottom of your foot
with water and step on a clean sheet of graph paper marked
with squares. Count the number of squares contained in your
footprint. Divide your weight by this area and you have the average pressure you exert on the ground while standing still on one
foot. How will this pressure compare with the pressure when
lerina,

and the pressure

is

you stand on two feet?

A dramatic illustration of pressure is shown in Figure 4-12.
The author applies appreciable force when he breaks the cement
block with the sledge hammer. Yet his friend, who is sandwiched
between two beds of sharp nails, is unharmed. This is because

Applying Force

4.6

Pressure

45

much of this force is distributed over the more than 200 nails
that make contact with his body. The combined surface area of
this many nail points results in a tolerable pressure that does
not puncture the skin. CAUTION: This demonstration is quite
dangerous. Do not attempt it on your own.

The author applies a force to fellow physics teacher Paul Robinson,

bravely sandwiched between beds of sharp nails. The driving force
per nail is not enough to puncture the skin. CAUTION: Do not attempt this on
Fig. 4-12

who

is

your own!

Questions
1.

In attempting to do the demonstration in Figure 4-12,

would it be wise to begin with a few

ward to more nails?
2.

nails,

and work up-

The massiveness of the cement block plays an important

role in this demonstration. Which provides more safety
a less massive or more massive block?

Answers
1.

No, no, no! There would be one fewer physics teacher if the demonstration
were performed with fewer nails, because of the resulting greater pressure.

2.

The greater the mass of the block, the smaller is the acceleration of the block
and bed of nails toward the friend. Much of the force wielded by the hammer
goes into moving this block and breaking it. It is important that the block be
massive and that it break upon impact.

46

4.7

Newton's Second Law of Motion

Free Fall Explained

Galileo

showed

that falling objects will accelerate equally, re-

gardless of their masses. This

negligible, that

is, if

is strictly

true

if

air resistance

the objects are falling freely.

It is

is

approxi-

mately true when air resistance is very small compared to the

weight of the falling object. For example, a 10-kg cannonball
and a 1 -kg stone dropped from an elevated position at the same
time will fall together and strike the ground at practically the
same time. This experiment, said to be done by Galileo from the
Leaning Tower of Pisa, demolished the Aristotelian idea that an
object that weighs ten times as much as another should fall ten
times faster than the lighter object. Galileo's experiment and
many others that showed the same result were convincing. But
Galileo couldn't say why the accelerations were equal. The explanation is a straight-forward application of Newton's second
law and is the topic of the cartoon "Backyard Physics" (next
separately here.
(a quantity of matter) and weight (the force
due to gravity) are proportional. A 2-kg bag of nails weighs twice
as much as a 1-kg bag of nails. So a 10-kg cannonball has 10
times the force of gravity (weight) of a 1-kg stone. Followers of
page). Let's treat

Recall that

Fig. 4-13

Galileo's

demonstration.

famous

it

mass

Aristotle believed that the cannonball should therefore acceler-

ate ten times as

much

as the stone because they considered only

the greater weight. But Newton's second law tells us to consider

the

mass as

much

well.

little

thought will show that ten times as

on ten times as much mass produces the same

acceleration as one-tenth the force acting on one-tenth the mass.
In symbolic notation,
force acting

where
stands for the force (weight) acting on the cannonball,
and til stands for the correspondingly large mass of the cannonball. The small Fand m stand for the smaller weight and mass of
the stone. We see that the ratio of weight to mass is the same for
these or any objects. All freely falling objects undergo the same
acceleration at the same place on earth. This acceleration, which
is due to gravity, is represented by the symbol g.
We can show the same result with numerical values. The
weight of a 1-kg stone (or 1 kg of anything) is 9.8 N at the earth's
surface. The weight of 10 kg of matter, such as the cannonball, is
98 N. The force acting on a falling object is the force due to gravity, or weight of the object. The acceleration of the stone is

4.7

Free Fall Explained

47

Newton's Second Law of Motion

48

and

F_

for the

__

m
F_

weight = 9.8 N = 9.8 kgm/skg

1
kg
m

9.8 m/s-

= g

cannonball,

_ weight

_.

98

__

10 kg

98 kgm/s J
= 9.8 m/s- = g
10 kg

The famous coin-and-feather-in-a-vacuum-tube demonstration

was discussed in Chapter 2, but not the reason for the equal accelerations.

Now we know

that the reason both

fall

with the

same acceleration (g) is because the net force on each is only

weight, and the ratio of weight to mass is the same for both.

its

_F_

Question
If

The ratio ot weight

(F) to mass (m) is the same
lor the 10-kg cannonball and
Fig. 4-14

you were on the moon and dropped a

feather from the

same

elevation at the

they strike the surface of the

moon

hammer and

same

time,

would

together?

the 1-ka stone.

4.8

Falling

and Air Resistance

fall w ith equal accelerations in a vacuum,
but quite unequally in the presence of air. When air is let into the
glass tube and it is again inverted and held upright, the coin
falls quickly while the feather flutters to the bottom. Air resisa tiny bit for the coin and very
tance diminishes the net forces
much for the feather. The downward acceleration for the feather
is very brief, for the air resistance very quickly builds up to counteract its tiny weight. The feather does not have to fall very long
or very fast for this to happen. When the air resistance of the
feather equals the weight of the feather, the net force is zero and
no further acceleration occurs. Acceleration terminates. The
feather has reached its terminal speed. If we are concerned with
direction, down for falling objects, we say the feather has reached

The feather and coin

its

terminal velocity.

air resistance on the coin does not have as much effect. At

small speeds the force of air resistance is very small compared
to the weight of the coin, and its acceleration is only slightly less

The

Answer

On the moon's surface the v would weigh on Iv one-sixth their earth weight,
which would be the only force acting on them since there is no atmosphere to
provide air resistance. The ratio of moonweight to mass for each would be the
same, and they would each accelerate at sg.
Yes.

4.8

Falling

and Air Resistance

49

than the acceleration of free fall, g. The coin might have to fall
for several seconds before its speed would be great enough for
the air resistance to increase to its weight. Its speed at that point,
perhaps 200 km/h or so, would no longer increase. It would have
reached its terminal speed.
The terminal speed for a human skydiver varies from about
150 to 200 km/h, depending on weight and position of fall. A heavier person will attain a greater terminal speed than a lighter person. The greater weight is more effective in "plowing through"
the air. A heavy and light skydiver can remain in close proximity
if the heavy person spreads out like a flying squirrel while the
light person falls head or feet first. A parachute greatly increases
air resistance, and terminal speed can be cut down to an acceptable 15 to 25 km/h.

Fig. 4-15

Terminal speed

is

reached

for the skydiver

when

air resistance

equals weight.

The

Fig. 4-16

increases

its

flying squirrel

area by spread-

ing out. This increases air re-

Question
If a heavy person and a light person parachute together
from the same altitude, and each wears the same size parachute, who should reach the ground first?

Answer
The heavy person reaches the ground

lust. This is because the light person,

reach terminal speed sooner while the heavy person continues to accelerate until a greater terminal speed is reached. So the heavy pel
son moves ahead of the light person, who is unable to catch up.
like the feather, will

sistance

and decreases the

speed of

fall.

Newton's Second Law of Motion

50

you hold a baseball and tennis

If

them

lease

time. But

ball at arm's length

and resame

together, vou'll note they strike the floor at the

if

you drop them from the top of a high building,

notice the heavier baseball strikes the ground

first.

This

you'll
is

be-

cause of the buildup of air resistance with higher speed (like the
parachutists in the check question). For low speeds, air resistance may be negligible. For higher speeds, it can make quite
a difference. Air resistance is more pronounced on the lighter
tennis ball than the heavier baseball, so acceleration of fall is
less for the tennis ball. The tennis ball behaves more like a parachute than the baseball does.

Question
If

ball

the force of air resistance

and a

falling tennis ball,

the same for a falling basewhich will have the greater

is

acceleration?

Fig. 4-17

A stroboscopic
a golf ball and a

photo of
Styrofoam ball falling in air.
The weight of the heavier golf
ball is more effective in overcoming air resistance, so its

acceleration

is

greater. Will

both ultimately reach a terminal speed? Which will do

so first?

When Galileo reportedly dropped the objects of different

weights from the Leaning Tower of Pisa, the heavier one did get
to the ground first
but only a split second before the other,
rather than the pronounced time difference expected by the followers of Aristotle. The behavior of falling objects was never
really understood until Newton announced his second law of
motion.
Isaac Newton truly changed our way of seeing the world.

Why?
Answer
Don't say the same!
doesn't

mean

It's

the net force

true the air resistance

is

the

same

is

the

same

for each, but this

for each, or that the ratio of net force to

mass is the same for each. The heavier baseball will have the greater net force,
and greater net force per mass, just like the heavier parachutist in the previous
question. (Convince yourself of this by considering the upper limit of air resistance

when

it is

equal to the weight of the tennis

tion of the tennis ball then?

at that point?

ball.

What

will

be the accelera-

that the baseball has a greater acceleration

And w ith more thought, do you see that the baseball has the greater
when the air resistance is less than the weight of the tennis

acceleration even
ball?)

Do you see

Chapter Review

51

Chapter Review

Concept Summary
An

object accelerates

direction

on

when

Important Terms

changes

there

is

speed and/or

air resistance (4.5)

a net force that acts

fluid (4.5)

inversely (4.2)

it.

The acceleration

proportional to the net force that acts on it.

The acceleration of an object is inversely
proportional to the mass of the object.

of an object

is

net force (4.1)

directly

Newton's second law

normal force (4.4)
pascal (4.6)
pressure (4.6)

Acceleration equals net force divided by

mass, and is in the same direction as the

support force (4.4)

terminal speed (4.8)
terminal velocity (4 8)

net force.

object remains at rest or continues to mov

with constant velocity when the net force on i
equals zero.
When an object is at rest, its weight is bal
anced by a support force of equal amount.
When an object is moving at constant velocity while an applied force acts on it, that
force must be balanced by an equal amount

(4.3)

An

D
2.

V-'/

of resisting force (usually friction).

(

ij

^-'

What

is

meant by the

an object?

on

N and 20 N in the same direcon an object. What is the net force

on the object?

net force that acts

(4.1)

Forces of 10
tion act

The application of a force over a surface produces pressure.

Pressure equals force divided by area of
application, where the force is perpen-

Distinguish between the relationship that

and the relationship
that states how it is produced. (4.1)
defines acceleration,

(4.1)

If the forces exerted on an object are 50 N in

J
/ one direction and 30 N in the opposite di-

4.

dicular to the surface area.

rection,

what

is

the net force exerted on the

object? (4.1)

falling object

pulls

is

acted on by gravity, which

a force equal to the weight

5.',

downward with

much does

of the object.

In free

fall

tion of all

(no air resistance), the acceleraobjects is the same, regardless of

When

there

is

air resistance, a falling ob-

ject will accelerate only until

6|

it

reaches

its

terminal speed.
At terminal speed, the force of air resistance balances the force of gravity.

its

is

being moved by a certain

is doubled, by how

acceleration change? (4.1)

moved by a certain
dumped into the cart so
doubled, by how much does the

Suppose a cart
net force.

mass.

Suppose a cart

net force. If the net force

its

mass

If

is

is

a load

being
is

acceleration change? (4.2)

Q7. jDistinguish

J proportional

between the concepts of directly

and inversely proportional.

Newton's Second Law of Motion

52

How

examples.

with

Support your statement

(4.1-4.2)

do the

speed
State Newton's second law in words, and in
the form of an equation. (4.3)

8,

m/s '?
-

What can be done so that

both terminal speeds are equal? (4.8)

a lighter skydiver?

(4.3)

'How much support force does a table exert

V_y on a book that weighs 5 N w hen the book is
placed on the table? What is the net force on

22/

the

book

in this

(4.8)

things being equal, why does a heavy

skydiver have a greater terminal speed than

9.

reached?

All

r>How much force must a 20 000-kg rocket develop to accelerate

is

and the weight of

compare when terminal

air resistance

object

falling

case? (4.4)

What

is

the net force acting on a 25-N freely

falling object?

What

encounters 15

enough

tails fast

the net force

is

of air resistance?
to

encounter 25

when

it

When

it

of air re-

sistance? (4.7-4.8)
1

When

a 100-N bag of nails hangs motionless

from a single vertical strand of rope, how

many newtons of tension are exerted in the
strand? How about if the bag is supported
by four vertical strands? (4.4)

What

121

is

the cause of friction,

and

it act with
respect
motion of a sliding object? (4.5)

direction does

13.

It

what

in

to

the

Activities
1

the force of friction acting on a sliding

/crate

100 N,

is

how much

force

must be ap-

plied to maintain a constant velocity?

What

be the net force acting on the crate?^

/hat will be the acceleration? (4.5)

will

Distinguish

between force and pressure.

(4.6)

you drop a sheet of paper and a book side

by side, the book will fall faster due to its
greater weight compared to the air resistance. If you place the paper against the
lower surface of the raised horizontally held
book and again drop them at the same time,
it will be no surprise that they will hit the
surface below at the same time. The book
has simply pushed the paper with it as it
falls. If you repeat this, only with the paper
on top of the book, which will fall faster? Try
it and see!
If

Which

produces more pressure on the

ground a narrow tire or a wide tire of the
same weight? (4.6)

2.

Drop two balls of different weights from the

same height, and for small speeds they practically fall together. Will they roll together

The

down

twice as great on a
2-kg rock as on a 1-kg rock. Why then, does
the 2-kg rock not fall with twice the acceleraforce of gravity

is

tion? (4.7)

How
tube

can a coin and a feather in a vacuum

with the same acceleration? (4.7)

fall

same

the

inclined plane?

If

each

is

3.

The net

on an object and the

force that acts

resulting acceleration are always in the

(

18./

Why

do a coin and a feather

fall

with

differ-

direction.

?.)

You can demonstrate

this

same
with

pulled toward you,

ent accelerations in the presence of air? (4.8)

a spool.

How much air resistance acts on a

ence if the string is on the bottom or the

top? Try it and maybe you'll be surprised.

If

which way
1

sus-

pended from the same size string and each

is made into a pendulum, and displaced
through the same angle, will they swing to
and fro in unison? Try it and see.

of nails thai falls at

its

00-N bag

terminal speed?

(4.8)

the spool
will

it

is

roll?

Does

it

make

a differ-

Chapter Review

/ffifaik

53

and Explain

v^_K What

is

10.

the difference between saying that

is proportional to another and
equal to another?

one quantity
saying

it is

2Ylf a four-engine jet accelerates down the

runway at 2 m/s 2 and one of the jet engines
fails,

how much

acceleration will the other

three produce?

Harry the painter swings year after year

from his bosun's chair. His weight is 500 N
and the rope, unknown to him, has a breaking point of 300 N. Why doesn't the rope
break when he is supported as shown in Figure B left? One day Harry is painting near a
flagpole, and for a change, he ties the free
end of the rope to the flagpole instead of to
his chair (Figure B right). Why did Harry
end up taking his vacation early?

a loaded truck can accelerate at 1 m/s

and loses its load so it is only' 3/4 as massive,
f

what acceleration can

it

attain for the

same

driving force?
rocket fired from its launching pad not
only picks up speed, but its acceleration increases significantly as firing continues.

'A

Why is this so? (Hint: About 90% of the mass

of a

newly launched rocketjs

/ 5yWhat

Fig.

What

is the advantage of the large flat sole

on the foot of an elephant? Why must the
small foot (hoof) of an antelope be so hard?

mass and what is the weight of a

10-kg object on the earth? What is its mass
and weight on the moon, where the force of

is

gravity

>\

The

the

is Ve

that of the earth?

little girl in

Figure

A hangs

(
at rest

12.

from

\When a rock
/Avhat

Vw

(Is

How

does the reading

on the scale compare to her weight?
the ends of the rope.

fuel.)

I
I

is its

is

thrown straight upward,

acceleration at the top of its path?

your answer consistent with a = F/m?)

13A As a skydiver falls faster and faster through

J the air, does the net force on her increase,
decrease, or remain unchanged? Does her
acceleration increase, decrease, or remain
unchanged? Defend your answers.
After she jumps, a skydiver reaches terminal

Fig.

speed after 10 seconds. Does she gain more

speed during the first second of fall or the
ninth second of fall? Compared to the first
second of fall, does she fall a greater or a
lesser distance during the ninth second?

why does a defending lineman often attempt to get his

In football-team blocking,

body under that of his opponent and push

upward? What effect does this have on the
friction force between the opposing lineman's feet and the ground?

Why

is

wide

tire

Whv

does a sharp knife cut better than a

the force of friction no greater for a

than for a narrow

duli knife?

,1^-

72

tire?

15J.

A regular

tennis ball and another filled with

heavy sand are dropped at the same time
from the top of a high building. Your friend
says that even though air resistance is present, they should hit the ground together
because they are the same size and "plow
through" the same amount of air. What do
vou say?

Newton's Third Law of MotionAction and Reaction

Fig. 5-1

When you push

on

the wall, the wall pushes

on you.

5.1

If you lean over too far, you'll fall over. But if you lean over with
your hand outstretched and make contact with a wall, you can
do so without falling. When you push against the wall, the wall
pushes back on you. That's why you are supported. Ask your
friends why you don't topple over. How many will answer, "Because the wall is pushing on you and holding you in place"?
Probably not very many people (unless they're physics types)
realize that walls can push on us every bit as much as we push
on them.*

Interactions Produce Forces

is a push or a pull. Looking closer,
however, we find that a force is not a thing in itself, but is due to
an interaction between one thing and another. For example, a
hammer hits a nail and drives it into a board. One object interacts with another. Which exerts the force and which receives the
force? Newton thought about questions like this, and the more
he thought, the more he came to the conclusion that neither object has to be identified as the "exerter" or "receiver." He reasoned that nature is symmetrical and concluded that both objects must be treated equally. The hammer exerts a force against

In the simplest sense, a force

Fig. 5-2

The

drives the nail

int.
is

.action that
tl

the one that brings

mer

54

to a halt.

same as
ham-

he

The terms push or pull usually invoke the idea of a living thing exerting a
So strictly speaking, to sa\ "the wall pushes on you" is to say "the wall
exerts a force as though it were pushing on you." As far as the balance of forces
is concerned, there is no observable difference between the forces exerted by
vou (alive) and the wall (nonliving).
force.

Law

Newton's Third

5.2

the nail, but

is itself

55

same interaction
hammer. Such observations
law of action and reaction.

halted in the process. The

that drives the nail also slows the

Newton

led

to his third

law

the

Newton's Third Law

5.2

Newton's third law

states:

Whenever one body exerts a force on a second body,

the second body exerts an equal and opposite force on
the

One

first.

force

is

called the action force.

The other

is

called the reac-

doesn't matter which force we call action and which

call reaction. They are equal. The important thing is that

tion force.

we

It

The action and reaction

third law is often stated
always opposed an equal reaction."

neither force exists without the other.

forces

make up a pair of forces. Newton's

as "to every action there

is

W-Nx

In every interaction, forces always occur in pairs. For ex-

ample, in walking across the floor you push against the floor,
and the floor in turn pushes against you. Likewise, the tires of a
car push against the road, and the road in turn pushes back on
the tires. In swimming you push the water backward, and the
water pushes you forward. There is a pair of forces acting in each
instance. The forces in these examples depend on friction; a person or car on ice, by contrast, may not be able to exert the action
force against the ice to produce the needed reaction force.

Fig. 5-3

the boat
to

Questions
1

2.

Does a

stick of

dynamite contain force?

A car accelerates along a

the force that

road. Strictly speaking,

what

is

moves the car?

Answers
1.

is not something a body has, like mass, but is an interaction between one object and another. A body may possess the capability of exerting a
force on another object, but it cannot possess force as a thing in itself. Later
you will see that something like a stick of dynamite possesses energy.

No, a force

2. It is

the road that pushes the car along. Really! Except for air resistance, only

How does it do this? The rotatThe road, in turn, pushes forward on

The next time you see a car moving along a road, tell your

the road provides a horizontal force on the car.

ing tires push back on the road (action).

the tires (reaction).
friends that

it is

server.

the road that pushes the car along.

at first they don't believe

If

them that there is more than meets

Turn them on to some physics.

you, convince

the eye of the casual ob-

What happens to
when she jumps

shore?

56

5.3

Newton's Third Law of Motion

and Reaction

Identifying Action

Often the identification of the pair of action and reaction forces

is not immediately obvious. As a example, what are the action
and reaction forces in the case of a falling boulder, say, when
there is no air resistance? You might say that the earth's gravitational force on the boulder is the action force, but can you identify
the reaction force? Is it the weight of the boulder? No, weight is
simply another name for the force of gravity. Is it caused by the
ground where the boulder strikes? No, the ground does not act
on the boulder until the boulder strikes it.
It turns out that there is a simple recipe for treating action
and reaction forces. It is this: Make a statement about one of the
forces in the pair, say the action force, in this form:

Body A

exerts a force on

Then the statement about the reaction

Body B

body

force

is

B.

simply

exerts a force on body A.

This is easy to remember; all you need to do is switch A and B

around. So, in the case of the falling boulder, the action is that
the earth (body A) exerts a force on the boulder (body B). Then
reaction is that the boulder exerts a force on the earth.

ACTION TIRE PUSHES ROAD

:

ACTION ROCKET PUSHES GAS

:

REACTION ROAD PUSHES TIRE

REACTION: 6AS PUSHES ROCKET

ACTION EARTH PULLS BAIL

REACTI0N:BALL PULLS EARTH

Fig. 5-4

Force-pair between bod\ A and body B. Note that

on B, the reaction is simply B exerts force on A.

exerts force

when

action

is

5.4

Action and Reaction on Bodies of Different Masses

57

up on the earth with

every bit as much force as the earth pulls down on it. The forces
are equal in strength and opposite in direction. We say that the
boulder falls to the earth; could we also say that the earth falls
to the boulder? The answer is yes, but not as much. Although the
forces on the boulder and the earth are the same, the masses are
quite unequal. Recall that Newton's second law states that the
respective accelerations will be not only proportional to the net
forces, but inversely proportional to the masses as well. Considering the massiveness of the earth, it is no wonder that we don't
sense its very small, or infinitesimal, acceleration. Strictly speaking, however, the earth moves up toward the falling boulder. So,
when you step off a curb, the street really does come up ever so
slightly to meet you!
A similar but less exaggerated example occurs in the firing
of a rifle. When the rifle is fired, the force the rifle exerts on the
bullet is exactly equal and opposite to the force the bullet exerts
on the rifle; hence the rifle kicks. At first thought, you might expect the rifle to kick more than it does, or wonder why the bullet
Interestingly enough, the boulder pulls

moves so fast compared to the rifle. According to Newton's

ond law, we must also consider the masses involved.

Fig. 5-6

The

force that

is

exerted against the recoiling

force that drives the bullet along the barrel.

more acceleration than the

Why

rifle is

sec-

just as great as the

then, does the bullet

undergo

rifle?

Suppose we let F represent both the action and reaction forces,

mass of the bullet, and 222 the mass of the rifle. Then the
acceleration of the bullet and rifle are found by taking the ratio
of force to mass. The acceleration of the bullet is given by

the

a =
while the acceleration of the

Fm

rifle is

_F_
222

Do you see why the change in motion of the bullet is so huge

compared to the change of motion of the rifle? A given force

Fig. 5-5
The earth is pulled
up by the boulder with just as

much
pulled
earth.

force as the boulder

downward by

the

is

58

Newton's Third Law of Motion

exerted on a small mass produces a large acceleration, while the

same force exerted on a large mass produces a small acceleration. Different sized symbols have been used to indicate the differences in relative masses

and resulting accelerations.

Question

Suppose you're riding in the front seat of a speeding bus

and you note that a bug splatters onto the windshield. Without question, a force has been exerted upon the unfortunate
bug and it has undergone a sudden deceleration. Is the corresponding force that the bug exerts against the windshield
greater, less, or the same? Is the resulting deceleration
of the bus greater than, less than, or the same as that of
the bug?

Fig. 5-7

Does the dog wag

the tail or does the

the dog?

5.5

Or both?

tail

wag

Have you ever noticed Newton's third law at work when a dog
wags its tail? If the tail is relatively massive compared to the
dog, note that the tail also wags the dog! The effect is less noticeable for dogs with tails of relatively small mass.

Why Action and Reaction Forces Don't Cancel

Action and reaction forces act on different bodies. If the action is
caused by A acting on B, then the reaction is caused by B acting
on A. The action force acts on B; the reaction force acts on A. The
action and reaction forces never act on the same object. Hence
action and reaction forces never cancel one another.
This is often misunderstood. Suppose for example that a friend
who hears about Newton's third law says that you can't move a
football by kicking it. The reason offered is that the reaction force
by the kicked ball is equal and opposite to your kicking force.
The net force would be zero. So if the ball was at rest to begin
with, it will remain at rest
no matter how hard you kick it!

What do vou sav

Answer
The force

to

vour friend?

the bug exerts against the windshield is just as great as the force the
windshield exerts against it. The two forces comprise an action-reaction pair.
The accelerations, however, are very different. This is because the masses involved are different. The bug undergoes an enormous deceleration, while the bus
undergoes a very tiny deceleration. Indeed, a person in the bus doesn't feel the
tiny slowing down of the bus as it is struck by the bug. Let the bug be more massive, like another bus for example, and the slowing down is quite evident!

The Horse-Cart Problem

5.6

You know

59

you kick a football, it will accelerate. Does

Newton's third law? No! Your kick
acts on the ball. No other force has been applied to the ball. The
net force on the ball is very real, and the ball accelerates. What
about the reaction force? Aha, that doesn't act on the ball; it acts
on your foot. The reaction force decelerates your foot as it makes
contact with the ball. Tell your friend that you can't cancel a
force on the ball with a force on your foot.
Now, if two people kick the same ball with equal and opposite
forces at the same time (Figure 5-9), then the net force on the
ball would be zero
but not for your single kick.
that

if

this acceleration contradict

When A acts on B, and C also

and cancellation can occur.
Fig. 5-9

5.6
A

acts on B,

two opposite forces

act

on B

The Horse-Cart Problem

is shown in the comic

"Horse Sense" (next page). The horse believes that its pull
on the cart will be cancelled by the opposite and equal pull by
the cart on the horse and make acceleration impossible. This is
the classical horse-cart problem which is a stumper for many
students at the university level. Through careful thinking, you
can understand it here.
The horse-cart problem can be looked at from different points
of view. First, there is the point of view of the farmer who is concerned only with his cart (the cart system). Then, there is the
point of view of the horse (the horse system). Finally, there is the

situation similar to the kicked football

strip

point of the horse and cart together (the horse-cart system).

First look at the farmer's point of view: The farmer is concerned only with the force that is exerted on his cart. A net force
on the cart, divided by the mass of the cart, will produce a very
real acceleration.

The farmer doesn't care about the reaction on

the horse.

Now look at the horse's point of view: It's true that the opposite reaction force by the cart on the horse restrains the horse.
Without

this force the horse could freely gallop to the market.

Fig. 5-8

If

the action force

acts on B, then the reaction

on A. Only a single
on each, so no cancellation can occur.
force acts

force acts

60

Newton's Third Law of Motion

Action Equals Reaction

5.7

This force tends to hold the horse back. So how does the horse
move forward? By pushing backward on the ground. The ground,
in turn, pushes forward on the horse. In order to pull on the cart,
the horse pushes backward on the ground. If the horse pushes
the ground with a greater force than its pull on the cart, there
will be a net force on the horse. Acceleration occurs. When the
cart is up to speed, the horse need only push against the ground
with enough force to offset the friction between the cart wheels

and the ground.

Finally, look at the horse-cart system as a whole. From this
viewpoint the pull of the horse on the cart and the reaction of
the cart on the horse are internal forces
forces that act and
react within the system. They contribute nothing to the acceleration of the horse-cart system. From this viewpoint they can
be neglected. The system can be accelerated only by an outside
force. For example, if your car is stalled, you can't get it moving
by sitting inside and pushing on the dashboard; you must get
outside and make the ground push the car. The horse-cart system
is similar. It is the outside reaction by the ground that pushes
the system.

Questions
1.

From Figure

5-10,

what

is

the net force that acts on the

cart? That acts on the horse? That tends to

make

the

ground recoil?
2.

Once the horse

gets the cart

up

to the desired speed,

the horse continue to exert a force

5.7

must

on the cart?

Action Equals Reaction

This chapter began with a discussion of how, when you push

against a wall, the wall in turn pushes back on you. Suppose for
some reason you punch the wall. Bam, your hand is hurt. Your

1.

Answers
The net force on the

2.

Yes, but only

cart

is

P -

/';

on the horse, F - P; on the ground,

F-f.

counteract the wheel friction of the cart and air resis

tance. Interestingly enough, air resistance would be absent il there were a
wind blowing in the same direction and just as last as the horse and cart. If
the wind blows just enough faster to provide a force to counteract friction, the

enough

horse could wear

market.

to

toilet

skates and simply coast with the cart

all

the

wav

to the

F<
Fig. 5-10

All the pairs of

on the horse
and cart are shown: (1) the
pulls P of the horse and cart
on each other; (2) the pushes
F of the horse and ground on
each other; and (3) the friction f between the cart wheels
and the ground. Notice that
there are two forces each applied to the cart and to the
horse. Can you see that the
forces that act

acceleration of the horse-cart

system

F-f?

is

due

to the net force

crv^

62

f
\

HONESTLY, THE WALL

HIT

It vou hit the wall,

you equally hard.

Fig. 5-11
it

MY HAND AND

SPRAINED MY WRIST ?

hits

Newton's Third Law of Motion

friends see your damaged hand and ask what happened. What
can you truthfully say? You can say that the wall hit your hand.
How hard did the wall hit your hand? Every bit as hard as you
hit the wall. You cannot hit the wall any harder than the wall
can hit back on you.
Hold a sheet of paper in midair and tell your friends that the
heavyweight champion of the world could not strike the paper
with a force of 200 N (nearly 50 pounds). You are correct. And
the reason is that the paper is not capable of exerting a reaction
force of 200 N. You cannot have an action force unless there can
be a reaction force. Now, if you hold the paper against the wall,
that is a different story. The wall will easily assist the paper in
providing 200 N of reaction force, and then quite a bit more if
need be!
For every action, there is an equal and opposite reaction. If
you push hard on the world, the world pushes hard on you. If
you touch the world gently, the world will touch you gently in
return. The way vou touch others is the way others touch vou.

Fig. 5-12

You cannot touch without being touched

Newton's third law.

63

Chapter Review

Chapter Review

Concept Summary
An

interaction between

8.\

9.)

two things produces a

pair of forces.

Each thing exerts a

The two forces are

force

on the

other.

called the action

and

^^

Action and reaction forces are equal in

strength and opposite in direction.

action

is

a bowstring acting on an arrow,

identify the reaction force. (5.3)

When you jump

up, the world really does

downward.

recoil

Why

can't this

motion of

the world be noticed? (5.4)

reaction forces.

If

When

rifle is fired,

how

does the size of the

on the bullet compare to the

force of the bullet on the rifle? How do the
accelerations of the rifle and bullet compare? Defend your answer. (5.4)
force of the rifle

Important Terms
action force (5.2)
Newton's third law (5.2)

1 1

J Since action and reaction are always equal

and opposite

in size

in direction,

why

don't

they simply cancel one another and make

net forces greater than zero impossible?

reaction force (5.2)

(5.5)

Review Questions

Questions 12-15 refer to Figure 5-10.

Besides the force of gravity, how
forces are exerted on the cart?
\^y
/lZ.Ja.

l.AVhat evidence can you cite to support the

idea that a wall can push on you? (5.1)

b.

Using the
figure,

\^2iWhat is meant by saying that a force

to an interaction? (5.1)

is

due

When a hammer interacts with a nail, which

exerts a force on which? (5.1)

v4^)When a hammer exerts a force on a

does the amount of force compare
the nail on the hammer? (5.2)

nail,

symbols shown

in the

the net force on the cart?

How many

forces besides gravity are ex-

erted on the horse?

b.

(K

is

(5.6)

13.J a.
3/

letter

what

many

how

What is the net force on the horse?

How many forces are exerted by the horse
on other objects?

(5.6)

to that of

I4\ a.

How many

forces are exerted

on the

horse-cart system?
(

5.)

Why

do we say that forces occur only

in

b.

pairs? (5.2)
/ffV

When you walk

v_>' pushes

What

is

system?
along a

you along?

floor,

what exactly

(5.2)

its speed, why must the

horse push harder against the ground than it

In order to increase

pulls on the

When swimming, you push the water backward call this action. Then what exactly
is

the reaction force? (5.3)

the net force on the horse-cart

(5.6)

wagon?

(5.6)

you

hit a

wall with a force of 200 N,

much

force

is

If

exerted on you? (5.7)

how

Why cannot

you

a force of 200

hit a

N?

feather in mid-air with

(5.7)

"6. jWhat is the reaction counterpart to a force

^-^ of 1000 N exerted by the earth on an orbit-

communications

ing

How does the saying "you get what

18
V

Newton's third law?

relate to

Newton's Third Law of Motion

satellite?

you give"

(5.7)

7. jlf

action equals reaction,

V^pulled

into orbit

why

isn't

the earth

around the communica-

tions satellite in the preceding question?

A pair of 50-N weights are attached to a

spring scale as shown in Figure B. Does the

Think and Explain

1.

Newton

0, 50, or 100 N? (Hint:

read any differently if one of the
strings were held by your hand instead of
being attached to the 50-N weight?)

realized that the sun pulls on the

\_y earth with

spring scale read

Would

and causes
around the sun.

the force of gravity

the earth to

move

in orbit

Does the earth pull equally on the sun? Defend vour answer.

Why

is it

easier to

walk on a carpeted

floor

than on a smooth polished floor?

you walk on a log that is floating

^"^water, the log moves backward. Why

\ 3. jlf

it

in the
is

this

If you

step off a ledge, you noticeably accelertoward the earth because of the gravitational interaction between the earth and
you. Does the earth accelerate toward you
ate

as well? Explain.
Fig.

rv^

Suppose you're weighing yourself while

^-^ standing next to the bathroom sink. Using
the idea of action and reaction, why will the
scale reading be less when you push downward on the top of the sink (Figure A)? Whywill the scale reading be more if you pull upward on the bottom of the sink?

9.

and a massive truck have a headupon which vehicle is the impact force greater? Which vehicle undergoes
the greater change in its motion? Defend

If a bicycle

\_/on

collision,

vour answers.
10.)

People used to think that a rocket could not

be fired to the moon because it would have
no air to push against once it left the earth's
atmosphere. Now we know that the idea is
mistaken, for several rockets have gone to
the

moon. What exactly

pels a rocket

in a

is

the force that pro-

vacuum?

Since the force that acts on a bullet when a

is fired is equal and opposite to the force
that acts on the gun, doesn't this imply a
zero net force, and therefore the impossibility of an accelerating bullet? Explain.

gun

Fig.

when

Suppose by chance you found yourself sitting beside a physicist

taking a long bus ride. Suppose the physicist was in a talkative mood and told you about some of the things he or she did
for a living. To explain things, your companion would likely doodle on a scrap of paper. Physicists love using doodles to explain
ideas. Einstein was famous for that. You'd see some arrows in
the doodles. The arrows would probably represent the magnitude
(how much) and the direction (which way) of a certain quantity.
The quantity might be the electric current that operates a minicomputer, or the orbital velocity of a communications satellite,
or the enormous force that lifts an Atlas rocket off the ground.
Whenever the length of an arrow represents the magnitude of a
quantity, and the direction of the arrow represents the direction
of the quantity, the arrow is called a vector.

when

Vector and Scalar Quantities

Some quantities require both magnitude and direction for a comThese are called vector quantities. A force, for
example, has a direction as well as a magnitude. So does a velocity. Force and velocity are the most familiar vector quantities,
but there are a few others treated in later chapters.
Many quantities in physics, such as mass, volume, and time,
can be completely specified by their magnitudes. They do not involve any idea of direction. These are called scalar quantities.
They obey the ordinary laws of addition, subtraction, multiplication, and division. If 3 kg of sand is added to kg of cement,
plete description.

65

66

Vectors

the resulting mixture has a mass of 4 kg. If 5 liters of water is

poured from a pail which initially had 8 liters of water in it, the
resulting volume is 3 liters. If a scheduled 1-hour trip runs into
a 15-minute delay, the trip ends up taking l\ hours. In each of
these cases, no direction is involved. We see that 10 kilograms
north, 5 liters east, or 15 minutes south have no meaning. Quantities that involve only magnitude and not direction are scalars.

6.2

Vector Representation of Force

It is easy to draw a vector that represents a force. The length,
drawn to some suitable scale, indicates the magnitude of the
force. The orientation on the paper and the arrowhead show the

direction.

shows a horse pulling a cart and a man pushing

The diagram shows vectors for these two
forces acting on the cart. The horse applies twice as much force
on the cart as the man. So the vector for the force supplied by
the horse is twice as long as the one for the force supplied by the
man. The vectors have been drawn to a scale on which 1 cm represents 100 N. The vectors are pointing in the same direction,
since the two forces are in the same direction.
Figure 6-1

left

the cart from behind.

The resultant of two

depends on the direc-

Fig. 6-1

forces

tions of the forces as well as

on their magnitudes.

MAN

-^ 100 N

+ HORSE

HORSE

-* 200 N

-^300N

200 N

MAN

MOON

RESULTANT
100 N-*

RESULTANT

The man pushes with 100 N and the horse pulls with 200 N.
Since the two forces act in the same direction, the resulting pull
is equal to the sum of the individual pulls and acts in the same
direction. The cart moves as if both forces were replaced by a
single net force of 300 N. This net force is called the resultant of
the two forces. We see it is represented by a vector 3 cm long.
Now suppose that the horse is pushing backwards with a force
of 200 N while the man is pulling forward with a force of 100 N
(Figure 6-1 right). The two forces then act in opposite directions.
The resultant (net force) is equal to the difference between them,
200 N - 100 N = 100 N, and acts in the direction of the larger
force. It is represented by a vector 1 cm long.

67

Vector Representation of Velocity

6.3

6.3

Vector Representation of Velocity

Speed is a measure of how fast something is moving; it can be in

any direction. When we take into account the direction of motion as well as the speed, we are talking about velocity. Velocity,
a vector quantity.
Consider an airplane flying due north at 100 km/h relative to
the surrounding air. There is a tailwind (wind from behind) that
also moves due north at a velocity of 20 km/h. This example
is represented with vectors in Figure 6-2 left. Here the velocity
vectors are scaled so that 1 cm represents 20 km/h. Thus, the
100-km/h velocity of the airplane is shown by the 5-cm-long vector and the 20-km/h tailwind is shown by the 1 -cm-long vector.
You can see (with or without the vectors) that the resultant velocity is going to be 120 km/h. Without the tailwind, the airplane
would travel 100 km in one hour relative to the ground below.
With the tailwind, it would travel 120 km in one hour.
Suppose, instead, that the wind is a headwind (wind headon), so that the airplane flies into the wind rather than with the
wind. Now the velocity vectors are in opposite directions (Figure 6-2 right). Their resultant is 100 km/h - 20 km/h = 80 km/h.
Flying against a 20-km/h headwind, the airplane would travel
only 80 km relative to the ground in one hour.
like force, is

Questions
1

In the cart, horse,

and man example shown

in Figure 6-1

suppose the man's kid brother assists by also pushing forward on the cart, but with a force of 50 N. What
would then be the resultant force the men and horse exert
on the cart?
left,

2.

Suppose

in the previous question that all parties exert

forces as stated in a

headwind

then be the resultant force the

of 10 km/h.

men and

What would

horse exert on

the cart?
Answers
(A reminder: are you reading this before you have thought about the questions

and come up with your own answers? Finding, seeing, and remembering the answer is not the way to do physics. Learning to think about the ideas of physics is
more important. Think first, then look! It will make a difference.)
1.

The resultant

will

be the

sum

of the applied forces:

200

N +

100

N +

50

N =

350 N.
2.

The resultant force

is

the same, 350 N. Be careful here. Adding apples to a pile

of oranges doesn't increase the

subtract velocity from force.

number

of oranges. Similarly,

we

can't

add or

120

Ji

100

20 k "}(

I00

=>

1?
Z0 km/h

WITH THE

A6AINST

WIND

THE WIND

The velocity of
an airplane relative to the
ground depends on its velocity relative to the air and
on the wind velocity.
Fig. 6-2

68

6.4

Vectors

Geometric Addition of Vectors

Consider the forces excited by the horses towing the barge in
Figure 6-3 left. When vectors act at an angle to each other, a simple geometrical technique can be used to find the magnitude
and direction of the resultant. The two vectors to be added are
drawn with their tails touching (see Figure 6-3 right). A projection of each vector is drawn (dashed lines) starting at the head of
the other vector. The four-sided shape that results is known as
a parallelogram because the opposite sides are parallel and of
equal length. The resultant of the two forces is the diagonal of
the parallelogram between the points where the tails meet and
the dashed lines meet."

F,>gCBt>

^ya%?
The barge moves under the action of the resultant of the two forces F,
and F The direction ol the resultant is along the diagonal of the parallelogram
constructed with sides F, and F
Fig. 6-3
,.

,.

We

can see that the barge will not move

in the direction of

either of the forces exerted by the horses, but rather in the direction of their resultant.

The resultant

is

found by using the

fol-

lowing rule for the addition of vectors:

The resultant

of

two vectors may be represented by

the diagonal of a parallelogram constructed with the

two vectors as

You can apply

common

When

sides.

on a
north and

this rule to other pairs of forces that act

shows

point. Figure 6-4

forces of 3

the angles in a parallelogram are 90,

four sides of the rectangle are the

same

length,

it

it is

to the

becomes a rectangle;

if

the

a square.

Since vector arrows only represent forces, it doesn't matter where you
them on a drawing so long as their directions and lengths are correct.
Another way to find their resultant is to rearrange the arrows in any order so
place

they are united

tail

to tip.

A new

to the tip of the last vector,

vector,

drawn from

the

tail

of the

represents the resultant or net force.

first

vector

Geometric Addition of Vectors

6.4

69

4 N to the east. Using a scale of 1 N:l cm, you construct a parallelogram, using the vectors as sides. Since the vectors are at right
angles, your parallelogram

is simply a rectangle. If you draw a

diagonal from the tails of the vector pair, you have the resultant.
Measure the length of the diagonal and refer to the scale, and

you have the magnitude of the resultant. The angle can be found
with a protractor.

31

The 3-N and 4-N

Fig. 6-4

forces

add

to

produce a force of

N.

Exercises
1

method construct the resultants of

and 4-N forces represented by the vectors shown.
They are drawn to a scale on which 1 cm:l N. Measure
your resultants with a ruler and compare them to the
correct answers given at the bottom of the page.

By

the parallelogram

the 3-N

>.

What

are the

ble for a 3-N

minimum and maximum

and a 4-N

force acting

resultants possi-

on the same object?

With the technique of vector addition you can correct for the
crosswind on the velocity of an airplane. Consider a
slow-moving airplane that flies north at 80 km/h and is caught
in a strong crosswind of 60 km/h blowing east. Figure 6-5 shows
vectors for the airplane velocity and wind velocity. The scale

effect of a

Answers
1.

2.

Left: 6

N; right: 4 N.

The minimum resultant occurs when the forces oppose each other: 4 N - 3 N =
N. The maximum resultant occurs when they are in the same direction: 4 N +
3 N = 7 N. (At angles to each other, 3 N and 4 N can combine to range between N and 7 N.)
1

Vectors

70

is
cm: 20 km/h. The diagonal of the constructed parallelogram (rectangle in this ease) measures 5 cm, which repreui^sents 100 km h. So the airplane moves at 100 km/h relative to
the ground, in a northeasterly direction.*

here

JO

80

kn

100 "%

RESULTANT

(SCALE
Fig. 6-5

speed

of

An 80-km

km

100

20^)

lew--

60 n,
6v-Km

h airplane living in a

h crosswind has a resul

h relative to the ground.

a special case of the parallelogram that often occurs.

vectors that are equal in magnitude and at right angles to one another are to be added, the parallelogram becomes
2 or
a square. Since for any square the length of a diagonal is

There

is

When two

The diagonal

Fig. 6-6

square
of

its

is

sides.

6.5

of a

\ 2 the length ol one

times one of the sides, the' resultant is \ 2 times one of the

vectors. For example, the resultant of two equal vectors of magnitude 100 acting at right angles to each other is 141 .4.
1

.414,

Equilibrium
of combining vectors by the parallelogram rule is an
experimental fact. It can be shown to be correct by considering
an example that is common and quite surprising the first time
the case of being able to hang safely from a vertical clothesline
but not being able to do so when the line is strung horizontally.
Invariably, it breaks (Figure 6-7).

The method

*
Whenever the vectors are at right angles to each other, their resultant can
be found by the Pythagorean Theorem, a well-known tool of geometry. It states
that the square of the hypotenuse of a right-angle triangle is equal to the sum
of the squares of the other two sides. Note that two right triangles are present
in the parallelogram (rectangle in this case) in Figure 6-5. From either one of

these triangles
resultant

we

get:

= (60 km h) - (80 km h)
= 3600 (kmh) + 6400 (km
= 10 000 (km h)
:

h)

The square

root of 10

000 (km

h)

is

100

km

h, as

expected.

6.5

Equilibrium

71

g^r^

You can safely hang

from a piece of clothesline
when it hangs vertically, but
you'll break it if you attempt
to make it support your
weight when it is strung
Fig. 6-7

horizontally.

We can understand this with spring scales that are used to

measure weight. Consider a block that weighs 10 N, about the
weight of ten apples. If we suspend it from a single scale, as in
Figure 6-8 left, the reading will be its weight, 10 N. Did you know
that if you stand with your weight evenly divided on two bathroom scales, each scale will record half your weight? This is because the springs in the scales are compressed and, in effect,
push up on you just as hard as gravity pulls down on you. If two
scales support your weight, each will have half the job. The same
is

true

if

you hang by a pair of scales. In

10

this case the springs that

support you are stretched, and each scale has half the job and
reads half your weight. So if we suspend the 10-N block from a
pair of vertical scales (Figure 6-8 right), each scale will read 5 N.
The scales pull up with a resultant force that equals the weight
of the block. The diagram shows a pair of 5-N vectors that have a
10-N resultant that exactly opposes the 10-N weight vector. The
net force on the block is zero, and the block hangs at rest; we say
it is in equilibrium. The key idea is this: if a 10-N block is to hang
in equilibrium, the resultant of the forces supplied by the pair of
springs must equal 10 N. For vertical orientation this is easy:
5 N + 5 N = 10 N. This is all Chapter 4 stuff.
Now let's look at a non-vertical arrangement. In Figure 6-9
left, we see that when the supporting spring scales hang at an
angle to support the block, the springs are stretched more, as indicated by the greater reading. At 60 from the vertical, the
readings are 10 N each
double what they were when the scales
were hanging vertically! Can you see the explanation? The result-

ant of the two scale readings must be 10 N upward to balance

the downward weight of the block. A pair of 5-N vectors will
produce a 10-N resultant only if they are parallel and acting in
the same direction. If the vectors have different directions, then
each vector must be greater than 5 N to produce a resultant of
ION. For equilibrium, the diagonal of the parallelogram formed
by the vector sides, whatever the angle between them, must remain the same. Why? Because the diagonal must correspond to
10 N, exactly equal and opposite to the 10 N weight of the block.

(Left) When a 10-N

block hangs vertically from a
single spring scale, the scale
pulls upward with a force of

Fig. 6-8

When it hangs
from two spring
scales, each scale pulls upward with a force of half the
10 N. (Right)

vertically

weight, or

N.

72

Vectors

As the angle between the spring scales in-

Fig. 6-9

creases, the scale readings

increase so that the resultant

(dashed-line vector) remains

at 10

N upward, which

is re-

quired to support the 10-N

block.

In Figure 6-9 right, where the angle from the vertical has been
increased to 75. 5, each spring must pull with 20 N to produce
the required 10-N resultant. As the angle between the scales is
increased, the scale readings increase. Can you see that as the
angle between the sides of the parallelogram increases, the magnitude of the sides must increase if the diagonal is to remain the
If you understand this, you understand why you can't
be supported by a horizontal clothesline without producing a
stretching force that is considerably greater than your weight.
The parallelogram rule turns out to be quite interesting.

same?

Questions
1

2.

If

the kids on the swings are of equal weight, which swing

is

more

pictures of equal weight are hung in a gallery as

shown. In which of the two arrangements is the wire
likely to

break?

Answers
The tension
likely to

2.

break?

Two

more

likelv to

is greater in the ropes that hang at an angle, so they are more

break than the vertical ropes.

The tension is greater in the picture to the left, because the supporting rope
makes a greater angle with respect to the vertical than the picture on the right
This

is

similar to the stretched clothesline.

73

Components of Vectors

6.6

Components of Vectors

6.6
Two

same object may be replaced by a

same effect upon
combined effects of the given vectors. The re-

vectors acting on the

single vector (the resultant) that produces the

the object as the

verse is also true: any single vector

two

may

be regarded as the

re-

on the body in some

direction other than that of the given vector. These two vectors
are known as the components of the given vector that they replace. The process of determining the components of a vector is
sultant of

vectors, each of

which

acts

called resolution.

A man pushing

lawnmower

applies a force that pushes the

machine forward and also against the ground. In Figure 6-10,

vector F represents the force applied by the man. We can separate this force into two components. Vector Y is the vertical component, which is the downward push against the ground. Vector
X is the horizontal component, which is the forward force that
moves the lawnmower.
We can find the magnitude of these components by drawing a
rectangle with F as the diagonal Since X and Y are the sides of a
parallelogram, vector F is the resultant of the vectors X and Y.
Hence the two components X and Y acting together are equivalent to the force F That is, the motion of the lawnmower is the
same whether we assume that the man exerts two forces, components X and Y, or only one force, F
The rule for finding the vertical and horizontal components of
any vector is relatively simple, and is illustrated in Figure 6-11.

Fig. 6-10

The

plied to the

force

F ap-

lawnmower may

be resolved into a horizontal

A vector V is drawn

in the proper direction to represent the force,

whatever vector is in question (Figure 6- 1 1 left). Then
and horizontal lines are drawn at the tail of the vector

velocity, or

vertical

drawn that encloses the vector

and the sides of the rectangle are the desired components. We see that the components
of the vector Vare then represented in direction and magnitude
bv the vectors X and Y.
(Figure 6-1

in

such a

right).

way

A rectangle

that

is

is

a diagonal

X
Fig. 6-1

The vector V has component vectors

X and

Y.

component, X, and a
component, Y.

vertical

74

Vectors

Any vector can be represented by a pair of components that

are at right angles to each other. This is neatly illustrated in the
explanation of a sailboat sailing against the wind. (See Appendix C, Vector Applications, at the back of this book.)

Exercise

With a ruler, draw the horizontal and vertical components of the two vectors shown. Measure the components
and compare your findings with the answers given at the
bottom of the page.

Components of Weight
We

know that a ball will roll faster down a steep hill than a
with a small slope. The steeper the hill is, the greater the acceleration of the ball We can understand why this is so with vector components. The force of gravity that acts on things gives
all

hill

them weight, which we represent

as vector

W. The force vector

W acts only straight down toward the center of the earth but
most often usecomponents of W may act in any direction.
It is

consider components that are at right angles to each other.

In Figure 6-12 we see
broken up into components A and B,
where A is parallel to the surface and B is perpendicular to the
ful to

It is component A that makes the ball move. Component

presses the ball against the surface. Pictures are better than
words, so study the figure and see how the magnitudes of the

surface.

components vary

for different slopes.

Answer
Left vector: the horizontal

component

Right vector: the horizontal component

is
is

cm; the

vertical

6 cm; the vertical

component
component

is

4 cm.

is

4 cm.

Projectile

6.8

75

Motion

SLOPE

SLOPE

@90
SLOPE @<?

(A-0)

(B=W)

w
Fig. 6-12

The weight

dicular components

of the ball

A and

is

represented by vector W, which has perpenA serves to change the speed of the ball,

B. Vector

while vector B presses it against the surface. Note

B vary from zero to W.

how

the magnitudes of

A and

see that only when the slope is zero

when the surhorizontal
is component A equal to zero? That's why
the speed of the ball does not change on a horizontal surface.
Note another thing when the surface is horizontal. Component
B is equal to W; the ball presses against the surface with the most
force. But when the slope is 90, component B becomes zero and
A equals W, so the ball has its maximum acceleration.

Can you

face

is

Question
At what angle will components

A and B

in Figure 6-12

have equal magnitudes? At what angle will A equal

what angle will A be greater in magnitude than W?

6.8

Projectile

W?

At

Motion

Chapter 2 discussed the vector quantities velocity and acceleration. Since only horizontal and vertical motion was considered,
we did not need to know about vector addition or the techniques
of vector resolution. But for objects projected at angles other
than straight up or straight down, we do.
A projectile is any object that is projected by some means and
continues in motion by its own inertia. A cannonball shot from
a cannon, a stone thrown into the air, or a ball that rolls off
the edge of the table are all projectiles. These projectiles follow
curved paths that at first thought seem rather complicated. However, these paths are surprisingly simple when we look at the
horizontal and vertical components of motion separately.

Answer
Components A and B have equal magnitudes
have a greater magnitude than Wat any angle.

at 45;

A =

W at 90;

A cannot

(A = W)

76

Vectors

The horizontal component of motion for a projectile is no

more complex than the horizontal motion of a bowling ball rolling freely along a level bowling alley.

If

the retarding effect of

can be ignored, the bowling ball moves at constant velocity. It covers equal distances in equal intervals of time. It rolls
of its own inertia, with no component of force acting in its direction of motion. It rolls without accelerating. The horizontal part
of a projectile's motion is just like the bowling ball's motion
along the alley (Figure 6-13 left).
friction

M
Fig. 6-13

Lett

accelerates

The

..

Roll a ball along a level surface,

cause no component
it

ol

and

its

velocity

is

constant be-

gravitational force acts horizontally. (Right) Drop

downward and

il,

and

covers greater vertical distances each second.

component

motion for a projectile following a

motion described in Chapter 2 for a
freely-falling object. Like a ball dropped in mid-air, the projectile moves in the direction of earth gravity and accelerates downward (Figure 6-13 right). The increase in speed in the vertical
direction causes successively greater distances to be covered in
each successive equal-time interval.
Interestingly enough, the horizontal component of motion for
a projectile is completely independent of the vertical component
vertical

curved path

is

of

just like the

Each acts independently of the other. Their combined

produce the variety of curved paths that projectiles follow.
The multiple-flash exposure of Figure 6-14 shows equallytimed successive positions for a ball rolled off a horizontal table.
Investigate the photo carefully, for there's a lot of good physics
there. The curved path of the ball is best analyzed by considering the horizontal and vertical components of motion separately. There are two important things to notice. The first is that
the balls horizontal component of motion doesn't change as the
falling ball moves sideways. The ball travels the same horizontal distance in the equal times between each flash. That's because there is no component of gravitational force acting horiof motion.
effects

downward, so the only acceleration of

downward. The second thing to note from the photo is
that the vertical positions become farther apart with time. The
distances traveled vertically are the same as if the ball were simzontally. Gravity acts only

the ball

is

ply dropped.

It is

of the ball

the

is

interesting to note that the

same

as that of free

fall.

downward motion

6.8

Projectile

Fig. 6-14

77

Motion

multiple-flash photograph of a ball rolling off a horizontal table.

Notice that in equal times

travels equal horizontal distances but increasingly

it

greater distances vertically.

Do you know why?

The path traced by a projectile that accelerates only in the

moving at a constant horizontal velocity

vertical direction while

When air resistance can be neglected ususlow-moving projectiles or ones very heavy compared to
the forces of air resistance
the curved paths are parabolic.
is

called a parabola.

ally for

Question
At the instant a horizontally held rifle is fired over a level
range, a bullet held at the side of the rifle is released and
drops to the ground. Which bullet the one fired downrange or the one dropped from rest strikes the ground

first?

Answer
Both bullets fall the same vertical distance with the same acceleration g due
to gravity and therefore strike the ground at the same time. Can you see that this
is consistent with our analysis of Figure 6-14? We can reason this another way by
asking which bullet would strike the ground first if the rifle were pointed at an
upward angle. In this case, the bullet that is simply dropped would hit the ground
first. Now consider the case where the rille is pointed downward. The fired bullet
hits first. So upward, the dropped bullet hits first; downward, the fired bullet
where both
hits first. There must be some angle at which there is a dead heat
hit at the same time. Can you see it would be when the rifle is neither pointing

upward nor downward

when

it

is

horizontal?

78

Upwardly Moving

Vectors

Projectiles

Consider a cannonball shot at an upward angle. Pretend for a

moment that there is no gravity; then according to the law of
inertia, the cannonball will follow the straight-line path shown
by the dashed line in Figure 6-15. But there is gravity, so this
doesn't happen. What really happens is that the cannonball continually falls beneath this imaginary line until it finally strikes
the ground. Get this: The vertical distance it falls beneath any
point on the dashed line is the same vertical distance it would
fall if it were dropped from rest and had been falling for the same
amount of time. This distance, as introduced in Chapter 2, is
given by d = igt 2 where t is the elapsed time.
,

Fig. 6-15

projectile

With no gravity the

would follow the

straight-line path (dashed

line).
it

But because of gravity,

beneath this line the

falls

same vertical distance it

would fall if released from
rest. Compare the distances

w ith Table 2-3 in Chap(With g = 9.8 m/s-, these

distances are more accuratelv
4.9 m, 19.6 m, and 44.1 m.)

45m

fallen
ter 2.

We

another way: Shoot a projectile skyward at

there is no gravity. After so many seconds t, it should be at a certain point along a straight-line path.
But because of gravity, it isn't. Where is it? The answer is, it's directly below this point. How far below? The answer in meters is

can put

this

some angle and pretend

5r 2 (or

more

accuratelv, 4.9 1 2 ). Isn't that neat?

Note another thing from Figure 6-15. The cannonball moves

equal horizontal distances in equal time intervals. That's because no acceleration takes place horizontally. The only acceleration is vertically, in the direction of earth gravity. The vertical
distance it falls below the imaginary straight-line path during
equal time intervals continually increases with time.

6.9

Upwardly Moving

Projectiles

Figure 6-16 shows vectors representing both horizontal and

components of velocity for a projectile following a parabolic path. Notice that the horizontal component is everywhere
the same, and only the vertical component changes. Note also
that the actual velocity is represented by the vector that forms
the diagonal of the rectangle formed by the vector components.
At the top of the path the vertical component vanishes to zero, so
the actual velocity there is the horizontal component of velocity
at all other points. Everywhere else the magnitude of velocity is
greater (just as the diagonal of a rectangle is greater than either
vertical

of

its sides).

Fig. 6-16

The velocity of a projectile at various points along its path. Note that
component changes and the horizontal component is the same

the vertical

everywhere.

Figure 6-17 shows the path traced by a projectile with the

same launching speed at a steeper angle. Notice that the initial
velocity vector has a greater vertical component than when the
projection angle is less. This greater component results in a
higher path. But the horizontal component is less so the range
is less.

Fig. 6-17

Path for a steeper projection angle.

79

80

Vectors

Figure 6-18 shows the paths of several projectiles all having

same initial speed but different projection angles. The figure
neglects the effects of air resistance, so the paths are all parabolas. Notice that these projectiles reach different altitudes, or
heights above the ground. They also have different horizontal
ranges, or distances traveled horizontally.
the

Ranges

Fig. 6-18
tile

shot at the

ol a

projec-

same speed

v.

at

s
/

different projection angles.

/

^\

"-

/
1

75

_\

-\

60

/
V

45

s,

''

30

**

\
\

15

L-'---

The remarkable thing to note from Figure 6-18 is that the

same range is obtained from two different projection angles
angles that add up to 90 degrees! An object thrown into the air
at an angle of 60 degrees, for example, will have the same range
as if it were thrown at the same speed at an angle of 30 degrees.
For the smaller angle, of course, the object remains in the air for
a shorter time.

Maximum

Fig. 6-19

attained

when

range

the ball

is

is

bat-

ted at an angle of nearly 45.

(In cases

where the weight of

is comparable

the projectile

to the applied force, as

a heavy javelin

is

when

thrown, the

applied force does not produce the same speed for different projection angles,

maximum

and

range occurs for

angles quite a bit less than
45.)

Upwardly Moving

6.9

81

Projectiles

Fig. 6-20

In the presence of

path of a
high-speed projectile falls
air resistance, the

short of a parabola (dashed

curve).

We have emphasized the special case of projectile motion

without air resistance. When there is air resistance, the range of
a projectile is somewhat shorter and is not a true parabola (Figure 6-20).

lO^s'^UWs

Questions
1

projectile

is

shot at an angle into the

tance is negligible, what is

horizontal acceleration?

its

air. If air resis-

downward

acceleration?

30<Vs
Q

Its
2.

At what part of

its

path does a projectile have

3(Ws
9

minimum

speed?

small enough to be negligible, a projectile

height in the same time it takes to fall
from that height to the ground. This is because its deceleration
by gravity while going up is the same as its acceleration by gravity while coming down. The speed it loses while going up is therefore the same as the speed it gains while coming down. So the
projectile arrives at the ground with the same speed it had when
it was projected from the ground.
If an object is projected fast enough so that its curvature
matches the curvature of the earth, and it is above the atmosphere so that air resistance does not affect its motion, it will fall
all the way around the earth and be an earth satellite. This interIf air

resistance

esting topic

is

maximum

will rise to its

is

treated in Chapter 12.

Answers
1. Its

downward

2.

The speed

is

of a projectile

vertically, its

is g because the force of gravity is downward;

zero because no horizontal forces act on it.

acceleration

horizontal acceleration

is

minimum

speed at the top

is

zero.

at the top of its path.

If

it is

If it

is

its

launched

projected at an angle, the vertical

component of velocity is zero at the top. leaving only the horizontal component. So the speed at the top is equal to the horizontal component of the projectile's velocity at any point. Isn't that neat?

50/s

Fig. 6-21

5CVS
o

q\'q^>

Without

air resis-

tance, speed lost while going

up equals speed gained while

coming down; time up equals
time down.

't

?V

H> OyJ D,

Vectors

($> Chapter Review

Concept Summary
Vector quantities

What

have both magnitude and

direction.

A vector

is

an arrow whose length repre-

sents the magnitude of a vector quantity

and whose direction represents the

direc-

4J

ties

is

in

equilibrium, the

sultant of all the forces supporting

exactly oppose

its

it

When

gravity

is

What

projectile, the horizontal

flies at

200 km/h

if it

experi-

(6.3)

is

a parallelogram? (6.4)

When

a parallelogram is constructed in order to add forces, what represents the resultant of the forces? (6.4)

/8.)What
the only force acting on a

is

the resultant velocity of an airplane

of forces or

velocities.

speed classified as a scalar, and ve-

is

wind?

weight.

components

vertical

they both act

if

re-

must

single vector can be replaced by two components that add to form the original vector.
It is often convenient to study the horizon-

and

N downward?

ences a 50-km/h tailwind? A 50-km/h head-

Any

tal

Why

that normally

to scale.

When something

75

What is their resultant

downward? (6.2)

What

of several forces or several veloci-

can be determined from a vector diagram

drawn

the resultant of a pair of forces,

locity as a vector? (6.3)

tion of the quantity.

The resultant

is

N upward and

100

component

of

^ two

the magnitude of the resultant of

at right angles to

its

velocity does not change.

is

vectors of magnitudes 4 and 3 that are

each other?

(6.4)

SWhat

is the magnitude of the resultant of a

pair of 100-N vectors that are at right angles

to each other? (6.4)

Important Terms
component
equilibrium

The tension in a clothesline carrying a load of

resultant (6.2)

(6.6)

wash

scalar quantity (6.1)

(6.5)

vector (6.1)
vector quantity (6.1)

projectile (6.8)

resolution (6.6)

line
r^

ll l)

is

is

appreciably greater when the clothesstrung horizontally than when it is

strung vertically.

What

is

Why?

(6.5)

the net force, or equivalently, the

resultant force that acts on an object

is

Review Questions
y\J

How

doe a vector differ from a scalar?

(6.1

in

equilibrium?

when

it

(6.5)

your weight, what is the stretcharm when you let yourself

hang motionless by one arm? By both arms

Compared

to

ing force in your

ll

If

a vector that

is

cm

long represents a

how many newtons does a

cm long drawn to the same scale,

force of 5 N,

vec-

tor 2

rep-

resent? (6.2)

vertically? Is this force greater or less if you

hang with your hands wide apart? Why?
(6.5)

13.)

Chapter Review

83

Distinguish between the

ric

method

of geomet-

(22 ) At

addition of vectors and vector resolution

/*)

What

are the magnitudes of the horizontal

components of a vector that

100 units long, and oriented at 45? (6.6)

and

maximum

altitude? For

maximum

hori-

zontal range? (6.9)

(6.4, 6.6)

(14/

what angle should a slingshot be oriented

for

vertical

Neglecting air resistance, if you throw a ball

upward with a speed of 20 m/s, how
fast will it be moving when you catch it?
straight

is

(6.9)
/
>

lS.

The weight

of a ball rolling

J plane can be broken

into

down an

inclined

two vector compoand

g),NNeglecting

you throw

the other acting perpendicular to the plane.

is

At what slope angle are these two com-

b.

At what slope angle

is

the

component

parallel to the plane equal to zero?

c.

who

base, will the catching speed

(6.9)

Activity

Why does a bowling ball move without accelwhen

eration

it

rolls

along a bowling alley?

(6.8)

17/ In the

absence of air resistance,

why

does

component of velocity for a

projectile remain constant, and why does
only the vertical component change? (6.8)
the horizontal

Place a coin at the edge of a smooth table so that

it overhangs slightly. Then place a second coin
on the table top some distance from the overhanging coin. Set the second coin sliding across
the table (such as by snapping it with your finger) so that it strikes the overhanging coin and
both coins fall to the floor below. Which if
either

does the

downward component

motion of a projectile compare

the floor first? Does your answer

on the speed of the sliding coin?

of the

to the

mo-

tion of free fall? (6.8)

from

rest

strikes the ground

What

the

maximum

and the other

b.

What

A boat

first? (6.8)

is

How

b.

far below an initial straight-line

path will a projectile fall in one second?
Does your answer depend on the angle of
launch or on the initial speed of the projectile? Defend your answer. (6.9)

a.

is

fired straight

upward

is

is

a.

What

b.

How

the

of

fired

is

the

upward

answer

il

the projectile

at 45 instead? (6.9)

magnitude
magnitude 4?

minimum

rowed
is

possible resultant?

km/h directly across a

km/h (Figure A).

at 8

the resultant speed of the boat?

and in what direction can the

boat be rowed to reach a destination directly across the river?

at

100 m/s. How fast is it moving at the instant it reaches the top ol its trajectory?

What

possible resultant

of a pair of vectors, one of

river that flows at 6

projectile

)>^i

ink and Explain

At the instant a ball is thrown horizontally

over a level range, a ball held at the side of
the first is released and drops to the ground.
If air resistance can be neglected, which
ball
the one thrown or the one dropped

hits

depend

How

b.

first

At what slope angle is the component

parallel to the plane equal to the weight?
(6.7)

a.

on

be greater than, equal to, or less than

20 m/s?
How about if air resistance is a factor?

ponents equal?

6/

if

a baseball at 20 m/s to your friend

a.

air resistance,

nents: one acting parallel to the plane,

is

Fig.

fast

84

3/B\ whatever means,

find the direction of the

'
.

.airplane in Figure 6-5.

which position is the tension the

arms of the weightlifter shown
ure B? The most?
In

Vectors

^Why

are the main supporting cables of suspension bridges designed to sag the way
they do (Figure D)?

least in

the

in Fig-

Fig.

8.)

The boy on the tower (Figure E) throws a

ball 20 m downrange, as shown. What is his
pitching speed?

Why

do

electric

power

lines

sometimes

break in winter when a small weight of

forms on them?
6.)

Why cannot
hard enough

the strong
to

make

man

in

Figure

ice

pull

the chain straight?

20
Fig.

Why

does a ball rolling down an incline

undergo more acceleration the steeper the
incline?

o.

Why

is

less force

required to push a barrel

F) than to lift it

up a sloping ramp (Figure

vertically?

Fig.C

Fig.

Momentum

Have you ever wondered how a karate expert can sever a stack
cement bricks with the blow of her bare hand? Or why a fall
on a wooden floor is not nearly as damaging as a fall on a cement
floor? Or why "follow through" is important in golf, baseball,
and boxing? To understand these things, you first need to recall
of

the concept of inertia, introduced in Chapter 3 in Newton's

first

law of motion, and developed further in Chapters 4 and 5 in

Newton's second and third laws of motion. Inertia was discussed
in terms of objects both at rest and in motion. Now we concern
ourselves only with the inertia of moving objects. The idea of inertia in motion is momentum, which refers to moving things.

Momentum

7.1

We

all know that a massive truck is harder to stop than a small

car moving at the same speed. We say the truck has more momentum than the car. By momentum we mean inertia in motion, or more specifically, the mass of an object multiplied by its

velocity.

That

is:

momentum = mass
or, in

x velocity

shorthand notation,

momentum = mv
When

direction

is

not an important factor,

momentum = mass
which we still abbreviate mv.
We can see from the definition that
a large momentum if either its mass is

we can

say:

x speed,

moving object can have

large, or its

speed

is

large,

85

<Q Q&i (maJ^.^

Momentum

both its mass and speed are large. A truck has a larger mothan a car moving at the same speed because its mass is
larger. An enormous ship moving at a small speed can have a
large momentum, whereas a small bullet moving at a high speed
can also have a large momentum. And, of course, a huge object
moving at a high speed, such as a massive truck rolling down a
steep hill with no brakes, has a huge momentum, whereas the
same truck at rest has no momentum at all.
or

il

mentum

A truck rolling

Fig. 7-1

down a hill has more momentum than a roller skate

moving

at the

same speed,

be-

cause the truck has more-

mass. But
rest

it

and the

the truck

is

at

Question

roller skate

Can you think of a case where the roller skate and truck
shown in Figure 7-1 both would have the same momentum?

moves, then the skate has

mote momentum because

onK

it

7.2

has speed.

Impulse Equals Change in

If

the

momentum

Momentum
mass or the
mass remains unchanged, as is

of an object changes, either the

velocity or both changes.

If

the

most often the case, then the velocity changes. Acceleration ocAnd what produces an acceleration? The answer is a force.
The greater the force that acts on an object, the greater will be
the change in velocity, and hence, the change in momentum.
But something else is important also: time how long the
force acts. Apply a force briefly to a stalled automobile, and you
produce a small change in its momentum. Apply the same force
over an extended period of time, and a greater change in momentum results. A long sustained force produces more change in
momentum than the same force applied briefly- So for changing
the momentum of an object, both force and time are important.
Interestingly enough, Newton's second law {a = Flm) can be
re-expressed to make the time factor more evident when the term
for acceleration is replaced by its defiruIiaa-rebaQge in ve locity
per time." Then the equation becomes ('force x time intervalyis
curs.

Answer
The

roller skate

roller skate

As

many

is

very

and truck can have the same momentum

much

greater than the speed of the truck.

times as the mass of the truck

is

if

the speed of the

How much greater?

greater than the mass of the roller

skate! Get it?

can state Newton's second law as F = ma. Since a = (change in velocity)

X [(change in v)lt\. Simple algebraic rewe can say F =
arrangement gives Ft = change in mv, or in delta notation, FSt = \mv.

We

(time interval),

Impulse Equals Change in

7.2

Momentum

^rf^Uij^^

equal to the change in "mass x velocity." The quantity "mass x

velocity" has already been defined as momentum. So Newton's
second law can be re-expressed in the form

= change

Ft

in

mv

which reads, "force multiplied by the time-during-which-it-acts

equals change in momentum."
The quantity "force x time interval" is called impulse. Thus,
;

change

in

momentum

The impulse-momentum relationship helps us to analyze a

where momentum is changed. We will
consider familiar examples of impulse for the cases of ( 1 ) increasing momentum, (2) decreasing momentum over a long time, and
variety of circumstances

(3)

decreasing

Case

momentum

Increasing

over a short time.

Momentum

that in order to increase the momentum of

an object, we should apply the greatest force we can. Also, we
should extend the time of contact as much as possible. A golfer
and baseball player do both when they swing as hard as possible
and "follow through" when hitting a ball.
The forces involved in impulses are usually not steady, but
vary from instant to instant. For example, a golf club that strikes
a ball exerts zero force on the ball until it comes in contact; then
It

makes good sense

the force increases rapidly as the club and ball are distorted (Figure 7-2); the force then diminishes as the ball comes up to speed
and returns to its original shape. So when we speak of such impact forces in this chapter, we mean the average force of impact.

Case 2: Decreasing Momentum over a Long Time

If you were in a car that was out of control, and you had your
choice of hitting a concrete wall or a haystack, you wouldn't have
to call on your knowledge of physics to make up your mind. But
knowing some physics helps you to understand why hitting something soft is entirely different from hitting something hard. In
the case of hitting either the wall or the haystack, your momentum will be decreased by the same impulse. The same impulse

means the same product of force and time, not the same force or
the same time. You have a choice. By hitting the haystack instead of the wall, you extend the time of impact
you extend the
time during which your momentum is brought to zero. The longer
time is compensated by a lesser force. If you extend the time

impact 100 times, you reduce the force of impact by 100. So

whenever you wish the force of impact to be small, extend the

of

time of impact.

Fig. 7-2

The

force of a golf

club against a golf ball varies

throughout the duration of

impact.

88

^S's^,

,^_ . x -i5^0O^^'< *^i

,!

Fig. 7-3

impact

If

is

Momentum

the change in

momentum

occurs over a long time, the force of

small.

Everyone knows that a padded dashboard in a car is safer

than a bare metal one, and that airbags save lives. Most people
also know that if you're going to catch a fast baseball with your
bare hand, you extend your hand forward so you'll have plenty
of room to let your hand move backward after you make contact
with the ball. You wouldn't intentionally hold your hand stationary, as in catching a ball with your bare hand against a hard
wall! In these cases you extend the time of impact and thereby
reduce the force of impact.
When you jump off something to the ground below, you don't
keep your legs straight and stiff (ouch!). Instead, you bend your
knees upon making contact. By doing this, you extend the time
during which your momentum is decreasing by 10 to 20 times
that of a stiff-legged abrupt landing. The forces your bones experience are thus reduced 10 to 20 times by such knee bending.
A wrestler thrown to the floor tries to extend his time of arrival on the mat by relaxing his muscles and spreading the impact into a series of smaller impacts, as his foot, knee, hip, ribs,
and shoulder fold onto the mat in turn. Of course, falling on a

mat

is

preferable to falling on a solid

floor, for this also

increases

the time of impact.

Everyone knows that it is less harmful to fall on a wooden

than on a concrete floor. And most people know that this is
because the wooden floor has more "give" than the concrete floor.
Ask most people why a floor with more give makes for an easier
fall, and you'll get a puzzled response. They may say, "Because it
has more give." But your question is," Why does a floor with more
give produce a safer fall?" The answer is, we know, because an
impulse is required to bring your momentum to a halt, and the
impulse is composed of two variables, impact force and impact
time. By extending the impact time as the floor gives, the impact
force is correspondingly reduced. The safety net used by acrobats provides an obvious example of small impact force over a
long time to provide the required impulse to reduce the momenfloor

tum of fall.
A boxer confronted with

high-momentum punch wishes

minimize the force of impact.

If

he cannot avoid being

to

hit, at

Impulse Equals Change in

7.2

Momentum

89

least he can control the length of time it takes for his body to
absorb the incoming momentum of his opponent's fist. So he
wisely extends the impact time by "riding or rolling with the
punch." This lessens the force of impact.

OMENTUM
jaw reduces the
moves away (rides with the
time. (Right) When the boxer moves

In both cases the impulse provided by the boxer's

Fig. 7-4

momentum

of the punch. (Left)

When

the boxer

punch), the chief ingredient of impulse

into the glove, the time

is

is

reduced and the chief ingredient of impulse

is

force.

Question
If

is able to make the duration of

times as long by riding with the punch, by how
will the force of impact be reduced?

the boxer in Figure 7-4

impact

much

five

Case 3: Decreasing Momentum over a Short Time

Ride a bicycle into a concrete wall, and you're in trouble. When
you catch a high-speed baseball, move your hand toward the
ball instead of away upon contact, and your hand is messed up.
When boxing, move into a punch instead of away, and your opponent has a better chance of scoring a knockout. In these cases
the principal ingredients of impulses required to reduce momentum are impact forces because the times of impact are brief.

Op

^
Fig. 7-5

force

is

If

the change in

momentum

occurs over a short time, the impact

large.

Answer
The force of impact

will be 5 times less than

if

he didn't pull back.

Momentum

90

The idea of short time of contact explains how a karate expert

can sever a stack of bricks with the blow of her bare hand (Figure 7-6). She brings her arm and hand swiftly against the bricks
with considerable momentum. This momentum is quickly re-

duced when she delivers an impulse

to the bricks. The impulse is

hand against the bricks multiplied by the time
her hand makes contact with the bricks. By swift execution she
makes the time of contact very brief, and correspondingly makes
the force of impact huge. If her hand is made to bounce upon

the force of her

impact, the force

Fig. 7-6

is

even greater.

large impulse to

the bricks in a short time

produces a considerable
force.

Question

A boxer being
for best results,

hit with a punch contrives to extend time

whereas a karate expert delivers a force in

a short time for best results. Isn't there a contradiction here?

7.3

Bouncing
a flower pot falls from a shelf onto your head, you may be in
it bounces from your head, you're certainly in trouble.
Impulses are greater when bouncing takes place. This is because
If

trouble. If

Answer
is no contradiction because the best results for each are quite different.
The best result for the boxer is reduced force, accomplished by maximizing time,
and the best result for the karate expert is increased force delivered in minimum

There

time.

7.4

Conservation of Momentum

and then, in
"throw it back again" is greater than the impulse required
merely to bring something to a stop. Suppose, for example, that
you catch the falling pot with your hands. Then you provide an
impulse to catch it and reduce its momentum to zero. If you were
to then throw the pot upward, you would have to provide additional impulse. So it would take more impulse to catch it and
throw it back up than merely to catch it. The same greater impulse is supplied by your head if the pot bounces from it.
The fact that impulses are greater when bouncing takes place
was employed with great success in California during the gold
rush days. The water wheels used in gold-mining operations were
inefficient. A man named Lester A. Pelton saw that the problem
had to do with their flat paddles. He designed curved-shape paddles that would cause the incident water to make a U-turn upon
impact to "bounce." In this way the impulse exerted on the
water wheels was greatly increased. Pelton patented his idea
and made more money from his invention, the Pelton wheel,
than any of the gold miners.
the impulse required to bring something to a stop

effect,

IMPULSE
The Pelton wheel. The curved blades cause water to bounce and make
a U-turn, which produces a greater impulse to/ turn the wheel.
Fig. 7-7

7.4

Conservation of Momentum

Newton's second law tells you that if you wish to accelerate an

you must apply a force to it. This chapter is saying much
the same thing, but in different language. If you wish to change
the momentum of an object, exert an impulse on it.
In either case, the force or impulse must be exerted on the object by something outside the object. Internal forces do not count.
object,

91

92

Momentum

For example, the molecular forces within a basketball have no

upon the momentum of the basketball, just as your push
against the dashboard of a car you're sitting in will have no effect in changing the momentum of the car. This is because these
forces are internal forces. They act and react within the object.
To change the momentum of the basketball or car, an outside
push or pull is required. If no outside force is present, then no

effect

change in momentum is possible.

Consider a rifle being fired. The force that pushes on the bullet

when

rifle barrel is equal to the force that makes

(Newton's third law, action and reaction). These
forces, interestingly enough, are internal to the "system" that
comprises the rifle and bullet. So they don't change the momentum of the rifle-and-bullet system. Before the firing, the system
is at rest and the momentum is zero. After the firing, the net, or
total, momentum is still zero. No net momentum is gained and
no net momentum is lost. Let's look at this carefullv.

the

it

is

inside the

rifle recoil

Fig. 7-8
is still

The momentum before

zero because the

firing

momentum

is

momentum
momentum of the

zero. After firing, the net

of the rifle cancels the

bullet.

Momentum,

like the quantities velocity

tion as well as a size

i
Fig. 7-9

The machine gun

from the bullets

and climbs upward.
recoils

it

hres

it is

and

force,

has a direc-

a vector quantity. Hence, like velocity

and force, it can be cancelled. So although the bullet in the preceding example has considerable momentum as it accelerates
within the rifle barrel and then continues at high speed outside
the barrel, and the recoiling rifle has momentum in and of itself,
the system of both bullet and rifle has none. The momenta (plural form of momentum) of the bullet and the rifle are equal in
size but opposite in direction. They cancel each other for the system as a whole. No external force acted on the system before or
during firing. When there is no net force, there can be no net acceleration. Or, when there is no net force, there is no net impulse
and therefore no net change in momentum. You can see that if
no net force acts on a system, then the momentum of that system
cannot change.
If you extend the idea of a rifle recoiling or "kicking" from the
bullet it fires, you can understand rocket propulsion. Consider a
machine gun recoiling each time a bullet is fired. The momen-

Conservation of Momentum

7.4

93

Questions

Newton's second law says that if there is no net force exerted on a system, no acceleration is possible. Does it follow from this that no change in momentum can occur?

1.

Newton's third law says that the force a

2.

rifle

exerts on

its

equal and opposite to the force the bullet exerts

on the rifle. Does it follow that the impulse the rifle exerts
on the bullet is equal and opposite to the impulse the bullet exerts on the rifle?
bullet

is

turn of recoil increases by an amount equal to the momentum of

each bullet fired. If the machine gun is fastened so it is free to
slide on a vertical wire (Figure 7-9), it will accelerate upward as
bullets are fired downward. A rocket accomplishes acceleration
by the same means. It is continually "recoiling" from the ejected
exhaust gases. Each molecule of exhaust gas can be thought of
as a tiny bullet shot

from the rocket (Figure

momentum of the
momentum of the exhaust

ing to note that the

posite to the

7-10). It

is

interest-

equal and opgases. So if we consider

rocket

is

and exhaust gases, like the rifle and

no net change in momentum. As a practical matwe are usually not concerned with the net momentum of

the total system of rocket

bullet, there is
ter,

rocket-exhaust, but of the rocket

The

momentum

external forces.

some

itself.

Fig. 7-10

prodded by
possessed by a system before

of a system cannot change unless

The momentum

internal interaction will be the

same

The rocket

recoils

from the "molecular bullets"

it fires and climbs upward.

momentum

as the

possessed by the system after the interaction. When the momentum (or any quantity in physics) does not change, we say it is
conserved. The idea that momentum is conserved when no external force acts is elevated to a central law of mechanics, called
the law of conservation of momentum:
In the absence of an external force, the

mome n,u mo f

V-TC^

a system remains unchanged.

If

a system undergoes changes wherein

all forces

are internal,

Answers
1

other line of reasoning

is

which means no change

in

momentum.

it

follows because the absence of acceleration

velocity,

2.

means

there is no change in
(mass X velocity). Ansimply that no net force means no net impulse.

Yes,

which

in turn

means no change

Yes, because the time during

momentum

rifle

acts on the bullet

and the

bullet

the same. Since time is equal for both, and force is equal
opposite for both, then force x time (impulse) is equal and opposite fol

acts on the

and

which the

in

rifle is

both. Impulse, like force,

is

a vector quantity

and can be cancelled.

J<

icnr\

Momentum

94

as for example, in atomic nuclei undergoing radioactive decay,

cars colliding, or stars exploding, the net momentum of the sys-

tem betore and

after the event

is

the same.

Question
About 50 years ago it was argued that a rocket wouldn't
operate in outer space because there is no air for it to push
against. But a rocket works even better in outer space precisely because there is no air. How can you explain this?

Collisions

7.5

The law
sions.

the total or net

net

Fig. 7-1

momentum

of conservation of

Whenever objects

is

neatly seen in colli-

collide in the absence of external forces,

momentum

never changes:

momentum,

net

momentum,

after collision

Conservation of

momentum

is

neatly

demon-

strated with the use of this air

track. Jets of air

many

from the

tiny holes in the track

provide an air cushion upon

which the glider

slides nearly

friction-free.

Elastic Collisions

When

moving

billiard ball

makes

a head-on collision with an-

other billiard ball at rest, the moving ball comes to rest and the
struck ball moves with the initial velocity of the colliding ball.
We see that momentum is simply transferred from one ball to

Answer
gun doesn't need air to push against in order to recoil, a rocket doesn't
push against in order to accelerate. The presence of air impedes acceleration by offering air resistance, so a rocket actually works better where
there is no air. A rocket is not propelled by pushing against air, but by pushing
Just as a

need air

against

to

its

own

exhaust.

Collisions

7.5

the other.

When

the colliding objects

bound or rebound without

lasting deformation or the generation of heat, the collision

is

said to be an elastic collision. Colliding objects bounce perfectly

in elastic collisions (Figure 7-12).

or

3w_

QT

TO

Elastic collisions,

The dark

ball strikes a

A head-on colbetween two moving

balls, (c) A collision of two
balls moving in the same diball at rest, (b)
lision

1Q

rection. In all cases,

tum

momen-

simply transferred or
redistributed without loss or
is

gain.

-Lj

Fig. 7-12
(a)

-vV^

XuS
Inelastic Collisions
Momentum conservation holds true even when the colliding
objects become distorted and generate heat during the collision.

Such

Whenever colbecome tangled or couple together, we have an

collision. The freight train cars shown in Figure 7-13

collisions are called inelastic collisions.

liding objects
inelastic

provide an illustrative example. Suppose the freight cars are of

equal mass m, and one moves at 4 m/s while the other is at rest.
Can we predict the velocity of the coupled cars after impact?
From the conservation of momentum.
(net momentum before
4 m/s) + (m x (J m/s)'

(m x

net
(2

momentum

after

m/s)

Since twice as much mass is moving after the collision, can you
see that the velocity must be half as much as the 4 m/s value before the collision? This

is

2 m/s, in the

same

direction as before.

-=
F

Fig. 7-13

u=4

an

u=o

nVs

on

UDL

-on:
f,

the freight car on the right.

w*

uxr

TTCT

u--2/s

uu

TTTT

Inelastic collision.

The momentum of the freight

car on the left is shared with

nn

frcr

Momentum

96

Then both sides of the equation are equal. The initial momentum is shared between both cars without loss or gain. Momen-

tum

conserved.

is

More

Fig. 7-14

and

^'-(Sj'W ",nL^4P

<HS.

inelastic collisions.

after collision are the

The net momenta

of the vehicles before

same.

Questions
The following questions

refer to the gliders

on the

air track

in Figure 7-11.

Suppose both gliders have the same mass. They move toward each other at the same speed and experience an
elastic collison. Describe their motion after the collision.

Suppose both gliders have the same mass and have Velcro
on them so that they stick together when they collide.
They move toward each other at equal speed. Describe
their motion after the collision.

3.

Suppose one of the gliders is at rest and is loaded so that

it has 3 times the mass of the moving glider. Again, the
gliders have Velcro on them. Describe their motion after
the collision.

Answers
1.

Since the collision

colliding

2.

elastic, the gliders will

simply reverse directions upon

same speed as before.

at the

Before the collision, the gliders had equal and opposite momenta, since they
had equal mass and were moving in opposite directions at the same speed.
The net momentum was zero. Since momentum is always conserved, their net

momentum
gether, this
3.

is

and move away from each other

after the collision must also be zero. As they

means they must slam to a dead halt.

Before the collision, the net

loaded,

moving

before, but

mass

now

the gliders are stuck together

of the stuck-together gliders

is

It is

in the

same

'/

is

now

to-

momentum of the unmomentum is the same as

and moving as

a single unit.

The

four times that of the unloaded glider.

that of the

unloaded glider before the

direction as before, since the direction of the

conserved as well as the amount.

stuck

equals the

glider. After the collision, the net

Thus, the velocitv can be only

sion.

momentum

are

colli-

momentum

Momentum Vectors

7.6

97

most collisions there are usually some external forces that

on a system. Billiard balls do not continue indefinitely with
the momentum imparted to them. The balls encounter some
friction with the table and the air they move through. These exIn

act

ternal forces are usually negligible during the collision

itself,

so

momentum does not change during the collision. The net

momentum of a couple of trucks that collide is the same before
the net

and just after collision. As the combined wreck slides along the
pavement, friction provides an impulse to decrease momentum.
For a pair of space vehicles docking in outer space, however, the
net momentum before and after contact is exactly the same, and
persists until the vehicles encounter external forces.
Another thing: perfectly elastic collisions are not common in
the everyday world. We find in practice that some heat is generated in collisions. Drop a ball, and after it bounces from the floor,
both the ball and the floor are a bit warmer. So even a dropped
superball will not bounce to its initial height. At the microscopic
level, however, perfectly elastic collisions are commonplace. For
example, electrically charged particles bounce off one another
without generating heat; they don't even touch in the classic
sense of the word. (As later chapters will show, the notion of
touching at the atomic level is different from at the everyday
level.)

7.6

Momentum Vectors

Momentum

is

conserved even when colliding objects move at an

angle to each other. To analyze momentum for angular directions, we use the vector techniques discussed in Chapter 6. It
will be enough in this chapter if you merely become acquainted
with momentum conservation for cases that involve angles, so
three examples that convey the idea will be considered briefly

without going into depth.

In Figure 7-15 you can see that car A has momentum directed
due east, and that car B has momentum directed due north. If
their individual momenta are equal in magnitude, then after
collision their combined momentum will be in a northeast direction. Just as the diagonal of a square is not simply the arithmetic sum of two of its sides, the momentum of the wreck will
not be twice the arithmetic sum of the individual momenta before collision.

'

It

will be

Vy times the momentum of either vehicle before collision, just as

the diagonal of a square

is

n/T the length

of a side.

98

Momentum

-,M
COMBINED

MOMENTUM

MOMENTUM

OF CAR A
;

Fig. 7-15

Momentum

equal to the vector

is

sum

a vector quantity.

of the

momenta

OF

A+B

The momentum of the wreck

A and B before collision.

is

of cars

Figure 7-16 shows a falling firecracker that explodes into two

The momenta of the fragments combine by vector rules
to equal the original momentum of the falling firecracker.

pieces.

<*^~
Fig. 7-16

When

the firecracker bursts, the

(by the vector parallelogram rule) to the

momenta

momentum

of

its

fragments add up

just before bursting.

is a photo of tracks made by subatomic particles

streamer spark chamber. The masses of these particles can
be computed by applying both the conservation of momentum and the conservation of energy (which is covered in the next
chapter). The conservation laws are extremely useful to experimenters in the atomic and subatomic realms. A very important

Figure 7-17

in a

is the fact that forces don't show up in

The forces of the collision processes, however com-

feature of their usefulness

Fig. 7-17

Momentum

is conserved for the high-speed ele-

the equations.

plicated, need not be of concern.

other things, by the paths

The law of conservation of momentum and, as the next chaplaw of conservation of energy are the two
most powerful tools of mechanics. Their application yields detailed information that ranges from understanding the interactions of subatomic particles to measuring the spin rates of

they take after co

entire galaxies.

mentary particles, as shown

by the tracks the\ le e in a
streamer spark chamber. The
relative masses of the particles is

determi

ted,

among

ter will discuss, the

Chapter Review

99

Chapter Review

Concept Summary
Momentum

of an object

4.; a.
is

the product of

The change

in

mass

momentum

depends on the
and on the length of time it

force that acts

momentum

acts.

Impulse is force multiplied by the time during which it acts.

The change in momentum equals the

net external force.

When

objects collide in the absence of ex-

momentum is conserved no
matter whether the collision is elastic or
ternal forces,

is

6.

If

^
u.

a vector quantity.
vector rules.

to

the collision

why is it advantageous for an

extend the time during which

is

taking place? (7.2)

the time of impact in a collision is extended

by four times, by how much is the force of
impact altered? (7.2)
If

a.

Why

advantageous for a boxer to

punch?
Why is it disadvantageous to move into
an oncoming punch? (7.2)
is

it

ride with the

b.

Important Terms
(7.4)

When you throw a ball, do you experience an

elastic collision (7.5)

impulse

in

In a car crash,

occupant

Momenta combine by

conserved

increased? (7.2)

how much is the impulse increased?

By how much is the resulting change
momentum increased? (7.2)

inelastic.

Momentum

the duration of

both the force that acts on an object

and the time of impact are doubled, by

or.

impulse.

According to the law of conservation of momentum, momentum is conserved when there is no

if

impact upon an object is doubled, by

how much is the impulse increased?
By how much is the resulting change in

times velocity.

For a constant force,

impulse? Do you experience an impulse if

you instead catch a ball of the same speed? If
you catch it, then throw it out again? Which
impulse is greatest? (Visualize yourself on a

(7.2)

inelastic collision (7.5)

momentum

law of conservation of

momentum

(7.4)

(7.1)

skateboard.) (7.3)

Review Questions
a

v-/

10.

a.

Which has

b.

Which has greater momentum?

truck at rest

mass

(7.1)

Why

is

the force of impact greater in a colli-

sion that involves bouncing? (7.3)

heavy
or a rolling skateboard?

the greater

11.

Why

is

the Pelton wheel design a better one

than paddle wheels with

/i.

When

\J

ject
v or

3.1

\J

is

the average force of impact on an obextended in time, does this increase

decrease the impul.se?

What

is

(7.2)

the relationship between impulse

and momentum?

(7.2)

12.

Lx

What does

it

mean

(lat

blades? (7.3)

to say that

momentum

is

a vector quantity? (7.4)

\i

In terms ol momentum conservation, whj

does a gun kick when fired? (7.4)

I^i

100

The

14.

text states

V/ a system,

hilt

it

Everybody knows that you will be harmed

less il you tall on a floor with "give" than a
rigid Moor. In terms of impulse and momen-

no net force acts on

momentum

then the

ol

that sys-

tem cannot change. It also states there is no

change in momentum when a rifle is Bred.
Doesn't the fact that a bullet undergoes a
considerable change in momentum as it ac-

tum, why isthis so?

it

mean

to say that

momentum

is

/conserved? (7.4)

How

The

a.

b.

mo-

c.

the car are the same.

(]
9.

Imagine that you are hovering next to a

Vy space shuttle in earth orbit and your buddy

mass who

moving

is

bumps

respect to the ship

at 4

km

20.^Js

If

conserved for colliding obthat are moving

" at angles to one an-

OLAvTciAT^
\fwrMJLV\J^u^L
Who is in greater trouble a person who

(7.6)

ride a bicycle at full speed,

the greater

momentum you

Lbike? (Does this help explain

SA

tt)'^'V

X^^VS*

over me handlebars
abrupt halt?)

if

the bike

brought

diesel engine weighs^? timesj&s

as a flatcar.

into a flatcar that

do

to

km/h
how fast

If

a diesel coasts at 5

is

initially at rest,

the.tvvo coast after thev couple together?

an

10.

\2.\

the ball? Explain.

vO^A<00\ tM^Gjuf/*

i<kw&-

An alpha particle is a nuclear particle

known mass. Suppose one is accelerated
a high speed in an atomic accelerator

is

collides

You can't throw

same speed
breaking

it.

it.

raw egg against a wall withbut you can throw it with the
a

into a sagging sheet without

Explain.

(v^V^C^

-Vfa^A.<b<\jL/ \ju

i^X

and

to

and

sticks to a target particle that

initially at rest.

the

out breaking

of

directed into an observation chamber. There

it

combined

comes to an abrupt halt when he falls to the

pavement below or a person who bounces
upon impact? Explain. } OlA/VU-cVS

much

why you go
is

A railroad

which
or the

For the least harm to your hand, should you

N
\_y catch a fast-moving baseball while your
hand is at rest, while it moves toward the
ball, or while it is pulling back away from
"

a space shuttle dips back into the atorbit, it turns to the left and

How does this maneuver increase the impulse delivered to the

shuttle s.o that it will slow before Landing?

When you
t

bug

\__

right in giant S-curves.

Think and Explain

l\

the

move

X^other? Explain.

0.J has

-V

mosphere from

he

momentum

fleets

When

h with

into you.

holds onto you, how fast do you both

with respect to the ship? (7.5)

"

The changes in momentum of

and ol the car are the same size.

d.

of equal

impact on the bug and on

the car are the

collision. (7.5)

the

forces of

same size. ~"f

The impulses on the bug and on the car
are the same size.
_|
The changes in speed oLthe bug and of

Distinguish between an elastic and inelastic

What effect does friction have on

mentum of an object? (7.5)

pv.

A bug and the windshield of a fast-moving

car collide. Tell whether the follow ing statements are true or false.

can a rocket be propelled above the

is no air to "push
against"? (7.4)

atmosphere where there

17.

^dAtOs.

backwards. Would you roll backwards if

you didn't actually throw a rock but went
through the motions ol throwing the rock?
Defenu your answer. fLO \/^\jCPA^+^W*V

Explain. (7.4)

What does

f\4*Y<

II you throw a heavy rock from your hands

while standing on a skateboard, you roll

celerates along the barrel contradict this?

i.

Momentum

As a result of the impact,

and alpha par-

target particle

ticle is observed to move at half the initial

speed of the alpha particle. Why do observers conclude that the target particle is itself
an alpha particle?

Energy

Energy

is

the most central concept that underlies

all

of science.

Surprisingly, the idea of energy was unknown to Isaac Newton,

and its existence was still being debated in the 1850s. The con-

cept of energy is relatively new, and today we find it ingrained

not only in all branches of science, but in nearly every aspect of
human society. We are all quite familiar with it energy comes
to us from the sun in the form of sunlight, it is in the food we eat,
and it sustains life. Energy may be the most familiar concept in

it is one of the most difficult to define. Persons, places,

and things have energy, but we observe energy only when something is happening only when energy is being transformed. We
will begin our study of energy by observing a related concept:

science; yet

work.

Work
chapter showed that changes in an object's motion are
and to how long the force acts. "How long"
meant time. The quantity "force x time" was called impulse.
But "how long" need not always mean time. It can mean distance as well. When we consider the quantity "force x distance,"
we are talking about a wholly different quantity a quantity

The

last

related both to force

called work.

We do work when we lift a load against the earth's gravity. The

heavier the load or the higher we lift the load, the more work
is done. Two things enter into every case where work is done:

(1) the exertion of a force

and

(2)

the

Energy

movement of something by

that force.
Let's look at the simplest case, in which the force is constant
and the motion takes place in a straight line in the direction of
the force. Then the work done on an object by an applied force is
defined as the product of the force and the distance through
which the object is moved. *Ip. shocker form:

If you lift two loads one story up, you do twice as much work
as you do in lifting one load, because the force needed to lift
twice the weight is twice as great. Similarly, if you lift a load two

stories instead of

the distance

is

one

story,

you do twice as much work because

twice as much.

Note that the definition of work involves both a force and a

A weightlifter who holds a barbell that weighs 1000 N
overhead does no work on the barbell. He may get really tired
doing so, but if the barbell is not moved by the force he exerts,
he does no work on the barbell. Work may be done on the muscles
by stretching and contracting, which is force times distance on a
biological scale, but this work is not done on the barbell. Lifting
the barbell, however, is a different story. When the weightlifter
raises the barbell from the floor, he is doing work.
Work generally falls into two categories. One of these is work
done to change the speed of something. This kind of work is done
in bringing an automobile up to speed or in slowing it down.
The other category of work is the work done against another
force. When an archer stretches her bowstring, she is doing work
against the elastic forces of the bow. When the ram of a pile driver
is raised, it is exerted against the force of gravity. When you do
pushups, you do work against your own weight. You do work on
something when you force it to move against the influence of an
opposing force.
The unit of measurement for work combines a unit of force
(N) with a unit of distance (m). The unit of work is the n ewtonmeter (NJrn\ ^Jso ca lled the joule khvmes with pool). One joule
(symbol J) of work is~3one whefTa force of 1 N is exerted over a
distance of 1 m, as in lifting an apple over your head. For larger
values we speak of kilojoules (kJ)
thousands of joules or megadistance.

Fig. 8-1

Work

ing the barbell.

is
If

done

in lift-

the barbell

could be lifted twice as high,

would have
expend twice as much

the weightlifter
to

energy.

"

millions

The weightlifter in Figure 8-1

does work on the order of kilojoules. The energy released by one
kilogram of fuel is on the order of megajoules.

joules (MJ)

of joules.

For more general cases, work

is

component
and the distance moved.

the product of only the

that acts in the direction of motion,

of force

l^A^)

Mechanical Energy

8.3

Power
work says nothing about how long it takes to do
load up some stairs, you do the same
amount of work whether you walk or run up the stairs. So why
are you more tired after running upstairs in a few seconds than
after walking upstairs in a few minutes? To understand this difference, we need to talk about how fast the work is done, or
power. Power is the rate at which work is done. It equals the
amount of work done divided by the amount of time during
which the work is done:
The

definition of

When carrying a

the work.

An engine of great power can do work rapidly. An automobile

engine with twice the power of another does not necessarily produce twice as much work or go twice as fast as the less powerful
power means it will do the same amount of
The main advantage of a powerful autoits acceleration. It can get the automobile up to

engine. Twice the

work

in half the time.

mobile engine

is

a given speed in less time than less powerful engines.

We can look at power this way: a liter of gasoline can do a
specified amount of work, but the power produced when we

burn it can be any amount, depending on how fast it is burned.

The liter may produce 50 units of power for a half hour in an
automobile or 90 000 units of power for one second in a supersonic jet aircraft.

The unit of power is the joule per second, also known as the
(in honor of James Watt, the eighteenth-century developer
of the steam engine). One watt (W) of power is expended when
one joule of work is done in one second. One kilowatt (kW) equals
1000 watts. One megawatt (MW) equals one million watts. In
watt

the United States

power and

we customarily

rate engines in units of horse-

electricity in kilowatts, but either

may be used.

In the

metric system of units, automobiles are rated in kilowatts. (One

horsepower is the same as 0.75 kilowatt, so an engine rated at
134 horsepower is a 100-kW engine.)

Mechanical Energy
WhejAvork
s
is

is

drawing a bow, the bent bow

work on the arrow. When work
the heavy ram of a pile driver, the ram acquires

done by an archer

in

the ability of being able to do

done to

raise

Fig. 8-2

The three main

engines of a space shuttle

can develop 33 000
of

MW

power when fuel is burned

at the enormous rate of
3400 kg/s. This is like emptying an average-size swimming pool in 20 seconds!

104

Energy

do work on an object beneath it

done to wind a spring mechanism,
the spring acquires the ability to do work on various gears to
run a clock, ring a bell, or sound an alarm.
In each case, something has been acquired. This "something"
which is given to the object enables the object to do work. This
"something" may be a compression of atoms in the material of
an object; it may be a physical separation of attracting bodies:
it may be a rearrangement of electric charges in the molecules
a substance. This "something" that enables an object to do work
is called energy.
Like work, energy is measured in joules. It
appears in many forms, which will be discussed in the following
chapters. For now we will focus on mechanical energy
the
energy due to the position or the movement of something. Mechanical energy may be in the form of either potential energy or
the property of being able to

when

it

falls.

When work

is

kinetic energy.

Potential

Energy
An

object

may

store energy by virtue of

its

position.

The energy

stored and held in readiness is called potential energy

(PE), because in the stored state it has the potential for doing
work. A stretched or compressed spring, for example, has the potential for doing work. When a bow is drawn, energy is stored in
that

is

the bow.
of

its

A stretched rubber band has

position, for

if it is

potential energy because

part of a slingshot,

it is

capable of doing

work.

The chemical energy

in fuels is potential energy, for

it is

actu-

energy of position when looked at from a microscopic point

of view. This energy is available when the positions of electric
charges within and between molecules are altered, that is, when
a chemical change takes place. Any substance that can do work
through chemical action possesses potential energy. Potential
energy is found in fossil fuels, electric batteries, and the food
ally

we eat.
Work is

required to elevate objects against earth's gravity. The

potential energy due to elevated positions is called gravitational

potential energy. Water in an elevated reservoir and the ram of a

have gravitational potential energy.

The amount of gravitational potential energy possessed by an
elevated object is equal to the work done against gravity in liftpile driver

Strictly speaking, that

which enables an object

to

do work

is

called

its

able energy, for not all the energy of an object can be transformed to work.

avail-

Potential Energy

8.4

it. The work done equals the force required to move it upward times the vertical distance it is moved (W = Fd). The upward force required is equal to the weight mg of the object, so
the work done in lifting it through a height h is given by the

ing

product mgh:
g/avitational potential energy

PE

= weight x height
=_

Note that the height h is the distance above some reference

such as the ground or the floor of a building. The potential
energy mgh is relative to that level and depends only on mg and
the height h. You can see in Figure 8-3 that the potential energy
of the boulder at the top of the ledge does not depend on the path
level,

taken to get

it

there.

fcOOJ]

|2m

rjL

The potential energy of the 100-N boulder with respect to the ground
same (200 J) in each case because the work done in elevating it 2 m
is the same whether it is (a) lifted with 100 N of force, (b) pushed up the 4-m
incline with 50 N of force, or (c) lifted with 100 N of force up each 0.5-m stair.
No work is done in moving it horizontally (neglecting friction).

Fig. 8-3

below

is

the

Questions
1.

How much work is done on a 100-N boulder that you

carry horizontally across a 10-m room?

2. a.

How much work is done on a

lift it

b.

100-N boulder when you

m?

What power
1

c.

is

expended

if

you

lift it

this distance in

s?

What

is its

gravitational potential energy in the lifted

position?

Answers
You do no work on the boulder moved
cept for the tiny bit to start

across the

room than

it

had

it)

You do 100 J of work when you

Power = (100 J)/(l s) = 100 W.

c.

It

to

motion.

It

has no more

PE

initially.

b.

2. a.

horizontally, for you apply no force (ex-

in its direction of

lift it

(since

Fd = 100

Nm

depends; with respect to its starting position, its PE is 100

some other reference level, it would be some other value.

J;

= 100

J).

with respect

Energy

Push on an object and you can set it in motion. If an object moves,

then by virtue of that motion it is capable of doing work. It has
energy of motion, or kinetic energy (KE). The kinetic energy of
an object depends on the mass of the object as well as its speed.
It is equal to half the mass multiplied by the square of the speed.
kinetic energy

KE
When you throw

= |mass x speed 2
= jmv 2

you do work on it to give it the speed

The moving ball can then hit
something and push against it, doing work on what it hits. The
kinelic energy of a moving object is equal to the work required
to bring it to that speed from rest, or the work the object can do
in being brought to rest:
it

has

when

it

a ball,

leaves your hand.

net force x distance

kinetic energy

or in shorthanc

Fd = ;mv'
important to noticeThat the speed is squared, so that if
is doubled, its kinetic energy is quadrupled
2
(2 = 4). This means that it takes four times as much work to double the speed of an object, and also that an object moving twice
as fast as another takes four times as much work to stop. Accident investigators are well aware that an automobile traveling
at 100 km/h has four times as much kinetic energy it would have
traveling at 50 km/h. This means that a car traveling at 100 km/h
will skid four times as far when its brakes are locked as it would if
traveling 50 km/h. This is because speed is squared for kinetic
It is

the speed of an object

The potential en-

Fig. 8-4

ergy of the drawn bow equals

the work (average force x distance)

done

in

drawing the

arrow into position.

leased,

it

will

When

become

re-

the ki-

netic energy of the arrow.

energy.

Kinetic energy underlies other seemingly different forms of energy such as heat, sound, and light.
Fig. 8-5

Typical stopping

distances for cars traveling at

various speeds. Notice how
the

work done

to stop the car

x distance of
slide) depends on the square
of the speed. (The distances
would be even greater if reaction time were taken ini

30K>Am 10-mSKID
60 *%|

40-mSKID

160-mSKID

(friction force

>

account.)

120

i**4ii

This can be derived as follows. If we multiply both sides of F = ma (Newsecond law) by d, we get Fd = mad. Recall from Chapter 2 that for motion
in a straight line at constant acceleration d = tat 1 so we can say Fd = maiiat 2 )
ton's

:iaat'

= im(at) 2 Substituting
.

at,

we

get

Fd = \mv 2

107

Conservation of Energy

8.6

Question

When
become

the brakes of a motorcycle traveling at 60

locked,

how much

farther will

it

skid than

km/h
if

the

brakes locked at 20 km/h?

More important than being able

to state what energy is is understanding how it behaves

how it transforms. You can understand
nearly every process or change that occurs in nature better if
you analyze it in terms of a transformation of energy from one

form to another.
As you draw back the stone in a slingshot, you do work in
stretching the rubber band; the rubber band then has potential
energy.

When released,

potential energy.

wooden

It

fence post.

the stone has kinetic energy equal to this

delivers this energy to

The

its

target,

slight distance the post

is

perhaps a

moved

multi-

by the average force of impact doesn't quite match the kinetic energy of the stone. The energy score doesn't balance. But
if you investigate further, you'll find that both the stone and
fence post are a bit warmer. By how much? By the energy difference. Energy changes from one form to another. It transforms
plied

without net loss or net gain.

Fig. 8-6

wound

Part of the

PE

of the

spring changes into

KE. The rest turns into heatmachinery and the

surroundings due to friction.
ing the

No energy

The study of the various forms of energy and their transformations from one form into another has led to one of the great-

Answer
Nine times farther: the motorcycle has nine times as much energy when it
=;w9v' = 9(jwv )- The friction force will ordinarily be the same in cither case; therefore, to do nine limes the work requires
travels three times as fast: 3w(3v) 2

nine times as

much

sliding distance.

is lost.

108

est generalizations in physics,

of energy:

known

Energy

as the law of conservation

PE'IOOOOJ

Energy cannot be created or destroyed; it may be

transformed from one form into another, but the total

KE0

amount

of energy never changes.

When you consider any system in its entirety, whether it be as

simple as a swinging pendulum or as complex as an exploding
galaxy, there is one quantity that does not change: energy. It maychange form, or it may simply be transferred from one place to
another, but the total energy score stays the same.
PE = 7500J

**

KE-250OJ

PE

+ ICE

PE
AND 50 ON

KE
pendulum. The PE of the pendulum bob
at its highest point is equal to the KE of the bob at its lowest point. Everywhere along its path, the sum of PE and KE is the same. (Because of the work
done against friction, this energy will eventually be transformed into heat.)
Fig. 8-8

PE=50OOJ
KE=50O0J

Energy transformations

in a

This energy score takes into account the fact that the atoms
make up matter are themselves concentrated bundles of energy. When the nuclei (cores) of atoms rearrange themselves,
enormous amounts of energy can be released. The sun shines because some of this energy is transformed into radiant energy. In
nuclear reactors much of this energy is transformed into heat.
that

PE--25O0J

KE=7500J

Powerful gravitational forces in the deep hot interior of the

sun crush the cores of hydrogen atoms together to form helium
atoms. This welding together of atomic cores is called thermo-

=0
E-10000J

Fig. 8-7

When

distress leaps

the ladv in

from the burn-

ing building, note that the

of her PE and KE remains constant at successive

positions i, i, 4, and all the
wav down.

sum

nuclear fusion. This process releases radiant energy, some of

which reaches the earth. Part of this energy falls on plants, and
part of this later becomes coal. Another part supports life in the
food chain that begins with plants, and part of this energy later
becomes oil. Part of the energy from the sun goes into the evaporation of water from the ocean, and part of this returns to the
earth as rain that may be trapped behind a dam. By virtue of its
position, the water in a dam has energy that may be used to
power a generating plant below, where it will be transformed to
electric energy. The energy travels through wires to homes,
where it is used for lighting, heating, cooking, and to operate
electric toothbrushes. How nice that energy is transformed from
one form to another!

109

Machines

8.7

Question

Suppose a car with a miracle engine

1

00%

lion joules per liter)

and overall
speed
liter

is

is

able to convert

when

gasoline burns (40 milto mechanical energy. If the air drag

of the energy released

frictional forces

2000 N, what

is

on the car traveling at highway

the upper limit in distance per

the car could cover at highway speed?

8.7

Machines

A machine

is

a device for multiplying forces or simply changing

machine is the conservation of energy concept. Consider one of the simplest machines,
the lever (Figure 8-9). At the same time we do work on one end
of the lever, the other end does work on the load. We see that the
the direction of forces. Underlying every

is changed, for if we push down, the load is

heat from friction forces is small enough to neglect, the work input will be equal to the work output.

direction of force
lifted up. If the

work input = work output

Since work equals force times distance, then input force x input
distance = output force x output distance.
(force x distance), nput

(force

x distance) output

thought will show that the pivot point, or fulcrum, of

Then a small input
force exerted through a large distance will produce a large output force over a correspondingly short distance. In this way, a
lever can multiply forces. But no machine can multiply work
nor multipy energy. That's a conservation of energy no-no!
Consider the idealized case of the weightless lever shown in
Figure 8-10. The child pushes down with a force of only 10 N and
is able to lift a load of 80 N. The ratio of output force to input

little

the lever can be relatively close to the load.

Answer
From the definition of work as
distance = work -* force. If all 40

force x distance, simple rearrangement gives

distance =

wrk
force

The important point here

limit of fuel

economy

is

liter were used

drag and frictional forces, the distance

million joules of energy in one

do the work of overcoming the

would be:
to

air

40 000 000
2000 N

= 20 000

= 20

km

that even with a perfect engine, there

dictated by the conservation oi energy.

is

an upper-

The

The work
you do
at one end equals the work
done to the load at the other
Fig. 8-9

lever.

(force times distance)

end.

110

Energy

machine is called the mechanical advantage. Here the

mechanical advantage is (80 N)/(10 N), or 8. Note that the load
is lifted only one eighth the distance that the input force moves.
In the absence of friction, the mechanical advantage can also be
determined by considering the relative distances through which
force for a

the forces are exerted.

Fig. 8-10

The output

the output distance

force (80 N)

m)

is

is

eight times the input force (10 N), while

one eighth the input distance

(1

m).

There are three different ways to set up a lever (Figure 8-11).

Type is the kind of lever commonly seen in a playground with
children on each end
the seesaw. The fulcrum is between the
input and output ends of the lever. Push down on one end, and
you lift a load at the other. You can increase force at the expense
1

of distance.

TYPE 3

TYPE
Fig. 8-11

The three basic types

of lever.

With the type 2 arrangement, the load is between the fulcrum

and the input end. To lift a load, you lift the end of the lever. An
example is the placing of one end of a long steel bar under an
automobile frame and lifting on the free end to raise the automobile. Again, force is increased at the expense of distance.
In the type 3 arrangement, the fulcrum is at one end and the
load is at the other. The input force is applied between them.
Your bicep muscles are connected to your forearm in this way.
(The fulcrum is vour elbow and the load is your hand.) The type

8.7

Machines

111

3 lever increases distance at the expense of force. When you move

your bicep muscles a short distance, your hand moves a much

greater distance.

A pulley is basically a kind of type

lever that

is

used to change

the direction of a force. Properly used, a pulley or system of pulleys can multiply forces as well.

The

single pulley in Figure 8-12

left

changes the direction of

and applied force have equal

magnitudes, since the applied force acts on the same single
the applied force. Here the load

strand of rope that supports the load. Thus, the mechanical advantage equals one. Also, the distance moved at the input equals
the distance the load moves.

A pulley can (a) change the direction of a force as effort is exerted

downward and load moves upward; (b) multiply force as effort is now half the
load, and (c) when combined with another pulley both change the direction
Fig. 8-12

and multiply

force.

In Figure 8-12 center, the load

The force the

man

is

now supported by two strands.

applies to support the load

is

therefore only

The mechanical advantage equals

two. The fact that there is only a single rope does not change
this. Careful thought will show that to raise the load 1 m, the
man will have to pull the rope up 2 m (or 1 m for each side).
The pulley system in Figure 8-12 right combines the arrangements of the first two pulleys. The load is supported by two
strands of rope; the upper pulley serves only to change the direchalf the weight of the load.

tion of the force. Try to figure out the

mechanical advantage of

the pulley system. Actually experimenting with a variety of pulley

systems

is

much more

beneficial than reading about

a textbook, so do try to get your hands on

some

them

in

pulleys, in or out

of class. They're fun.

The pulley system shown in Figure 8-13 is a bit more complex,

but the principles of energy conservation are the same. When
the rope is pulled
with a force of 50 N, a 500-N load will be
lifted 1 m. The mechanical advantage is (500 N)/(50 N), or 10.
Force is multiplied at the expense of distance. The mechanical
1

Energy

112

Fig. 8-13

In

an idealized

pulle\ system, applied force

x input distance = output

output distance.

Force

advantage can also be found from the ratio of distances that is,
(input distance) * (output distance)
which is also 10.
No machine can put out more energy than is put into it. No
machine can create energy; a machine can only transfer it from
one place to another, or transform it from one form to another.

8.8

Efficiency
The previous examples of machines were considered to be ideal;
100% of the work input was transfered to work output. An ideal
machine would operate at 100% efficiency. In practice, this does
not happen, and we can never expect it to happen. In any ma-

some energy is dissipated to atomic or molecular kinetic

which makes the machine warmer.
Even a lever rocks about its fulcrum and converts a small fraction of the input energy into heat. We may do 100 J of work and
get out 98 J of work. The lever is then 98% efficient, and we waste
only 2 J of work input on heat. In a pulley system, a larger fracchine,

energy

tion of input energy goes into heat. If we do 100 J of work, the

forces of friction acting through the distances through which the
pulleys turn and rub about their axles may dissipate 40 J of

energy as heat. So the work output is only 60 J and the pulley

system has an efficiency of 60%. The lower the efficiency of a machine, the greater is the amount of energy wasted as heat.
Efficiency can be expressed as the ratio of useful work output
to total

work

input:
effic tencv

= useful work output

total work input

An inclined plane serves

as a machine. Sliding a load

incline requires less force than lifting

it

up an

vertically. Figure 8-14

113

Efficiency

8.8

Fig. 8-14
tical

times farther up the incline than the vers its weight. Whether pushed up the
gains the same amount of PE.

Pushing the block of

ice 5

distance lifted requires a force of only

plane or simply

lifted,

it

shows a 5-m inclined plane with its high end elevated 1 m.

we use the plane to elevate a heavy load, we will have to push

If
it

times farther than if we simply lifted it vertically. A little

thought will show that if the plane is free of friction, we need
apply only one fifth the force required to lift the load vertically.
The inclined plane provides a theoretical mechanical advantage
of 5. A block of ice sliding on an icy plane may have a near theoretical mechanical advantage and approach 100% efficiency, but
when the load is a wooden crate and it slides on a wooden plank,
both the actual mechanical advantage and the efficiency will
be considerably less. Efficiency can also be expressed as the
five

ratio of actual

mechanical advantage to theoretical mechanical

advantage:
cretnciencv ~
.

actual mechanical advantage

theoretical mechanical advantage

Efficiency will always be a fraction less than 1 To convert it to

percent we simply express it as a decimal and multiply by 100%.
For example, an efficiency of 0.25 expressed in percent is 0.25 x
100%, or 25%.
The jack shown in Figure 8-15 is actually an inclined plane
.

16 cm.

cylinder. You can see that a single turn of the

handle raises the load a relatively small distance. If the circular
distance the handle is moved is 500 times greater than the pitch
(the distance between ridges), then the theoretical mechanical
advantage of the jack is 500." No wonder a child can raise a
loaded moving van with one of these devices! In practice there is

wrapped around a

To

raise a load

by

mm the handle has to be moved once around, through a

distance equal to the circumference of the circular path of radius 16 cm. This
distance is 100 cm (since the circumference is 2nr = 2 x 3.14 x 16 cm =

show that the 100-cm work-input distance

500 times greater than the work-output distance of 2 mm. If the jack were
100% efficient, then the input force would be multiplied by 500 times. The theoretical mechanical advantage of the jack is 500.
100 cm). A simple calculation will

is

The auto jack

an inclined planewrapped around a cylinder.
Every time the handle is
turned one revolution, the
Fig. 8- 1 5
is

like

load

is

raised a distance of

one pitch.

114

Energy

Questions

A child on a sled (total weight 500 N) is pulled up a 10-m

slope that elevates her a vertical distance of
m.
1

a.

What

is

the theoretical mechanical advantage of the

slope?
slope is without friction, and she is pulled up the
slope at constant speed, what will be the tension in the

b. If the

rope?
c.

Considering the practical case where friction is present,

suppose the tension in the rope were in fact 100 N. Then
what would be the actual mechanical advantage of the
slope? The efficiency?

a great deal of friction in this type of jack, so the efficiency

may

be around 20%. Thus the jack actually multiplies force by about

100 times. The actual mechanical advantage therefore approximates an impressive 100. Imagine the value of one of these devices during the days when the great pyramids were being built!
An automobile engine is a machine that transforms chemical
energy stored in fuel into mechanical energy. The molecules of
the petroleum fuel break up when the fuel burns. Burning is a
chemical reaction in which atoms combine with the oxygen in
the air. Carbon atoms from the petroleum combine with oxygen
atoms to form carbon dioxide, and energy is released.
The converted energy is used to run the engine. It would be
nice if all this energy were converted to mechanical energy, but
a 100% efficient machine is not possible. Some of the energy goes
into heat. Even the best designed engines are unlikely to be more
than 35% efficient. Some of the energy converted to heat goes
into the cooling system and is wasted through the radiator to
the air. Some goes out the exhaust, and nearly half is wasted in
the friction of the moving engine parts. In addition to these inefficiencies, the fuel does not even burn completely
a certain
amount goes unused. We can look at inefficiency in this way: in

a.

b.

Answers
(output
The ideal or theoretical mechanical advantage is (input distance)
distance) = (10 m)/(l m) = 10.
50 N. With no friction, ideal mechanical advantage and actual mechanical advantage would be the same
10. So the input force, the rope tension, will be
10 the output force, her 500-N weight.
The actual mechanical advantage would be (output force) * (input force) =
(500 N)/(100 N) = 5. The efficiency would then be 0.5 or 50%, since (actual
mechanical advantage)
(theoretical mechanical advantage) = 5/10 = 0.5.
The value for efficiency is also obtained from the ratio (useful work output) *
(useful work input).
-s-

c.

-=-

8.9

115

Energy for Life

FUEL ENER6Y
100 %

IN =

COOLING WATER LOSSES

35 % _/

-^

EN6INE OUTPUT* EXHAUST HEAT

30 7.
35 %
/

Only 30% of the energy produced by burning gasoline

automobile becomes useful mechanical energy.

Fig. 8-16
efficient

any transformation there

is

a dilution of the

in a typically

amount

of useful

becomes heat. Energy is not desimply degraded. Heat is the graveyard of useful

energy. Useful energy ultimately

stroyed
energy.

8.9

it is

Energy for

Life

Every living cell in every organism is a living machine. Like any

machine, it needs an energy supply. Most living organisms on
this planet feed on various hydrocarbon compounds that release
energy when they react with oxygen. Like the petroleum fuel
previously discussed, there is more energy stored in the molecules in the fuel state than in the reaction products after combustion. The energy difference is what sustains life.
The principle of combustion in the digestion of food in the
body and the combustion of fossil fuels in mechanical engines is
the same. The main difference is the rate at which the reactions
take place. In digestion the rate is much slower, and energy is
released as it is required. Like the burning of fossil fuels, the reaction is self-sustaining once started. Carbon combines with oxygen to form carbon dioxide.
The reverse process is more difficult. Only green plants and
certain one-celled organisms can make carbon dioxide combine
with water to produce hydrocarbon compounds such as sugar.
This process is photosynthesis and requires an energy input,
which normally comes from sunlight. Sugar is the simplest

food; all others

carbohydrates, proteins, and fats are also
synthesized compounds of carbon, hydrogen, and oxygen. How
fortunate we are that green plants are able to use the energy of
sunlight to make food that gives us and all other food-eating organisms our energy. Because of this there is life.

Energy

116

Chapter Review

Concept Summary

Review Questions

When

muxes an object in the diwork done equals the

product ot the force and the distance through
which the object is moved.
Power is the rate at which work is done.
a constant force

rection ol the force, the

A force

sets

an object

motion.

in

When

the

multiplied by the time of its application, we call the quantity impulse, which
force

is

changes the momentum of that object. What

do we call the quantity force x distance, and
what quantity does this change? (8.1)

The energy of an object enables it to do work.

Mechanical energy is due to the position of
something (potential energy) or the movement ot something (kinetic energy).

is required to lift a barbell. How many

times as much work is required to lift the
barbell three times as high? (8.1)

Work

Which requires more work lifting a 10-kg

sack a vertical distance ot 2 m, or lifting a
5-kg sack a vertical distance of 4 m? (8.1)

According to the law of conservation ol energy,

energy cannot be created nor destroyed.
Energy can be translormed trom one form
to another.
4.

A machine

is

a device tor multiplying forces or

How many joules of work are done on

when

ject

a force of 10

pushes

it

an oba dis-

tance of 10 m?(8.1)

changing the direction of forces.

Levers,
pulleys, and inclined planes are
simple machines.
Normally, the useful work output of a machine is less than the total work input.

How much power

is

required to do 100

J of

work on an object in a time of 0.5 s? Howmuch power is required if the same amount
of work is done in
s? (8.2)
1

What
a.

Important Terms
efficiency (8.8)

energy (8.3)
fulcrum (8.7)

is

mechanical energy?

(8.3)

you do 100 J of work to elevate a

bucket of water, what is its gravitational
potential energy relative to its starting
If

position?
b.

What would

it

be

if it

were raised twice

as high? (8.4)

joule (8.1)
kinetic energ\ (8.5)

Jaw

of conservation ot

energy

(8.6)

8\

machine

(8.2)

pulley (8.7)

watt

work

(8.2)

(8.1)

is raised above the ground so

potential energy with respect to the

a boulder
its

ground

(8.7)

mechanical advantage (8.7)

mechanical energ\ iS.3)
potential energy (8.4)

power

If

that

"lever (8.7)

is

200

J,

and then

it is

dropped, what

will its kinetic energy be just before

it

hits

the ground? (8.5)

9\ Suppose an automobile has a kinetic energy

of 2000 J. If it moves with twice the speed,
what will be its kinetic energy? Three times
the speed? (8.5)

117

Chapter Review

What

will be the kinetic energy of

shot from a

bow having

an arrow

2.

of 50 J? (8.6)

What does

it

mean

to say that in

the total energy score stays the

any system
same? (8.6)
3.

In

what sense

is

capable of bringing a car

in 10 s. All other things
being equal, if the engine has twice the
power output, how many seconds would be
required to accelerate it to this speed?
certain engine

from

a potential energy

energy from coal actually

If

20

solar energy? (8.6)

to 100

a car that travels at 60

will

when
it

when

Why

there an upper limit on

is

how

far

automobile can be driven on a tank of gaso-

In

what two ways can a machine

put force?

alter

an

4.

in-

what way

a machine subject to the law

of energy conservation? Is it possible for a
machine to multiply energv or work input?
In

it mean to say that a certain machine has a certain mechanical advantage?

6.

is

how

traveling at 120

far

km/h

A hammer

falls off a rooftop and strikes the

ground with a certain KE. If it fell from a
roof twice as tall, how would its KE of impact compare?

Most earth

Why

satellites follow an oval shaped

path rather than a circular path

will a lighter car generally

ter fuel

What

it

around the earth. The PE increases when

the satellite moves farther from the earth.
According to energy conservation, does a
satellite have its greatest speed when it is
closest or farthest from the earth?

What does
(8.7)

17.

will skid

brakes are locked?

(elliptical)

is

(8.7)

16.

its

if

km/h

brakes are locked,

(8.7)
5.

15.

skid

its

an

line? (8.6)

14.

is

km/h

economy than a

have bet-

larger car? Will a

streamlined design improve fuel economy?

are the three basic tvpes of levers?

(8.7)
7.

18.

What

is

quires 100 J
useful

19.

(8.8)

Distinguish between theoretical mechanical

advantage and actual mechanical advantage.
is

20.

work?

How

100%

will these

compare

if

efficient? (8.8)

is

How many kilometers per liter will a car

obtain if its engine is 25% efficient and it
encounters an average retarding force of
1000 N at highway speed? Assume the energy content of gasoline is 40 MJ per liter.

cyclist

of 100

8.

machine

the efficiency of the body when a

of power to deliver
expends 1000
mechanical energy to her bicycle at the rate

What

Does an automobile consume more fuel

its air conditioner is turned on? When
its lights are on? When its radio is on while
it is sitting at rest in the parking lot? Explain in terms of the conservation of energy.

when

machine that reof input energy to do 35 J of

the efficiency of a

W?

(8.8)

9.

The pulley shown on the left in Figure A has

a mechanical advantage of 1. What is the
mechanical advantage when it is used as
shown on the right? Defend your answer.

Think and Explain

1.

When

a rock

there are
faster

tance.

if

is

projected with a slingshot,

two reasons why the rock

the rubber

What

is

will

go

stretched an extra dis-

are the two reasons? Defend

your answer.
Fig.

118

Energy

more energy than

What do you saj ?

reactor puts out

into

J3=
Fig.

the theoretical mechanical advan10. What

tage ol the three lexer systems shown in Figis

ure B?
11.

You

your friend that no machine can

more energy than is put
and your friend states that a nuclear

tell

possibly put out

into

it.

\\{\A r

(K

+^

(Wis

/)

cU.1

12.

it.

The energy we require

tor existence

is

put

comes

from the chemically stored potential energy

in food, which is transformed into other
forms by the process ol digestion. What happens to a person whose work output is less
than the energy consumed? When the person's work output is greater than the energy
consumed? Can an undernourished person
perform extra work without extra food? Defend vour answers.

Center of Gravity

Why doesn't the famous Leaning Tower of Pisa topple over? How
far can it lean before it does topple over? Why is it impossible to
stand with your back and your heels against a wall and bend
over and touch your toes without toppling forward? To answer
these questions, you first need to know about center of gravity.
Then you need to know how this concept can be applied to balancing and stability. Let's start with center of gravity.
f

[ffl

Heykmh^M kxS

^V^c/-f cVl<j7W^nf ^Wc

Center of Gravity
Throw a baseball into the air, and it follows a smooth parabolic
path. Throw a baseball bat into the air with a spin, and its path
is not smooth. The bat seems to wobble all over the place. But it
wobbles about a special point. This point follows a parabolic
path, even though the rest of the bat does not (Figure 9-1). The
motion of the bat is the sum of two motions: (1) a spin with this
point at the center and (2) a movement through the air as if all
the weight were concentrated at this point. This point is the cen-

ter of gravity of the bat.

13 CjU/vCxyi

bAdFig. 9-1
ity of

The centers of gravand of the

the baseball

spinning baseball bat each

follow parabolic paths.

120

)(

'4

Lif

i/0/A -\/^j>

|/(/V^

ffiCfa
v^

The center ut gravity of an object is the point located at the

ier ^ an OD J ect s we 'ght distribution. For a symmetrical ob-

i/l^P

such as the baseball, this point is at the geometrical center.

rre g u ar Lv shaped object, such as the baseball bat, has
more weight at one end, so the center of gravity is toward the

-Ject,

^ ut an

OfrnlAAf^:

tf*&>
*p

'

heavier end. The center of gravity of a piece of

1

JIa&\\

~r\(n
/laA ( L/l
_3CjJA^
/

Center of Gravity

tile

cut into the

shape of a triangle is one third of the way up from its base. A

CV solid cone has its center of gravity one fourth of the way up from
its

base.

ws. i?^ <^y

Fig. 9-2

The center

Objects not

of gravity for each object

made

of the

objects of varying densities)

Fig. 9-3
ity of

The center of gravis below its geo-

the toy

is

shown by

the colored dot.

same material throughout

may have

(that

far from the geometrical center. Consider a hollow ball half filled
with lead. The center of gravity would be not at the geometrical
center but within the lead. The ball would always roll to a stop
in the same position. Make it the body of a lightweight toy clown,
and whenever it is pushed over, it will come back right-side-up

(Figure 9-3).

metric center.

The multiple-flash photograph in Figure 9-4 shows a top view

wrench sliding across a smooth horizontal surface. Notice
that its center of gravity, which is marked by the dark X, follows
a straight-line path. Other parts of the wrench rotate about this
point as the wrench moves across the surface. Notice also that
the center ot
tne
of gravity moves equal distances in tne
the equal time inmf
tne
flashes
(because
no
net
force
acts
in
the
direction
of
/T^jjCCtj&f^^
*!
motion). The motion of the wrench is a combination of straightline motion of its center of gravity and rotational motion about
of a

fl

/i

\/^

c
f

f\

C_^

"

is,

the center of gravity quite

+ C.

tev\fce<vjfl}

its

center of gravity.

& * *
Fig. 9-4

path.

nv*

The center of gravity of the rotating wrench follows a

straight-line

Center of Mass

9.2

121

wrench were instead tossed into the air, no matter how

its center of gravity would follow a smooth parabola.
The same is true even for an exploding projectile, such as a fireworks rocket (Figure 9-5). The internal forces that take place in
the explosion do not change the center of gravity of the projecIf

it

the

rotates

tile. Interestingly enough, if air resistance can be neglected, the

center of gravity of the dispersed fragments upon reaching the
ground will be where the unexploded rocket would have hit.

The center of gravity of the fireworks rocket and

along the same path before and after the explosion.
Fig. 9-5

9.2

its

fragments move

Center of Mass

Center of gravity is also called center of mass, which is the point

of average mass distribution for an object. For almost all objects
on and near the earth, the terms are interchangeable. An object
in outer space, where gravity forces are practically zero, has a
center of mass but no center of gravity. There is also a small difference between center of gravity and center of mass when an object
is large enough for gravity to vary from one part to another. The
center of gravity of the moon, for example, is a bit closer to the
earth than the center of mass is. This is because the nearer parts
of the moon are pulled by earth gravity more than the far parts.
For everyday objects we can use the terms center of gravity and
center of mass interchangeably.
If you threw a wrench so that it rotated as it moved through
the air, you would see it wobble about its center of gravity. The
center of gravity itself would follow a parabolic path. Now suppose you threw a lopsided ball one with its center of gravity

off-center.

wobbles

wobble as it rotated. The sun itseli

a similar reason. The center of mass of the solar sys-

You would

for

see

it

122

tem can

Center of Gravity

outside the massive sun, not at its geometrical cenbecause the masses of the planets contribute to the
overall mass of the solar system. As they orbit their respective
distances, the sun actually wobbles off center. Astronomers are
seeking similar wobbles in nearby stars, which may indicate
that our sun is not the only star with a planetary system.
ter.

This

lie
is

CG OF SOLAR SYSTEM

<&

Fig. 9-6

The center

of the sun.

If all

of

2 solar radii

from the sun's center.

Locating the Center of Gravity

9.3

The center

The weight

of the

entire stick behaves as

were concentrated

if it

at its

Fine

the

CG

for

iped object
b

at the

pendulum, for example) at a single

hang directly below (or at) the
point of suspension. To locate the CG, construct a vertical line
beneath the point of suspension. The CG lies somewhere along
that line. Figure 9-8 shows how a plumb line and bob can be used
to construct a line that is exactly vertical. You can locate the CG
by suspending the object from some other point and construct a
second vertical line. The CG is where the two lines intersect.
The CG of an object may be where no actual material exists.
The CG of a ring lies at the geometrical center where no matter
exists. The same holds true for a hollow sphere such as a basketball. The CG of even half a ring or half a hollow ball is still outside the physical structure. There is no material at the CG of an
empty cup, bowl, or boomerang.
you suspend any object

point, the

an irregularly
with a plumb

is

force directed at this point.

If

center.

CG from

here on) of a uniform

midpoint, its geometrical
center. The CG is the balance point. Supporting that single point
supports the whole object. In Figure 9-7 the many small vectors
represent the force of gravity all along the meter stick. All of these
can be combined into a resultant force that acts at the CG. The
effect is as if the weight of the stick were concentrated at this
point. That's why you can balance the stick by a single upward
of gravity (called the

object (such as a meter stick)

mHM

Fig. 9-8

mass of the solar system is not at the geometrical center

were lined up on one side of the sun, the center of

the planets

mass would be about

Fig. 9-7

CG

(a

of the object will

123

Toppling

9.4

Fig. 9-9

There

is

no material

at the

CG

of these objects.

Questions

CG

1.

Where

2.

Can an object have more than one CG?

is

the

of a donut?

Toppling
Pin a plumb line to the center of a heavy wooden block and tilt
the block until it topples over (Figure 9-10). You can see that the
block will begin to topple when the plumb line extends beyond
the supporting base of the block.

Fig. 9-10

The block topples when the CG extends beyond

its

support base.

Answers
1

2.

In the center of the hole!

A
is

rigid object has

one CG.

If it is

nonrigid, such as a piece of clay or putty, and

its CG may change as
any given shape.

distorted into different shapes, then

changed. Even then,

it

has one

CG

for

its

shape

is

124

Fig. 9-11

Center of Gravity

A "Londoner"

double-decker bus undergoing a tilt test. The bus must

not topple

when

the chassis

is

tilted 28

with the top deck

lully loaded and only the
driver and conductor on the
lower deck. Because so much
weight ol the vehicle is
lower part, the load of
the passengers on the upper
deck raises the CG onl\ a
little, so the bus can be tilted

of the

in the

well

bevond

this 28 limit

uithout toppling.

Fig. 9-12

The Leaning Tower

of Pisa does not topple over

because
base.

its C

lies

above

its

The rule for toppling is: If the CG of an object is above the

area of support, the object will remain upright. If the CG extends
outside the area of support, the object will topple. This principle
is dramatically employed in Figure 9-11.
This is why the Leaning Tower of Pisa does not topple. Its CG
does not extend beyond its base. As shown in Figure 9-12, a vertical line below the CG falls inside the base, and so the Leaning
Tower has stood for centuries. If the tower leaned far enough so
that the CG extended beyond the base, the tower would topple.
A support base of an object does not have to be solid. The four
legs of a chair bound a rectangular area that is the support base
for the chair (Figure 9-13). Practically speaking, supporting
props could be erected to hold the Leaning Tower up if it leaned
too far. Such props would create a new support base. An object
will remain upright if the CG is above its base of support.
Try balancing the pole end of a broom or mop on the palm of
your hand. The support base is very small and relatively far

125

Stability

9.5

Questions
1

When you carry a heavy load such as a pail of water

with one arm, why do you tend to hold your free arm out
horizontally?

2.

To

being toppled, why does a wrestler stand with

wide apart and (b) his knees bent?

resist

(a) his feet

3.

How

will the support base of the chair in Figure 9-13

change

if

one of the front legs

is

removed? Will the chair

The shaded area

bounded by the bottom of the
Fig. 9-13

chair legs defines the support

topple?

base of the chair.

beneath the CG, so it is difficult. But after some practice you can
do it if you learn to make slight movements of your hand to exactly respond to variations in balance. You learn to avoid underresponding or over-responding to the slightest variations in
balance. Similarly, high-speed computers help massive rockets
remain upright when they are launched. Variations in balance
are quickly sensed. The computers regulate the firings at multiple nozzles to make corrective adjustments, quite similar to
the way your brain coordinates your adjustive action when balancing a long pole on the palm of your hand. Both feats are truly
amazing.

[Ei

Stability

It is nearly impossible to balance a pen upright on its point but

easy to stand it upright on its flat end. This is because the base of
support is inadequate for the point and adequate for the flat end.
But there is a second reason. Consider a solid wooden cone on a
level table. You cannot stand it on its tip (Figure 9-14). Even if
you position it so that its center of gravity is exactly above its tip,
the slightest vibration or air current will cause the cone to topple.

Answers
You tend

to hold your free arm outstretched to shift the CG of your body away
from the load so your combined CG will more easily be above the base of support. To really help matters, divide the load in two if possible, and carry half
in each hand. Or, carry the load on your head!

2. (a)

3.

Wide-apart

feet increase the

support base,

(b)

Bent knees lower the CG.

The support base for four legs is a rectangle. With three legs it's a triangle
The CG will be toward the rear of the chair because of the
weight of the back and loss of weight of the front leg, and be within and
above the triangular support base. So it will not topple until somebody sits

of half the area.

on

it!

126

When

it

Center of Gravity

docs, will the center of gravity be raised, be lowered, or

The answer to this question provides the second reason for stability. A little thought will show that the CG
will always be lowered by the movement. We say that an object
balanced so that any displacement lowers its center of gravity is
not change at all?

in

unstable equilibrium.

Equilibrium is (a) unstable when the CG is lowered with displacement, (b) stable when work must be done to raise the CG, and (c) neutral when
displacement neither raises nor lowers the CG.
Fig. 9-14

A cone balances easily on its base. To make it topple, its CG

must be raised. This means its potential energy must be increased, which requires work. We say an object that is balanced
so that any displacement raises its center of gravity is in stable

Fig. 9-15

For the pen to

when

it is on its flat
must rotate over one
edge. During the rotation, the
CG rises sliahtlv and then

topple

end,

falls.

it

equilibrium.
Place the cone on its side and its CG is neither raised nor lowered with displacement. An object in this configuration is in neutral equilibrium.
Like the cone, the pen is in unstable equilibrium when it is on
its point. When it is on its flat end, it is in stable equilibrium because the CG must be raised slightlv to topple it over (Figure
9-15).

Consider the upright book and the book lying flat in Figure
Both are in stable equilibrium. But you know the flat book
is more stable. Why? Because it would take considerably more
work to raise its CG to the point of toppling than to do the same
for the upright book. An object with a low CG is usually more
stable than an object with a relatively high CG.
9-16.

'-\

Fig. 9-16

Toppling the upright book requires only a slight raising of

book requires a relatively large raising of

quires more work?
toppling the

flat

its

its

CG. Which

CG;
re-

127

Stability

9.5

The pencil shown balancing horizontally in Figure 9-17 left is

barely in stable equilibrium. Its CG must be raised only a slight
distance to topple it. But suspend a couple of potatoes from its
ends and it is much more stable (Figure 9-17 right). Why? Because the

CG

raises the

CG, even

is

now below
if

the point of support. Displacement

the pencil

is

tipped almost vertical.

POTATOES
Fig. 9-17

When

its

because

(Left)

The balancing pencil

is

barely in stable equilibrium. (Right)

ends are stuck into long potatoes that hang below, it is very stable
new CG remains below the point of support even when it is tipped.

its

Question
Explain

remain on

why
its

the toy clown

Some well-known balancing

Their secret

is

shown

in Figure 9-3

cannot

side.

toys

depend on

this principle.

that they have been weighted so that the

CG

lies

underneath the point of support while most of the remainder of the toy is above it (Figure 9-18). Any object that hangs
with its CG below its point of support is in stable equilibrium.
vertically

Fig. 9-18

The

CG

of the toy

is

below the point of support, so the toy

is in

stable equilibrium.

Answer
The

CG

is

lowest

when

the toy

is

upright.

back with the aid of gravity until the

CG

is

If it is

pushed over,

the lowest

it

can be.

it

will easily rol

128

Center of Gravity

The CG of a building is lowered if much of the structure is

below ground level. This is important for tall, narrow structures.
An extreme example is the state of Washington's tallest freestanding structure, the Space Needle in Seattle. This structure is so
'deeply rooted" that its center of gravity is actually below ground
level. It cannot fall over intact. Why? Because falling would not
lower its CG at all. If the structure were to tilt intact onto the
ground, its CG would be raised]
The tendency for the CG to take the lowest position available
can be seen by placing a very light object, such as a table-tennis
ball, at the bottom of a box of dried beans or small stones. Shake
the box, and the beans or stones tend to go to the bottom and
force the ball to the top.

tem takes a lower

By

this process the

CG

of the

whole

sys-

position.

Fig. 9-19

The Space Needle

can no more fall
over than can a floating iceberg. In both, the CG is below

Fig. 9-20

in Seattle

dried beans. (Right)

the surface.

The same thing happens in water when a light object rises

and floats. If the object weighs less than an equal
volume of water, the CG of the whole system will be lowered

nudged

(Left)

to the top.

table-tennis ball

When

The

is

placed at the bottom of a container of

the container

result

is

is

shaken from side to

side, the ball

is

a lower center of gravity.

to the surface

when

the light object

is

forced to the surface. This

is

because

the heavier (more dense) water can then occupy the available

lower space.
water,

The

Fig. 9-21

water

is

CG

higher

table-tennis ball
to

of the glass

when

the

anchored
the bottom (left) and lower

when

is

the ball floats (right).

If

will be

the object

is

heavier than an equal volume of

more dense than water and

sink. In either case,

whole system is lowered. In the case where the object weighs the same as an equal volume of water (same density), the CG of the system is unchanged whether the object rises
or sinks. The object can be at any level beneath the surface without affecting the CG. You can see that a fish must weigh the same
as an equal volume of water (have the same density); otherwise
it would be unable to remain at different levels in the water. We
will return to these ideas in Chapter 19, where liquids are treated
the

of

CG

it

in

of the

more

detail.

Shake a box of stones of

the small stones to slip

different sizes.

down

The shaking enables

between the larger

into the spaces

stones and in effect lower the CG. The larger stones therefore tend
to rise to the top. The same thing happens when a tray of berries
is gently shaken
the larger berries tend to come to the top.

9.6

129

Center of Gravity of People

9.6

Center of Gravity of People

When you stand erect with your arms hanging at your sides, your
CG is within your body. It is typically 2 to 3 cm below your navel,
and midway between your front and back. The CG is slightly
lower in women than in men because women tend to be proportionally larger in the pelvis and smaller in the shoulders. In children, the CG is approximately 5% higher because of their proportionally larger heads and shorter legs.

Raise your arms vertically overhead. Your CG rises 5 to 8 cm.

Bend your body into a U or C shape and your CG may be located
outside your body altogether. This fact is neatly employed by a
high jumper who clears the bar while his CG passes beneath the

bar (Figure 9-22).

Fig. 9-22

CG

Dwight Stones executes a "Fosbury

flop" to clear the bar while his

passes beneath the bar.

CG is somewhere above your support

bounded by your feet (Figure 9-23). In unstable
as in standing in the aisle of a bumpy-riding bus, you

When you

stand, your

base, the area

situations,

place your feet farther apart to increase this area. Standing on

one foot greatly decreases this area. In learning to walk, a baby
must learn to coordinate and position the CG above a supporting foot. A pigeon does this by jerking its head back and forth
with each step.

When you

Fig. 9-23

the

CG

stand,

somewhere above
area bounded by your

your
Beet.

is

130

Center of Gravity

If you are in reasonable shape, you can probably bend over

and touch your toes without bending your knees. In doing so,
you unconsciously extend the lower part of your body, as shown
in Figure 9-24 left. In this way your CG, which is now outside
your body, is nevertheless above your supporting feet. If you try
it while standing with your back and heels to a wall, you may be
in for a surprise. You cannot do it! This is because you are unable
to adjust your body, and your CG protrudes beyond your feet.
You are off balance and you topple over.

You can lean over and touch your toes without toppling only
above the area bounded bv vour feet.

Fig. 9-24

CG

is

if

your

You don't need to take a course in physics to know where to

balance a baseball bat, or how to stand a pencil upright on its
flat end. With or without physics, everybody knows that it is
easier to hang by your hands below a supporting rope than it is
to stand on your hands above a supporting floor. And you don't
need a formal study of physics to balance like a gymnast. But
maybe it's nice to know that physics is at the root of many things
you already know about.
Knowing about things is not always the same as understanding things. Understanding begins with knowledge. So we begin
by knowing about things, and then progress deeper to an understanding of things. That's where a knowledge of physics is very
helpful.

Chapter Review

131

Chapter Review

Concept Summary
The center of gravity (CG) of an object

is

Cite an

8.

Why does

the point

weight distribution.
an object is thrown through the

at the center of its

An

When
its CG

follows a

air,

CG

is

above

in stable equilibrium

when

object will remain upright

if its

9.

An

object

is

any displacement raises

its

10.

1 1

12.

CG

Distinguish between unstable, stable, and

Is

of a baseball bat not at

its

14.

What

What part of an object follows a smooth path

when the object is made to spin through the
smooth surface?

15.

Describe the motion of the CG of a projectile,

it explodes in mid-air. (9.1)

is

When

5.

What

the

is

CG and center of mass of an obWhen are they different? (9.2)

same.

is

CG

of an object

the "secret" of balancing toys that

be unstable?

What accounts

16.

If

(9.5)

for the stability of the

17.

Space

(9.5)

a container of dried beans with a table

tennis ball at the bottom

happens
ject the

the

is it easier to hang by your arms below

a supporting cable than to do handstands on
a supporting floor? (9.5)

Needle in Seattle?

(9.1)

before and after

4.

when

exhibit stable equilibrium while appearing

(9.1)

air or across a flat

3.

or unchanged

raised? (9.5)

Why

to
2.

the gravitational potential energy more,

13.

Review Questions
the

is balancing the pole end of a broom in

an upright position on the palm of your hand
similar to the launching of a space rocket?

is

unstable equilibrium (9.5)

is

How

less,

neutral equilibrium (9.5)

stable equilibrium (9.5)

midpoint?

How far can an object be tipped before it top-

neutral equilibrium. (9.5)

center of gravity (9.1)

center of mass (9.2)

Why

top-

(9.4)

CG.

Important Terms

1.

Tower of Pisa not

ples over? (9.4)

the area of support.

the Leaning

ple? (9.4)

smooth parabolic path,

even if the object spins or wobbles.

For everyday objects, the center of gravity
is the same as the center of mass.

example of an object that has a CG

where no physical material exists. (9.3)

7.

to the

CG

is

shaken, what

of the container? (9.5)

What happens to the CG of a glass of water

when a table-tennis ball is poked beneath
the surface? (9.5)

suggested bv a star that wobbles?

(9.2)
18.
6.

How can
ject

CG

an irregularly shaped obbe determined? (9.3)

the

of

Why do some high-jumpers arch

into a U-shape

bar?

(9.6)

when

their bodies
passing over the high

132

19.

do vou spread your Feet farther apart

in a bumpy-riding bus? (9.6)

Whv

Think and Explain

when standing

20.

Center of Gravity

not successfully bend over and

touch your toes when you stand with your
back and heels against a wall? (9.6)

Why can you

To balance automobile wheels, particularly

when tires have worn unevenly, lead weights
are fastened to their edges.

CG
2.

Where should

the

of the balanced wheel be located?

Why

does a washing machine vibrate vioif the clothes are not evenly dis-

lently

tributed in the tub?

Activities
1

wire that
is bent as shown in Figure A. Why does the
belt balance as it dues.

Suspend a

belt

from a piece of

3.

stiff

Why do we speak of the center of mass of the

sun, rather than the center of gravity of the

sun?
4.

Which

glass in Figure

is

unstable and will

topple?

Fig.

2.

Hang

hammer on

a loose ruler as

Figure B. Then explain

why

it

shown

doesn't

Fig.

in

fall.

5.

RULER

TABLE TOP-

act in Figure E is in stable equilibrium? In unstable equilibrium?

Which balancing

Nearly at neutral equilibrium?

STRIN
Fig.E
Fig.

3.

Rest a meter stick on two fingers as shown

in Figure C. Slowly bring your fingers together. At what part of the stick do your fingers meet? Can you explain why this always

happens, no matter where you start your

fingers?

can the three bricks in Figure F be

stacked so that the top brick has maximum
horizontal overhang above the bottom
brick? For example, stacking them as the
dotted lines suggest would be unstable and
the bricks would topple. (Hint: Start with

How

the top brick

and think your way down. At

every interface the CG of the bricks above

must not extend beyond the end of the supporting brick.)

Fig.C

Fig.

^
Chapter Review

7.

Why

8.

In terms of CG,

don't

tall

133

floating icebergs simply tip

why is work required

to

Give two reasons why females can generally

do this and males cannot.

push

a table-tennis ball beneath the surface of a

glass of water?

9.

Why

does a pregnant

late stages of

large

woman

pregnancy or a

paunch tend

to lean

during the
with a

man

backward when

walking?
10.

Try the following with a group of males and

females. Stand exactly two footlengths
away from a wall. Bend over with a straight
back and let your head lean against the wall,

shown in Figure G. Then lift a chair that

placed beneath you while your head is
still leaning against the wall. With the chair
in lifted position, attempt to straighten up.
as
is

FOOTLEN6THS

<M

Universal Gravitation

Things such as leaves, rain, and satellites fall because of gravity.

Gravity is what holds tea in a cup and what makes bubbles rise.
It made the earth round, and it builds up the pressures which
kindle every star that shines. These are things that gravity does.
But what is gravity? We do not know what gravity is at least
not in the sense that we know what sound, heat, and light are.
We have names for things we understand, and we also have
names for things we do not understand. Gravity is the name we
give to the force of attraction between objects, even though we
do not thoroughly understand it. Nevertheless, we understand
how gravity affects things such as projectiles, satellites, and the
solar system of planets. We also understand that gravity extends
throughout the universe; it accounts for such things as the shapes
of galaxies. This chapter covers the basic behavior of gravity. In
the next chapter we shall investigate more of its consequences.

.1

The

Falling Apple
The idea that gravity extends throughout the universe is credited to Isaac Newton. According to popular legend, the idea occurred to Newton while he was sitting underneath an apple tree
on his mother's farm pondering the forces of nature. Newton
understood the concept of inertia developed earlier by Galileo;
he knew that without an outside force, moving objects continue
to move at constant speed in a straight line. He knew that if an
object undergoes a change in speed or direction, then a force is

responsible.

134

10.2

The

Falling

Moon

135

A falling apple triggered what was to become one of the most

far-reaching generalizations of the human mind. Newton saw the
apple

fall,

or

maybe even

this is not clear.

felt it fall on his head

Looking up through the apple

the story about

tree

branches

to-

i>

ward the origin of the apple, did Newton notice the moon? Newton had been giving a lot of thought to the fact that the moon
does not follow a straight line path, but instead circles about the
earth. Now, circular motion is accelerated motion, which requires a force. But what was this force? Newton had the insight
to see that the force that pulls between the earth and moon is the
same force that pulls between apples and everything else in our
universe. This force is the force of gravity.

Fig. 10-1

not

fall, it

moon

did

Because of

attraction to the earth,

falls

Falling

the

straight-line path.
its

The

If

would follow the

it

along a curved path.

Moon

Newton developed

this idea further. He compared the falling

apple with the falling moon. Does the moon fall? Yes, it does.
Newton realized that if the moon did not fall, it would move off
in a straight line and leave the earth. His idea was that the moon
must be falling around the earth. Thus the moon falls in the
sense that it falls beneath the straight line it would follow if no
force acted on it. He hypothesized that the moon was simply a
projectile circling the earth under the attraction of gravity.
This idea is illustrated in an original drawing by Newton,
shown in Figure 10-2. He compared motion of the moon with a
cannonball fired from the top of a high mountain. He imagined
that the mountaintop was above the earth's atmosphere, so that
air resistance would not impede the motion of the cannonball. If
a cannonball were fired with a small horizontal speed, it would
follow a parabolic path and soon hit the earth below. If it were
fired faster, its path would be less curved and it would hit the
earth farther away. If the cannonball were fired fast enough,
Newton reasoned, the parabolic path would become a circle and
the cannonball would circle indefinitely. It would be in orbit.
Both the orbiting cannonball and the moon have a component of velocity parallel to the earth's surface. This sideways
velocity, or tangential velocity, is sufficient to insure motion
around the earth rather than into it. If there is no resistance to
reduce its speed, the moon "falls" around and around the earth

Fig. 10-2

The original drawNewton showing

ing by Isaac

indefinitely.

how

Newton's idea seemed correct. But for the idea to advance

from hypothesis to the status of a scientific theory, it would have
to be tested. Newton's test was to see if the moon's "fall" beneath

enough would fall around the

earth and become an earth

its

otherwise straight-line path was

in correct

proportion to the

a projectile fired fast

satellite. In the

moon

falls

and

an earth

is

same way,

the

around the earth

satellite

136

Universal Gravitation

10

r9CT

TAN6ENTIAL

)f

VELOCITY

PULL OF 6RAVITY
Fig. 10-3

Tangential velocity is the "sideways" velocity. That

ponent of velocity that is perpendicular to the pull of gravity.
fall

of

an apple or any object

that the
earth.

at the earth's surface.

mass of the moon should not

mass has no

effect

is, it is

how

com-

He reasoned
falls, just

as

freely-falling objects

on

affect

on the acceleration of

the

it

How far the moon falls, and how far an apple at the earth's

falls, should relate only to their respective distances

from the earth's center. If the distance of fall for the moon and

surface

the apple are in correct proportion, then the hypothesis that

earth gravity reaches to the moon must be taken seriously.

DISTANCE
IN

MOON

TRAVELS

ONE SECOND

MOON

HERE

WITHOUT 6RAVITY
IT SHOULD BE HERE
IN

or l/(60) 2 So in one second

the moon should fall l/(60) of 4.9 m, which is 1.4 millimeters."
Using geometry, Newton calculated how far the circle of the
moon's orbit lies below the straight-line distance the moon other-

of gravity should be diluted

-^

of

ONE SECOND
WITH 6RAVITY

The moon was already known to be 60 times farther from the

center of the earth than an apple at the earth's surface. The apple will fall nearly 5
or more accuin its first second of fall
rately, 4.9 m. Newton reasoned that gravitational attraction to
the earth must be "diluted" by distance. Does this mean the
force of earth gravity would reduce to ^ at the moon's distance?
No, it is much less than this. As we shall soon see, the influence

IT

SHOULD FALL TO
HERE IN ONE SEC

r^h

Fig. 10-4

If

the force that

pulls apples off trees also

pulls the

moon

circle of the

into orbit, the

moon's orbit

mm

should fall 1.4

below a
point along the straight line

where the moon would otherwise be one second latei

Fig. 10-5

An apple falls 4.9 m during

Newton asked how far

earth's surface.
it

its first

the

were 60 times farther from the center

units)

.4

second of

of the earth.

mm.

Or working backwards, (0.0014 m) x

(60)

4.9

fall

when

it

is

near the

same time if
His answer was (in today's

moon would

m.

fall in

the

The

10.3

137

Falling Earth

wise would travel in one second (Figure 10-4). The distance should
have been 1.4 mm. But he was disappointed to end up with a
large discrepancy. Recognizing that a hypothesis, however elegant, is not valid if it cannot be backed up with tests, he placed
his papers in a drawer, where they remained for nearly 20 years.
During this period he laid the foundation and developed the
field of geometrical optics for which he first became famous.
It turns out that Newton used an incorrect figure in his calculations. When he finally returned to the moon problem at the
prodding of his astronomer friend Edmund Halley (of Halley
Comet fame) and used a corrected figure, he obtained excellent
agreement. Only then did he publish what is one of the greatest
achievements of the human mind the law of universal gravita-

tion."

Newton generalized

his

moon

finding to all objects,

and

stated that all objects in the universe attract each other.

Newton's theory of gravitation confirmed the Copernican theory

of the solar system.

was now
same way

It

clear that the earth

that the moon orbits the earth.

sun in the
The mighty sun and planets simply pull each other so that the
planets continually "fall" around the sun in closed paths. Why
don't the planets crash into the sun? They don't because of their
tangential velocities. What would happen if their tangential
velocities were reduced to zero? The answer is simple enough:
their motion would be straight toward the sun and they would
indeed crash into it. Any objects in the solar system with insufficient tangential velocities have long ago crashed into the sun.
What remains is the harmonv we observe.

orbit the

-Q-

and planets

TAN6ENTIAL VELOCITY

Fig. 10-6

into

it. If

velocity

Since the moon is gravitationally attracted to the earth,

does it not simply crash into the earth?

earth?

Answer
The moon would crash
zero, but because of

rather than into

it.

We

its

into the earth

if its

tangential velocity were reduced to

tangential velocity, the

will return to this idea in

moon

more

falls

around the earth

detail in the next chapter.

This is a dramatic example of the painstaking effort and crosschecking that

go into the formulation of a scientific theory. Compare this with the lack of "doing one's homework," the hasty judgements, and the absence of cross-check examinations that so often characterize the pronouncement of less-than-scientific
theories.

ve-

about the
sun allows it to fall around
the sun rather than directly

Question

why

The tangential

locity of the earth

this tangential

were reduced to zero,

what would be the fate of the

138

10.4

Universal Gravitation

10

The Law of Universal

Gravitation

Newton did not discover gravity. What Newton discovered was

was universal. Everything pulls on everything else
in a beautifully simple way that involves only mass and distance.
\cu ton s law of universal gravitation states that every object at-

that gravity

any two objects is

The greater the
masses, the greater the force of attraction between them. Newton deduced that the force decreases as the square of the distance between the centers of mass of the objects. The farther
away the objects are from each other, the less the force of attraction between them.
The law can be expressed symbolically as
tracts every other object with a force that for

directly proportional to the

mass of each

object.

m m
,

d-

where m, is the mass of one object, m, is the mass of the other,

and d is the distance between their centers of mass. The greater
the masses m, and m 2 the greater the force of attraction between them. The greater the distance d between the objects, the
weaker the force of attraction.
We can express the symbolic statement as an exact equation if
,

The force of gravbetween objects depends

on the distance between their
Fig. 10-7
ity

centers oi mass.

we introduce

the quantity G, the universal constant of gravitaa conversion factor needed to change the units of mass
and distance on the right side of the equation to the units of force
on the left (just as we use conversion factors to change from miles
tion.

is

to kilometers, for example).

Then we can express the law

of uni-

versal gravitation as the exact equation

m.m,
w

'

d'-

In words, the force of gravity

between two objects

is

found by

multiplying their masses, dividing by the square of the distance

between them, then multiplying this result by the constant G.
The value of G was determined by actual measurement long
after

Newton's time.

It is

Nm

0.0000000000667
or, in scientific

kg^

notation,"

G =

Scientific notation

is

6.67 x

discussed

in

10" N-m

Appendix A

/kg 2

at the

end of

this

book.

Gravity and Distance: The Inverse-Square

10.5

The value

of

force. It is the

tells

Law

us that the force of gravity

139

is

a very

weak

weakest of the four fundamental forces found in

nature. (The other three are the electromagnetic force and two
kinds of nuclear forces.) We sense gravitation only when masses
like that of the earth are involved. The force of attraction between
a pair of 1-kg masses with their centers of gravity 1
apart is
only 6.67 x 10"" N, too tiny for ordinary measurement. The
force of attraction between you and the earth, however, can be

measured. It is called your weight.

Your weight is the gravitational attraction between your mass
and the mass of the earth. If you gain mass, you gain weight. Or
if the earth somehow gained mass, you'd also gain weight. Your
weight also depends on your distance from the center of the
earth. At the top of a mountain your mass is no different than it
is anywhere else, but your weight is slightly less than at ground
level. This is because your distance from the center of the earth
is

Your weight is less

mountain because you are farther from
Fig. 10-8

at the top of a

the center of the earth.

greater.

Question
there is an attractive force between all objects, why do
not feel ourselves gravitating toward massive buildings

If

we
in

our vicinity?

Gravity

10.5

and Distance: The Inverse-Square Law

We can understand how gravity is reduced with

distance by con-

sidering an imaginary "butter gun" used in a busy restaurant for

buttering toast (Figure 10-9). Imagine melted butter sprayed

through a square opening in a wall. The opening is exactly the
size of one piece of square toast. And imagine that a spurt from
the gun deposits an even layer of butter 1
thick. Consider
the consequences of holding the toast twice as far from the butter
gun. You can see in Figure 10-9 that the butter would spread out

mm

Answer

We

and everything else in

The 1933 Nobel prize-winning physicist Paul A.M. Dirac put it this
way: "Pick a flower on earth and you move the farthest star!" How much you are
influenced by buildings or how much inter action there is between (lowers, is another story. The forces between you and buildings are relatively small because
the masses are small compared to the mass ol the earth. The forces due to the
are gravitationally attracted to massive buildings

the universe.

stars are small because of their great distances.

tice

when

they are

overwhelmed by

These

tins forces

escape our no-

the ovei powering attraction to the earth.

140

10

Universal Gravitation

and would cover twice as much toast vertoast horizontally. A little thought will
show that the butter would now spread out to cover four pieces
of toast. How thick will the butter be on each piece of toast?
for twice the distance
tically

and twice

as

much

Since it has been diluted to cover four times as much area,

thickness will be one-quarter as much, or 0.25 mm.

its

BUTTER HERE IS ONLY

/9 mm THICK
1

BUTTER ON TOAST HERE

IS \ mm THICK
BUTTER ON TOAST HERE
IS 1mm THICK

BUTTER 6UN
(U.S.

PATENT APPLIED FOR)

^ TOAST

IS

SIDE OF

PLACED ON BACK
SQUARE HOLE

The inverse-square law. Butter spray travels outward from the

nozzle of the butter gun in straight lines. Like gravity, the "strength" of the
spray obeys the inverse-square law.
Fig. 10-9

Note what has happened. When the butter gets twice as far
from the gun, it is only \ as thick. More thought will show that if
it gets 3 times as far, it will spread out to cover 3 x 3, or 9, pieces
of toast. How thick will the butter be then? Can you see it will be
9 as thick? And can you see that \ is the inverse square of 3? (The
inverse of 3 is simply \; the inverse square of 3 is (\) or |.) This
law applies not only to the spreading of butter from a butter
gun, and the weakening of gravity with distance, but to all cases
where the effect from a localized source spreads evenly throughout the surrounding space. More examples are light, radiation,
and sound.
The greater the distance from the earth's center, the less an
object will weigh (Figure 10-10). If your little sister weighs 300 N
at sea level, she will weigh only 299 N atop Mt. Everest. But no
matter how great the distance, the earth's gravity does not drop
to zero. Even if you were transported to the far reaches of the
universe, the gravitational influence of the earth would be with
you. It may be overwhelmed by the gravitational influences of
nearer and/or more massive objects, but it is there. The gravitational influence of every object, however small or far, is exerted
through all space. Isn't that amazing?
,

141

Universal Gravitation

10.6

Question

Suppose that an apple at the top of a tree is pulled by

earth gravity with a force of 1 N. If the tree were twice as
tall, would the force of gravity on the apple be only \ as
strong? Explain your answer.

APPLE WEI6HS
N HERE

^ \

APPLE WEI6HS
HERE
1 N

Fig. 10-10

An apple

that weighs

APPLE WEI6HS (?) N HERE

at the earth's surface

weighs only 0.25

when

located twice as far from the earth's center because the pull of gravity

only

as strong.

What would

it

When

weigh

3 times as far, it weighs only 5 as much, or

times the distance? Five times?

it is

at 4

0.1

N
is

N.

Universal Gravitation

10.6

We all know that

It

is

the earth is round. But why is the earth round?

round because of gravitation. Since everything attracts

everything

earth has attracted itself together before it

of the earth have been pulled in so
that the earth is a giant sphere. The sun, the moon, and the earth
are all fairly spherical because they have to be (rotational effects

became

else, the

solid.

Any "corners"

make them somewhat wider

at their equators).

everything pulls on everything else, then the planets must

pull on each other. The net force that controls Jupiter, for example, is not just from its interaction with the sun, but with the
planets also. Their effect is small compared to the pull of the
more massive sun, but it still shows. When the planet Saturn is
near Jupiter, for example, its pull disturbs the otherwise smooth
path of Jupiter. Both planets deviate from their normal orbits.
This deviation is called a perturbation.
If

Answer
No, because the twice-as-tall apple tree is not twice as far from the earth's center. The taller tree would have to have a height equal to the radius of the earth
(6370 km) before the weight of the apple reduces to N. Before its weight decreases by one percent, an apple or any object must be raised 32 km
nearl)
4 times the height of Mt. Everest, the tallest mountain in the world. So as a practical matter we disregard the effects of everyday changes in elevation.
:,

142

10

Up

until the

middle of the

last

Universal Gravitation

century astronomers were puz-

zled by unexplained perturbations of the planet Uranus.

when

Even

the influences of the other planets were taken into account,

Uranus was behaving stangely. Either the law of gravitation was

from the sun, or some unknown influence such as another planet was perturbing Uranus.
The source of Uranus's perturbation was uncovered in 1845
and 1846 by two astronomers, John Adams in England, and
Urbain Leverrier in France. With only pencil and paper and the
application of Newton's law of gravitation, both astronomers independently arrived at the same conclusion: a disturbing body
beyond the orbit of Uranus was the culprit. They sent letters to
failing at this great distance

their local observatories with instructions to search a certain

The request by Adams was delayed by misunderstandings at Greenwich, England, but Leverrier 's request to the
director of the Berlin observatory was heeded right away. The
planet Neptune was discovered within a half hour.
Other perturbations of Uranus led to the prediction and discovery of the ninth planet, Pluto. It was discovered in 1930 at
the Lowell observatory in Arizona. Pluto takes 248 years to make
a single revolution about the sun, so it will not be seen in its discovered position again until the year 2178.
The perturbations of double stars and the shapes of distant
galaxies are evidence that the law of gravitation extends beyond
the solar system. Over still larger distances, gravitation dictates
part of the sky.

the fate of the entire universe.

Fig. 10-11

Gravitation dictates the shape of the spiral arms in a galaxy.

Current scientific speculation

is

that the universe originated

in the explosion of a primordial fireball

some

15 to 20 billion

years ago. This is the "Big Bang" theory' of the origin of the universe. All the matter of the universe was hurled outward from
this event and continues in an outward expansion. We find ourselves in an expanding universe.

10.6

Universal Gravitation

143

This expansion may go on indefinitely, or it may be overcome

by the combined gravitation of all the galaxies and come to a
stop. Like a stone thrown upward, whose departure from the
ground comes to an end when it reaches the top of its trajectory,
and which then begins its descent to the place of its origin, the
universe may contract and fall back into a single unity. This
would be the "Big Crunch." After that, the universe would presumably re-explode to produce a new universe. The same course
of action might repeat itself and the process may well be cyclic.
If this is true, we live in an oscillating universe.
We do not know whether the expansion of the universe is cyclic
or indefinite, because we are uncertain about whether enough
mass exists to halt the expansion. The period of oscillation is esti-

mated

to be somewhat less than 100 billion years. If the universe

does oscillate, who can say how many times it has expanded and
collapsed? We know of no way a civilization could leave a trace
of ever having existed during a previous cycle, for all the matter
in the universe would be reduced to bare subatomic particles
during the collapse. All the laws of nature, such as the law of
gravitation, would then have to be rediscovered by the higher
evolving life forms. And then students of these laws would read
about them, as you are doing now. Think about that.

Few theories have affected science and civilization as much as

Newton's theory of gravity. The successes of Newton's ideas ushered in the Age of Reason or Century of Enlightenment. Newton
had demonstrated that by observation and reason, people could
uncover the workings of the physical universe. How profound
that all the moons and planets and stars and galaxies have such
a beautifully simple rule to govern them, namely
f _
The formulation of

m m
l

one of the major reasons

it provided hope that
other phenomena of the world might also be described by equally
simple laws.
This hope nurtured the thinking of many scientists, artists,
writers, and philosophers of the 1700s. One of these was the English philosopher John Locke, who argued that observation and
reason, as demonstrated by Newton, should be our best judge
and guide in all things. Locke urged that all of nature and even
society should be searched to discover any "natural laws" that
might exist. Using Newtonian physics as a model of reason,
Locke and his followers modeled a system of government that
found adherents in the 13 British colonies across the Atlantic.
These ideas culminated in the Declaration of Independence and
the Constitution of the United States of America.
this

simple rule

is

for the success in science that followed, for

144

Universal Gravitation

10

Chapter Review

10

Concept Summary

6.

What

is

required before a hypothesis (an

educated guess) advances to the status of a

scientific theory (organized knowledge)?

The moon and other objects in orbit around the

earth are actually falling toward the earth but
have great enough tangential velocity to avoid
7.

hitting the earth.

Since the planets are pulled to the sun by

why don't they simply crash into the sun? (10.3)

gravitational attraction,

According to Newton's law of universal gravitation, everything pulls on everything else with a
force that depends upon the masses of the objects

and the distances between

8.

their centers of

The greater the masses, the greater

is

not discover gravity but, rather,

the
9.

What does

the very small value of the gravi-

force.

The greater the distance, the smaller

is

dis-

covery? (10.4)

mass.

Newton did

something about gravity. What was his

tational constant

the

G tell us about the strength

of gravitational forces? (10.4)

force.
10.

Important Terms

Exactly what are the two specific masses

and the one specific distance that determine

your weight?

law of universal gravitation (10.4)

perturbation (10.6)

11.

In

(10.4)

what way

gravity reduced with dis-

is

tance from the earth? (10.5)

tangential velocity (10.2)

universal constant of gravitation (10.4)

12.

What would be
if

you were

five

the difference in your weight

times farther from the cenvou are now? Ten times?

ter of the earth as

Review Questions
1

Why did Newton

on the moon?

2.

What was

it

(10.5)

think that a force must act

Newton discovered about

that

the force that pulls apples to the ground and

the force that holds the moon in orbit? (10.1)
3.

If

the

moon

falls,

why doesn't

it

What

is

What makes

14.

What causes planetary

15.

Distinguish between the "Big Bang" and the

"Big Crunch." (10.6)

meant by

How

did

there

is

Newton check

his hypothesis that

an attractive force between the earth

and moon?

(10.2)

perturbations? (10.6)

Think and Explain

tangential velocity? (10.2)

1.

The moon
this

5.

the earth round? (10.6)

get closer to

the earth? (10.2)

4.

13.

(10.1)

mean

"falls" 1.4

that

it

mm each second. Does

mm closer to the

gets

earth each second?

.4

Would

it get closer if its

tangential velocity were reduced? Defend

your answer.

10

2.

Chapter Review

145

Comparing a large boulder with a small

stone, which is attracted to the earth with
the greater force? Which undergoes the

center,
its

how would your weight compare

to

present value?

when falling without

Defend both answers.

greater acceleration
air resistance?

3.

If

the gravitational forces of the sun on the

planets suddenly disappeared, in what kind

of paths
4.

If

the

would they move?

moon were

twice as massive, would

the attractive force between the earth and

moon be twice as large? Between the moon

and earth? Defend your answer.
5.

a.

b.

the gravitational force between two

massive bodies were measured and divided by the product of their masses ^and
then multiplied by the square of the distance between their centers of mass, what
If

number would result?

Would this number differ

8.

the

9.

moon were

twice as far

if

Evidence indicates that the present expanis slowing down. Is this

consistent with, or contrary to, the law of

for different

gravity? Explain.

away and

Some
tific

mained in circular orbit about the earth, by

what distance would it "fall" each second?

ries.

people dismiss the validity of scien-

theories by saying they are "only" theo-

The law

theory.

doubt

you stood atop a ladder that was so tall

that you were twice as far from the earth's

7. If

would your weight change

mass?

re-

If

factor

sion of the universe

10.

the

By what

the earth were twice as big and had twice

masses at different distances? Defend

your answer.
6.

Fig.

Does

its

of universal gravitation
this

mean

that scientists

validity? Explain.

is

still

Gravitational Interactions

Everyone knows that objects

before the time of Isaac

fall

because of gravity. Even people

this. Contrary to popular

Newton knew

belief, Newton did not discover gravity. What Newton discovered was that gravity is universal that the same force that pulls
an apple off a tree holds the moon in orbit, and that the earth
and moon are similarly held in orbit about the sun. And the sun
revolves as part of a cluster of other stars about the center of the
galaxy, the Milky Way. Newton discovered that all objects in the
universe attract each other. This was discussed in the last chapter. In this chapter we shall investigate the role of gravity at, below, and above the earth's surface; in the earth's oceans and its
atmosphere; and in stellar objects called black holes. We begin
with the simple case of free fall.

Acceleration

Due

to Gravity

Recall from earlier chapters that an object in free fall (that is,
with only the force of gravity acting on it) accelerates
downward at the rate of 9.8 m/s 2 This acceleration is known as
falling

Some people get mixed up beg and big G. Big G is not acceleration, but is the universal gravitational constant in the equation for the gravitag,

the acceleration due to gravity.

tween

little

tional force

between any two objects:

m.m,
Thus g and

G are quite different quantities that represent quite

different things. Nevertheless, there

between the two. The value of

little

is

an interesting relationship

g was found from measure-

146

Acceleration

11.1

ments of

Due

147

to Gravity

But why should it be 9.8m/s 2 and not

can find the answer from the equation for

falling objects.

some other value? We

gravity.

The weight of any object is the gravitational force of attracbetween that object and the earth. This force depends on
the mass of the object, m, and the mass of the earth, which we
will call M. At the surface of the earth the distance between their
centers is simply the radius of the earth, which we will call R. If
we make these substitutions (m, = m, m = M, d = R) in the law
tion

of gravity,

we

get an equation for the weight of an object at the

earth's surface:

From Newton's second

mg. Substituting

is

we

mg

law,

the weight of an object

for weight in the preceding equation,

get

GmM

mg

Notice that m, the mass of the object, appears on both sides.

can divide it out, which leaves

We

GM
b

Suppose you use a calculator

M, and R:

to evaluate g, using the follow-

ing values for G,

6.67 x

5.98 x 10^ kg
6.37 x 10"

R =

since

as m/s 2

/kg 2

two numbers and divide two times by the third.

two digits, your answer will be 9.8 N/kg. And,
equals 1 kg-m/s 2 the combination unit N/kg is the same

Multiply the

Rounded

10" N-m

G =

first

off to

Thus, you can see that the numerical value of g depends on

mass of the earth and its radius. If the earth had a different
mass or radius, g would have a different value. If you know the
mass and radius of any planet, you can calculate the value of g
the

for that planet.

Interestingly enough, the mass of the earth was itself found

from this equation after the value of G was experimentally determined in 1798 by an English scientist named Henry Cavendish. In fact, his procedure for determining G with heavy lead

weights was called the "weighing-the-earth experiment." You

can see that if g, G, and R are known, then the mass of the earth

is

easilv found.

is

The weight

of the

the gravitational force

between the load

and the earth.

F = ma,

load

of attraction

GmM

weight

1-1

Fig.

So the combined mass

of all the rock,

oceans,

148

Gravitational Interactions

1 1

TWIST N6 FORCE
I

mountains, trees, sticks, and twigs that make up the planet

earth can be easily calculated. The power of a little algebra is

enormous!

Questions
1

The acceleration of objects on the surface of the moon

due to universal gravity is only i of 9.8 m/s 2 From this
fact, is it correct to say that the mass of the moon is therefore i the mass of the earth?
.

Cavendish deby measuring the

force ol gravit) between lead
masses. Onee he had the value

Fig. 11-2

termined

2.

of the force, the masses,

had the same radius, "g" for Jupiter would

be 300 earth g.) But the radius of Jupiter is about 10
times greater than the radius of earth, so an object at its
surface is 10 times farther from Jupiter's center than it
would be on earth. So about how much greater is " g" on
Jupiter compared to earth g?

means

and

their distances apart, ean you

see how he determined G?

11.2

Estimate "g" on the surface of the planet Jupiter by comparing its mass and size with that of the earth. First, it
is about 300 times more massive than the earth. (This
that

if it

Gravitational Fields
The earth

pulls on the

moon. We regard this as action at a disand moon interact with each other even
contact. Or we can look at this in a dif-

tance, because the earth

though they are not in

way: we can regard the moon as interacting with the gravi-

ferent

The properties of the space surroundmass can be considered to be altered in such a way that
another mass introduced to this region will experience a force.
tational field of the earth.

ing any

r*

This alteration of space is the gravitational field. It is common to

think of distant rockets and space probes as interacting with
gravitational fields rather than with the masses of the earth and
other planets or stars that are the sources of these fields. The
field concept plays an in-between role in our thinking about the
forces between different masses.

#*

Answers
Fig. 11-3

the rocket

We can
is

say that

it is

field of

the earth. Both points ol

are equivalent.

could assume the mass of the

moon

to

be

that of earth only

if

both

The radius of the moon 1 .74 x 10* m)

in fact less than one third the earth's radius, and its mass (7.36 x 10" kg)
about ^ the mass of the earth.

interacting

with the gravitational

We

moon and

attracted to the

earth, or that

No.

2.

earth had the

The value of "g" on

3GM/R 1 = 3g.

same

radius.

Jupiter's surface

is

is
is

G(300yW)/(10fl) 2 = 300GjW/(100K 2 ) =

1 1 .2

149

Gravitational Fields

A gravitational field is an example of a force field, for something in the space experiences a force due to the field. Another
force field you may be familiar with is a magnetic field. You have
probably seen iron filings lined up in patterns around a magnet.
(Look ahead to Figure 36-4 on page 541, for example.) The pat-

and direction of the magaround the magnet.

Where the filings are closest together, the field is strongest. The
direction of the filings shows the direction of the field at each
tern of the filings shows the strength

netic field at different points in the space

point.

The pattern of the earth's gravitational field can be represented by field lines (Figure 1 1-4). Like the iron filings around a
magnet, the field lines are closer together where the gravitational field is stronger. At each point on a field line, the direction
of the field at that point is along the line. Arrows show the field
direction. A particle, astronaut, spaceship, or any mass in the vicinity of the earth will be accelerated in the direction of the field

Fig. 11-4

Field lines repre-

sent the gravitational field

about the earth. Where the

field lines

the field

is

are closer together,

stronger. Farther

away, where the

field lines are

farther apart, the field

is

weaker.

line at that location.

The strength
its force on

of the earth's gravitational field, like the strength

objects, follows the inverse-square law." It is

strongest near the earth's surface, and weakens with increasing
distance from the earth.

of

The gravitational field of the earth exists inside the earth

To investigate the field beneath the surface,
imagine a hole drilled completely through the earth, say from
the North Pole to the South Pole. Forget about impracticalities
such as lava and high temperatures, and consider the kind of motion you would undergo if you fell into such a hole. If you started
at the North Pole end, you'd fall and gain speed all the way down
to the center, then overshoot and lose speed all the way "up" to
the South Pole. You'd gain speed moving toward the center, and
lose speed moving away from the center. Without air drag, the
one-way trip would take nearly 45 minutes. If you failed to grab
the edge, you'd fall back toward the center, overshoot, and return to the North Pole in the same time.
Repeat this trip with an accelerometer of some kind. Your
as well as outside.

acceleration at the beginning of fall is g, but you'd find it progressively less as you continue toward the center of the earth.

Fig. 11-5

Why? Because

and

of the earth, the pull

completely through the earth,

your acceleration diminishes
because the mass you leave
behind retards it. At the

as you are being pulled "downward" toward the

earth's center, you are also being pulled "upward" by the part of
the earth that is "above" you. In fact, when you get to the center

"down"

is

balanced by the pull "up." You

As you

earth's center
tion
ries

The strength of the gravitational field at any point is measured by the force
on a unit mass placed there. If a force F is exerted on a mass m, the field
strength is Flm, and its units are nevvtons per kilogram (N/kg).

fall

faster in a hole

is

zero.

faster

bored

your accelera-

Momentum

car-

you against a growing

acceleration past the center

to the opposite end of the tunnel

where

it is

again

g.

150

11

Gravitational Interactions

are pulled in every direction equally, so the net force on you is

zero. There is no acceleration as you whiz with maximum speed
past the center of the earth.

center

its

Fig.

1-6

is

The gravitational

field of the

earth at

zero!

In a cavity at the center of the earth your weight

because you are pulled equally by gravity

field at the earth's

center

is

in all directions.

would be zero
The gravitational

zero.

Interestingly enough, you'd gain acceleration during the first

few kilometers beneath the earth's surface, because the density
of surface material is much less than the condensed center. The
mass of earth "above you" exerts a small backward pull compared to the disproportionally greater pull of the denser core
material. This means you'd weigh slightly more for the first few
kilometers beneath the earth's surface. Further in, your weight
would decrease and would diminish to zero at the earth's center.

Questions
1

2.

If you stepped into a hole through the center of the earth

and made no attempt to grab the edges at either end,
what kind of motion would you experience?

Halfway to the center of the earth, would you weigh more

or weigh less than you weigh at the surface of the earth?

Answers
You would oscillate back and forth, in what is called simple harmonic motion.
Each round trip would take nearly 90 minutes. Interestingly enough, we will
see in the next chapter that an earth satellite in close orbit about the earth
to make a complete round trip. (This is no coyou study physics further, you'll learn about an interesting
relationship between simple harmonic motion and circular motion at con-

also takes the

incidence, for

same 90 minutes
if

stant speed.)
2.

You would weigh less, because the part of

"down" is counteracted by mass above you

the earth's

mass that

that pulls you "up."

If

pulls

you

the earth

were of uniform density, halfway to the center your weight would be exactly
half your surface weight. But since the earth's core is so dense (about 7 times
the density of surface rock) your weight would be somewhat more than half
surface weight. Exactly how much depends on how the earth's density varies
with depth, information that

is

not

known

today.

1 1 .3

Weight and Weightlessness

11.3

151

Weight and Weightlessness

The force of gravity, like any force, causes acceleration. Objects

under the influence of gravity accelerate toward each other. We
are almost always in contact with the earth. For this reason, we
think of gravity primarily as something that presses us against
the earth rather than as something that accelerates us.
ing against the earth

is

the sensation

we

The press-

interpret as weight.

Stand on a bathroom scale that is supported on a stationary

The gravitational force between you and the earth pulls
you against the supporting floor and scale. By Newton's third
law, the floor and scale in turn push upward on you. Located in
between you and the supporting floor are springs inside the bathroom scale. The springs are compressed by this pair of forces.
The weight reading on the scale is linked to the amount of comfloor.

pression of the springs.

If you repeat this weighing procedure in a moving elevator, you
would find your weight reading would vary not during steady
motion, but during accelerated motion. If the elevator acceler-

upward, the bathroom scale and floor push harder against

your feet, and the springs inside the scale are compressed even
more. The scale shows an increase in your weight.
ates

NO WEIGHT
equal to the force that you exert against
the supporting floor. If the floor accelerates up or down, your weight seems to
vary. You feel weightless when you lose your support in free fall.

Fig. 11-7

The sensation

of weight

is

If the elevator accelerates downward, the scale shows a decrease in your weight. The support force of the floor is now less.
If the elevator cable breaks and the elevator falls freely, the scale

152

11

Gravitational Interactions

reading would register zero. According to the scale, you are

weightless. Would you really be weightless? The answer to this
question depends on your definition of weight.
If you define weight as gravitational force acting on an object,
then you still have weight whether or not you are freely falling.
Astronauts in earth orbit who float freely inside their capsules
are still being pulled by gravity and therefore have weight. But
you in the falling elevator and the astronauts in the "falling" satellites don't feel the usual effects of weight. You feel weightless.
That is, your insides are no longer supported by your legs and
pelvic region. Your organs respond as though gravity were absent. You are in a state of apparent weightlessness.

Fig. 11-8

The astronaut

is

in a state of

apparent weightlessness

all

the time in

orbit.

To be truly weightless, you would have to be far out in space

away from the earth and other attracting stars and planets
where gravitational forces are negligible. In this truly weightless condition, you would drift in a straight-line path rather than
in the curved path of an orbit.
In a more practical sense, we define the weight of an object as
the force it exerts against a supporting floor (or weighing scales).
According to this definition, you are as heavy as you feel. The
well

condition of weightlessness is then not the absence of gravity,

but the absence of a support force. That queasy feeling you get
when you are in a car that seems to leave the road momentarily
when it goes over a hump, or worse, off a cliff, is not the absence
of gravity. It is the absence of a support force. Astronauts in orbit
are without a support force and are in a sustained state of weightlessness. That causes "spacesickness" until they get used to it.

153

Ocean Tides

11.4

-*r

Fig. 11-9

-<r

Both people experience weightlessness.

11.4

Ocean Tides

Seafaring people have always known there was a connection

between the ocean tides and the moon, but no one could offer
a satisfactory theory to explain the two high tides per day. Newton showed that the ocean tides are caused by differences in the
gravitational pull between the moon and the earth on opposite
sides of the earth. The gravitational force of the moon on the
earth is stronger on the side of the earth nearer to the moon, and
weaker on the side of the earth farther from the moon. This is
simply because the gravitational force is weaker with increased
distance.

LOW

TIDE

HI6H TIDE

Twice a day, every point along the ocean shore has a high
between the high tides is a low tide.
Fig. 11-10

tide. In

Consider a big spherical ball of gooey taffy. If you exert the

force on every part of the ball, it would remain spherical
as it accelerates. But if you pulled harder on one side than the
other, its shape would become elongated. Imagine that you attach a rope to the ball of taffy and swing it in a circle around you
(Figure 11-11). The ball will bulge outward in two places. One

same

154

1 1

Gravitational Interactions

bulge will be toward the center of the circular path. The other
bulge will be on the opposite side, on the faster-moving outer
part that is "thrown" outward. For an initially spherical ball of
uniform taffy, these oppositely facing bulges would be of equal
size.

Fig.

1-1

spun

An

initially spherical ball of

in a circular

gooey

taffy will

be elongated when

it is

path.

This is what happens to this big ball we're living on. You might
tend to think of the earth at rest and the moon circling around

you lived on the moon, you'd likely say that the moon
and the earth circles about the moon. It turns out that
both the earth and the moon orbit about a common point their
earth-moon center of mass. So the water covering the circling
earth is distorted and will bulge like the circling taffy in Figus.
is

But

if

at rest

ure 11-11.

O
Fig. 11-12

remain

Two

relativ

respect to the n

earth spins dail

them.

tidal

bulges

fixed with

while the
beneath

ion

The earth makes one complete spin per day beneath these
ocean bulges. This produces two sets of ocean tides per day. Any
part of the earth that passes beneath one of the bulges has a high
tide. On a world average, a high tide is about 1 m above the average surface level of the ocean. When the earth makes a quarter
turn, 6 hours later, the water level at the same part of the ocean
is about 1 m below the average sea level. This is low tide. The
water that "isn't there" is under the bulges that make up the high
tides. A second high tidal bulge is experienced when the earth
makes another quarter turn. So we have two high tides and two
low tides daily. It turns out that while the earth spins, the moon
moves in its orbit and appears at the same position in our sky
every 24 hours and 50 minutes, so the two-high-tide cycle is actually at 24-hour-and-50-minute intervals. That is why tides do
not occur at the same time every day.
The sun also contributes to ocean tides, although it is less than
half as effective as the
zling
is

when

it

is

moon

in raising tides.

This

may seem

puz-

and earth
between the moon and

realized that the pull between the sun

about 180 times stronger than the pull

Why, then, does the sun not cause tides 180 times greater

earth.

Ocean Tides

1 1 .4

155

than lunar tides? The answer has to do with a key word: differBecause of the sun's great distance from the earth, there is
not much difference in the distances from the sun to the near

ence.

part of the earth and the far part. This means there is not much
difference in the gravitational pull of the sun on the part of the
ocean nearest it and on the part farthest from it. The relatively

small difference in pulls on opposite sides of the earth only

slightly elongates the earth's shape and produces tidal bulges
less than half those produced by the moon."

DISTANCES TO FAR SIDE OF THE EARTH

DISTANCES TO NEAR SIDE OF THE EARTH

Fig. 11-13

If

you stand close

to the globe (as the

part. If

moon

is

in relation to the

noticeably nearer to you than the farthest

(as the sun is in relation to the earth), the difference

earth), the closest part of the globe

is

you stand far away

between the closest and farthest part of the globe

in distance

is

less significant.

When

the sun, earth, and moon are all lined up, the tides due
sun and the moon coincide. Then we have higher-thanaverage high tides and lower-than-average low tides. These are
called spring tides (Figure 11-14). (Spring tides have nothing to
do with the spring season.)
to the

Fig.

1-14

When

the attractions of the sun

and the moon are

lined

up with

each other, spring tides occur.

The

relative differences in distance

is

only hinted at in Figure 11-13, which

way off scale. The actual distance between the earth and the sun is nearly 12 000
earth diameters. So for an ordinary globe of] meter in diameter, to judge the
is

between closest and farthest parts of the globe from the

meters away; and from the sun's position, you'd
stand "across town," 4 kilometers away!

relative difference

moon, you'd have

have to

to stand 10

156

1 1

Gravitational Interactions

It the alignment is perfect, we have an eclipse. A lunar eclipse

produced when the earth is directly between the sun and moon
(Figure 11-1 5a). A solar eclipse is produced if the moon is directlj between the sun and earth (Figure 1-1 5c). The alignment
is usually not perfect. Instead, we have a full moon when the
earth is in the middle, and a new moon when the moon is in the
middle. So spring tides occur at the times of a new or full moon.

is

LUNAR ECLIPSE
IN

MOON

IS

THE EARTH

COMPLETELY

SHADOW

'5

6y
Detail of sun-earth-moon alignment, (a) Perfect alignment produces a lunar eclipse, (b) Non-perfect alignment produces a full moon, (c) Perfect alignment produces a solar eclipse, (d) Non-perfect alignment produces a
new moon. Can you see that from the daytime side of the world, the new moon
cannot be seen because the dark side faces the earth, and from the nighttime
side of the world, the moon is out of view altoeether?
Fig. 11-15

^1/

When the moon is half way between a new moon and a full
moon, in either direction (Figure 1-16), the tides due to the sun
and the moon partly cancel each other. Then, the high tides are
lower than average and the low tides are not as low as average
low tides. These are called neap tides.
1

Fig. 11-16

Wl

tions of the sun

are at right angi

the attrac-

d the
to

moon

one an-

(at the time of a half

moon), neap tides occur.

other

Another factor that

axis (Figure 11-17).

affects the tides

is

the

tilt

Even though the opposite

of the earth's

tidal bulges are

1.5

Tides in the Earth and Atmosphere

equal, the earth's

in

tilt

157

causes the two daily high tides experienced

to be unequal most of the time.

most parts of the ocean

Fig. 11-17

The

earth's

tilt

causes the two daily high tides

in the

same place

to

be unequal.

E3
The earth

Tides in the Earth and Atmosphere

not a rigid solid but, for the most part, is molten

by a thin solid and pliable crust. As a result, the
moon-sun tidal forces produce earth tides as well as ocean tides.
Twice each day the solid surface of the earth rises and falls by as
much as 25 cm! As a result, earthquakes and volcanic eruptions
have a slightly higher probability of occurring when the earth is
experiencing an earth spring tide that is, near a full or new
is

liquid covered

moon.

We

bottom of an ocean of air that also experiences

Because of the low mass of the atmosphere, the atmospheric tides are very small In the upper part of the atmosphere is
the ionosphere, so named because it is made up of ions, electrically charged atoms that are the result of intense cosmic ray
bombardment. Tidal effects in the ionosphere produce electric
currents that alter the magnetic field that surrounds the earth.
These are magnetic tides. They in turn regulate the degree to
which cosmic rays penetrate into the lower atmosphere. The
cosmic ray penetration affects the ionic composition of our atmosphere, which in turn, is evident in subtle changes in the behaviors of living things. The highs and lows of magnetic tides
are greatest when the atmosphere is having its spring tides
again, near the full and new moon.
live at the

tides.

This

moon!

may

be

why some

of

vour friends seem a

bit

weird

at the

time

ol a kill

158

11.6

11

Gravitational Interactions

Black Holes
There are two main processes that are going on all the time in
stars such as our sun. One is gravitation, which tends to crunch
all solar material toward the center. The other process is nuclear
fusion. The core of the sun is continuously undergoing hydrogenbomb-like explosions that tend to blow its material out from the
center. The two processes balance each other, and the result is
the sun of a given size.

Fig. 11-18

The

size of the

sun

is

the result of a "tug-of-war" between

two op-

posing processes: nuclear fusion, which tends to blow it up (colored arrows),

and gravitational contraction, which tends to crunch it together (black arrows).
If

the fusion rate increases, the sun gets bigger;

if

the fusion

What happens when

the sun
runs out of fusion fuel (hydrogen)? The answer is, gravitation
dominates and the sun collapses. For our sun, this collapse will
ignite the nuclear ashes of fusion (helium) and fuse them into
carbon. During this fusion process, the sun will expand to become the type of star known as a red giant. It will be so big that it
will extend beyond the earth's orbit and swallow the earth. Fortunately, this won't take place until 5 billion years from now.
When the helium is all fused, the red giant will collapse and die
out. It will no longer give off heat and light. It will then be the
type of star called a black dwarf a cool cinder among billions
rate decreases, the sun gets smaller.

of others.

a bit different for stars more massive than the

more than four solar masses, once gravitational
it doesn't stop! The
fusion or no fusion
collapse takes place
stars not only cave in on themselves, but the atoms that compose the stellar material also cave in on themselves until there

The story

is

sun. For stars of

are no

empty

spaces.

What

is left is

compressed

densities. Gravitation near the surfaces of these

to unimaginable
shrunken config-

159

Black Holes

11.6

is so enormous that nothing can get back out. Even

cannot escape. They have crushed themselves out of visible
existence. They are called black holes.
Interestingly enough, a black hole is no more massive than
the star from which it collapsed. The gravitational field near the
black hole may be enormous, but the field beyond the original
radius of the star is no different after collapse than before (Figure 11-19). The amount of mass has not changed, so there is no
change in the field at any point beyond this distance. Black holes
will be formidable only to future astronauts who venture too

urations

FIELD STRENGTH

light

HERE

15

G~

--J
<*

STAR OF MASS

FIELD

BLACK HOLE
OF MASS

STRENGTH

IS

STILL

r m

close.

Question

strength near a giant

field

If

the sun were to collapse from its present size and behole, would the earth be drawn into it?

come a black

The gravitational

Fig. 11-19

become
same (top)
and (bottom)

star that collapses to

a black hole

is

before collapse

the

after collapse.

The configuration of the gravitational field about a black hole

represents the collapse of space itself. The field is usually represented as a warped two-dimensional surface, as shown in Figure
1 1-20. Astronauts could enter the fringes of this warp and, with
a powerful spaceship, still escape. After a certain distance, however, they could not, and they would disappear from the observable universe. Don't go too close to a black hole!

Fig.

1-20

A two-dimensional

representation of the gravita-

around a black
Anything that falls into

tional field
hole.

the central

warp disappears

from the observable universe.

Answer
No, the gravitational force between the solar black hole and the earth would
not change. The earth and other planets would continue in their orbits. Observers outside the solar system would see the planets orbiting about "nothing" and
likely deduce the presence of a black hole. Observations of astronomical bodies
that orbit unseen partners indicate the presence of black holes.

160

Gravitational Interactions

1 1

11

Chapter Review

Concept Summary

3.

Why was Henry

Cavendish's experiment
determine the value of G called the
"weighing-the-earth experiment"? (1 1.1)
to

The acceleration due

to gravity at the earth's

surface can be derived from the law of universal

gravitation.

4.

The earth can be thought of as being surrounded

by a gravitational

field that interacts

and causes them

jects

with ob-

is more correct
to say that a distant
rocket interacts with the mass of the earth
or to say that it interacts with the gravita-

tional field of the earth? Explain.

(1 1.2)

to experience gravita-

tional forces.
5.

Objects in orbit around the earth have a gravitational force acting on them even though they

may appear

Which

to be weightless.

distance? (11.2)
6.

it

you

was bored comwould you accelthe way through and shoot like a

fell

erate all

into a hole that

projectile out the other side?

Whv

or

why

not? (11.2)
7.

a star runs out of fuel for fusion,

lapses under gravitational forces.

If

pletely through the earth,

Ocean tides (and even tides in the solid earth

and in the atmosphere) are caused by differences
in the gravitational pull of the moon on the earth
on opposite sides of the earth.

When

How does the gravitational field strength

about the planet earth vary with increasing

col-

What

is

the value of the gravitational field of

the earth at
8.

Where

is

its

center? (11.2)

your weight greatest at the surbelow the surface,

face of the earth, slightly

Important Terms

or above the surface?

apparent weightlessness

(1

9.

force field (11.2)

lunar eclipse

neap

( 1 1

spring tide

.2)

Does your apparent weight change when

you ride an elevator at constant speed, or
only while

.2)

it is

accelerating? Explain.

(1

.2)

(1 1.4)

10.

tide (1 1.4)

solar eclipse

.3)

black hole (11.6)

gravitational field

(1

Distinguish between apparent weightlessness and true weightlessness. (11.3)

(1 1.4)

(1 1.4)
1 1

Why

is

difference a key

word

in explaining

tides? (11.4)

Review Questions
Distinguish between g and G. Which is a
variable (that is. something that can vary in

the gravitational pull between the moon

and the earth were the same over all parts of
the earth, would there be any tides? Defend

value)? (11.1)

your answer.

12.
1.

2.

Upon what quan

of gravity

depend?

on the

(11.1)

ties
b

does the acceleration

riaces of various planets

13.

If

Which
the

(1 1.4)

force-pair

moon and

and earth?

is

(11.4)

that between
between the sun

greater

earth, or that

1 1

14.

Chapter Review

Which
tides

15.

is

more

161

effective in raising

the moon or the sun?

Why are

chapters will show, an electric field affects

electric charges, and a magnetic field affects
magnetic poles. What does a gravitational

ocean

(1 1 .4)

tides greater at the times of the full

and new moons and

and solar eclipses? (1

field affect?

at the times of lunar

1

5.

.4)

The weight of an apple near the surface of

the earth

16.

Distinguish between spring tides and neap

is

N.

What

is

the weight of the

earth in the gravitational field of the apple?

tides. (11.4)
6.

17.

Why

are ocean tides greater than atmos-

in effect

pheric tides? (11.5)

18.

What two major

you stand on a shrinking planet, so that

you get closer to its center, your
weight increases. But if you instead burrow
into the planet and get closer to its center,
your weight decreases. Explain.
If

processes determine the

size of a star? (11.6)

7.

19.

Distinguish between a stellar black dwarf

and a black hole. (11.6)

20.

Why would

you were unfortunate enough to be in a

you might notice the
bag of groceries you were carrying hovering

If

freely-falling elevator,

in front of you, apparently weightless. Are

sun

if it

the groceries falling? Defend your answer.

the earth not be sucked into the

evolved to become a black hole?

(11.6)

8.

When would

the

moon be

its fullest

just

before a solar eclipse or just before a lunar

eclipse? Defend your answer.

Think and Explain

1.

the earth were the

9.

same

but twice as
massive, what would be the value of G? What
would be the value of g? (Why are your answers different? In this and the following
2
question, let the equation g = GM/R guide
If

size

The human body is about 80% water.

likely that the moon's gravitational

Is it

pull

causes biological tides cyclic changes in

water flow among the fluid compartments
of the body? (Hint: Is any part of your body
appreciably closer to the moon than any
other part? Is there a difference in lunar

your thinking.)

pulls?)
2.

If

the earth

somehow shrunk

to half size

without any change in its mass, what would

be the value of g at the new surface? What
would be the value of g above the new surface
at a distance equal to the present radius?
3.

10.

A black hole
star

A man named Philipp von Jolly suspended a

spherical vessel of mercury of known mass

shown in Figure A. He rolled a 6-ton lead

sphere beneath the mercury and had to readjust the balance. Why? And how did this
enable him to calculate the value of G?
as

We

can think of a force field as a kind of extended aura that surrounds a body, spreading its influence to affect things. As later

no more massive than the

it

collapsed.

Why

then,

gravitation so intense near a black hole?

on one arm of a very sensitive balance that

he put in equilibrium with counterweights,

4.

is

from which

Fig.

is

Satellite

If

Motion

you drop a stone from a

ground below, it
you move your hand horizonwill follow a curved path to the

rest position to the

will fall in a straight-line path. If

you drop the stone, it

If you move your hand faster, the stone lands farther
away and the curvature of the path is less pronounced. What
would happen if the curvature of the path matched the curvature of the earth? The answer is simple enough: if air resistance
can be neglected, you would have an earth satellite.
tally as

ground.

The greater the

Fig. 12-1

the arc of

its

stone's horizontal

motion when released, the wider

curved path.

Earth Satellites

12.1

An earth

simply a projectile that falls around the earth

is easier to see if you imagine yourself on
a "smaller earth," perhaps an asteroid (Figure 12-2). Because of
the small size and low mass, you will not have to throw the stone
very fast to make its curved path match the surface curvature of
the asteroid. If you toss the stone just right, it will follow a cirsatellite is

rather than into

Fig. 12-2

If

you

tov. the

stone horizontally with the

proper speed,

its

pat;

match the surface cu

of the asteroid.

162

will

ature

it.

This

cular orbit.

How

fast would the stone have to be thrown horizontally

order to orbit the earth? The answer depends on the rate at
which it falls and the degree to which the earth curves. Recall
in

12.1

163

Earth Satellites

from Chapter 2 that a stone dropped from rest will accelerate

9.8 m/s 2 and fall a vertical distance of 4.9 m during the first second of fall. Also recall from Chapter 6 that the same is true
of anything already

moving

a projeca vertical distance of 4.9

from where it would have gone without
gravity. (It may be helpful to refresh your memory and review
Figure 6-15 on page 78.) So in the first second after a stone is
thrown, it will be 4.9
below the straight-line path it wpuld
have taken if there were no gravity.
tile.

as

it

starts to fall, that

is,

Remember that in the first second a projectile will

fall

Fig. 12-3
4.9

Throw a stone

m below where

it

A geometrical

at any speed and one second later

would have been without gravity.

it

will

have fallen

about the curvature of our earth is that its

for every 8000
tanmeans that if you were
swimming in a calm ocean, you would be able to see only the top
of a 4.9-m mast on a ship 8 km away.
fact

surface drops a vertical distance of 4.9

gent to the surface (Figure 12-4). This

8000 m

Fig. 12-4

The

earth's curvature (not to scale).

If a stone could be thrown fast enough to travel a horizontal

distance of 8 km during the time (1 s) it takes to fall 4.9 m, then
can you see it will follow the curvature of the earth? Isn't this
speed simply 8 km/s? So we see that the orbital speed for close
orbit about the earth is 8 km/s. If this doesn't seem to be very
fast, convert it to kilometers per hour; you'll see it is an impressive 29 000 km/h (or 18 000 mi/h). At this speed, atmospheric
friction would burn a stone to a crisp. That is why a satellite
must stay about 150 km or so above the earth's surface to keep
from burning up like a "falling star" against the friction of the

atmosphere.

164

12

Satellite

Motion

165

Circular Orbits

12.2

Circular Orbits

12.2

Interestingly enough, in circular orbit the speed of a satellite

not changed by gravity.

is

We can understand this by comparing a

with a bowling ball rolling along a bowldoesn't the gravity that acts on the bowling ball

satellite in circular orbit

ing alley.

Why

change its speed? The answer is that gravity is pulling neither a

bit forward nor a bit backward: gravity pulls straight downward. The bowling ball has no component of gravitational force
along the direction of the

alley.

DIRECTION OF MOTION

Fig. 12-5

(Left)

because there

same
is

is

is

The force of gravity on the bowling ball does not affect its speed
no component of gravitational force horizontally. (Right) The

true for the satellite in circular orbit. In both cases, the force of gravity

at right angles to the direction of motion.

The same

is

true for a satellite in circular orbit. In circular or-

always moving perpendicularly to the force of

does not move in the direction of gravity, which would
increase its speed, nor does it move in a direction against gravity, which would decrease its speed. Instead, the satellite exactly
"criss-crosses" gravity, with the result that no change in speed
occurs only a change in direction. A satellite in circular orbit
around the earth moves parallel to the surface of the earth at
constant speed.
For a satellite close to the earth, the time for a complete orbit
about the earth, called the period, is about 90 minutes. For higher
altitudes, the orbital speed is less and the period is longer. Communications satellites located in orbit 5.5 earth radii above the
surface of the earth, for example, have a period of 24 hours. Their
period of satellite rotation matches the period of daily earth rotation. Their orbit is around the equator, so they stay above the
bit a satellite is

gravity.

It

8K

Fig. 12-6

satellite in cir-

cular orbit close to the earth

moves tangentially

at 8 km/s.

During each second, it falls

4.9 m beneath each successive

8-km tangent.

166

12

Satellite

Motion

the equator. The moon is even farther away and

has a period of 27.3 days. The higher the orbit of a satellite, the
less its speed and the longer its period.*

same point on

Questions
1.

There are usually alternate explanations for things. Is

the following explanation valid? Satellites remain in orbit instead of falling to the earth because they are be-

yond the main

2. Satellites in

pull of earth's gravity.

close circular orbit

each second of orbit.

How can

about 4.9
during
be if the satellite does

fall

this

not get closer to the earth?

Recall from the last chapter that satellite motion was understood by Isaac Newton. He stated that at a certain speed a cannonball would circle the earth and coast indefinitely, provided
air resistance could be neglected. Newton calculated the required speed to be equivalent to 8 km/s. Since such a cannonball
speed was clearly impossible, he did not foresee people launching satellites. Newton did not consider multi-stage rockets.

12.3

Elliptical Orbits
a projectile just above the drag of the atmosphere is given a
horizontal speed somewhat greater than 8 km/s, it will overshoot a circular path and trace an oval-shaped path, an ellipse.
If

Answers
1.

2.

No, no, a thousand times no! If any moving object were beyond the pull of
gravity, it would move in a straight line and would not curve around the earth.
Satellites remain in orbit because they are being pulled by gravity, not because they are beyond it.
In each second, the satellite falls about 4.9
it

would have taken

if

beneath a straight-line tangent that

8 km/s,

it

"falls" at the

there were no gravity.

same

is

km

below the straight-line tangent

The earth's surface curves 4.9 m
long. Since the satellite moves at

rate that the earth "curves."

you continue with your study of physics and take a follow-up c ourse, y ou'll
s given by v = VGM/d and
the period T of satellite motion is given by T = 2n VcWGM where G is the universal gravitational constant (see Chapter 10), M is the mass of the earth (or
whatever body the satellite orbits), and d is the altitude of the satellite measured from the center of the earth or parent body.
'

If

learn that the speed v of a satellite in circular orbit

12.3

167

Elliptical Orbits

An ellipse is a specific curve: the closed path taken by a point

moves in such a way that the sum of its distances from two

that

fixed points (called foci)

is

constant. In the case of a satellite

is at one focus (singular of foci); the other focus is empty. An ellipse can be easily
constructed by using a pair of tacks, one at each focus, a loop of
string, and a pencil, as shown in Figure 12-7. The closer the foci
are to each other, the closer the ellipse is to a circle. When both
foci are together, the ellipse is a circle. A circle is actually a special case of an ellipse with both foci at the center.

orbiting a planet, the center of the planet

Fig. 12-7

A simple method
an

of constructing

Fig. 12-8

The shadows

of the ball are all ellipses with one focus

where the

ellipse.

ball

touches the table.

Whereas the speed of a satellite is constant in a circular orspeed varies in an elliptical orbit. When the initial speed

bit,
is

greater than 8 km/s, the satellite overshoots a circular path

and moves away from the earth, against the force of gravity. It
therefore loses speed. Like a rock thrown into the air, it slows to
a point where it no longer recedes, and it begins to fall back
toward the earth. The speed it lost in receding is regained as
it falls back toward the earth, and it finally crosses its original
path with the same speed it had initially (Figure 12-9). The procedure repeats over and over, and an ellipse is traced each cycle.
Fig. 12-9

When

8 km/s,
(left)

Elliptical orbit.

the satellite exceeds

it

and

overshoots a circle
travels

away Irom

the earth against gravity- At

its

maximum

ter)

ward

come back

the earth.

lost in
in

separation (cen-

starts to

it

The speed

going away

returning,

cycle repeats

and

is

toit

gained

(right) the

itself.

12

168

Satellite

Motion

Question

The orbital path of a satellite is shown in the sketch. In

which of the marked positions A through D does the satellite

12.4

have the greatest speed? Lowest speed?

Energy Conservation and

Satellite

Motion

Recall from Chapter 8 that an object in motion possesses kinetic

energy (KE) by virtue of its motion. An object above the earth's
surface possesses potential energy (PE) by virtue of its position.
Everywhere in its orbit, a satellite has both KE and PE with respect to the body it orbits. The sum of the KE and PE will be a
constant all through the orbit. The simplest case occurs for a satcircular orbit.

ellite in

PE+ KE

PE+KE

PE+KE

PE+KE
Fig. 12-10

The

force of gravity on the satellite

is

always toward the center of

For a satellite in circular orbit, no component of force acts

along the direction of motion. The speed, and thus the KE, cannot change.
the

at

body

it

orbits.

Answer
The satellite has its greatest speed as it whips around A.
position C. Beyond C it gains speed as it falls back to A

It

has

its

lowest speed

to repeat its cycle.

Energy Conservation and

12.4

Satellite

169

Motion

In circular orbit the distance between the body's center

the satellite does not change. This

the

lite is

means

that the

PE of the

and

satel-

same everywhere in orbit. Then, by the conservation of

KE must also be constant. In other words, the speed

energy, the
is

In elliptical orbit the situation

is

different.

Both speed and

distance vary. The PE is greatest when the satellite is farthest

away (at the apogee) and least when the satellite is closest (at the
perigee). Correspondingly, the KE will be least when the PE is
most; and the KE will be most when the PE is least. At every
point in the orbit, the sum of the KE and PE is the same.
points along the orbit there is a component of gravitational force in the direction of motion of the satellite (zero at the
apogee and perigee). This component changes the speed of the
satellite. Or we can say: (this component of force) x (distance

At

all

moved) = change

KE+PE

KE'PE

constant in any circular orbit.

Fig. 12-11

and PE

KE

The sum of

for a satellite

is

a con-

stant at all points along

an

elliptical orbit.

COMPONENT OF
FORCE DOES WORK
ON THE SATELLITE

THIS

KE. Either way we look at it, when the satellite gains altitude and moves against this component, its speed
and KE decrease. The decrease continues to the apogee. Once
past the apogee, the satellite moves in the same direction as the
component, and the speed and KE increase. The increase continues until the satellite whips past the perigee and repeats the
in

cycle.

Questions
1

The

orbital path of a satellite is shown in the sketch in

the previous question box (opposite page). In which

marked positions A through D does the satellite have the

KE? Greatest PE? Greatest total energy?

greatest
2.

How can
lite

the force of gravity change the speed of a satel-

when

it is

in

an

elliptical orbit,

but not

when

it is

in a

circular orbit?

Fig. 12-12

In elliptical orbit,

component

of force exists

along the direction of the satellite's motion. This component changes the speed and,
thus, the KE. (The perpendicular

component changes

only the direction.)

Answers
The KE is
total

2.

maximum at

energy

the

is

At any point on
to its path. If a

its

the perigee A; the

same everywhere

PE is maximum at

path, the direction of motion of a satellite

component

the apogee C; the

in the orbit.
is

always tangent

of force exists along this tangent, then the accelera-

tion of the satellite will involve a

cular orbit the gravitational force

change in speed as well as direction. In ciris always perpendicular to the direction of

motion of the satellite, just as every part of the circumference of a circle is

perpendicular to the radius. So there is no component of gravitational force
along the tangent, and only the direction of motion changes, not the speed.
But when the satellite moves in directions that are not perpendicular to the
force of gravity, as in an elliptical path, there is a component of force along the
direction of motion which changes the speed of the satellite. From a workenergy point of view, a component of force along the direction the satellite
moves does work to change its KE.

170

12

Satellite

Motion

Escape Speed

12.5

When

a payload

is

put into earth orbit by a rocket, the speed and

rection of the rocket are important. For example,

pen

di-

what would hap-

the rocket were launched vertically

and quickly achieved a

km/s? Everyone had better get out of the way, because
it would soon come crashing back at 8 km/s. To achieve orbit,
the payload must be launched horizontally at 8 km/s once above
the drag of the atmosphere. Launched only vertically, the old
saying "What tioes up must come down" becomes a sad fact
if

speed

of

ol 8

life.

isn't there some vertical speed that is sufficient to insure

what goes up will escape and not come down? The answer
is yes. Fire it at any speed greater than
.2 km/s, and it will continually leave the earth, traveling more and more slowly, but

But

that

CV>*

will never be

brought to a stop by earth gravity. Let's look at

from an energy point of view.
How much work is required to move a payload against the

this

force of earth gravity to a distance very, very far ("infinitely far")

away? We might think

distance

is infinite.

that the PE would be infinite because the

But gravity diminishes with distance via the

inverse-square law. The force of gravity is strong only close to

Most of the work done in launching a rocket, for example, occurs near the earth's surface. It turns out that the value
of PE for a 1 -kilogram mass infinitely faraway is 60 million joules
(MJ). So to put a payload infinitely far from the earth's surface
requires at least 60 MJ of energy per kilogram of load. We won't
the earth.

Fig. 12-13

The

of the rocket

Another thrust
vertical course.

moving

initial thrust

lifts it

tips

vertically.
it

When

horizontally,

from
it
it

its

is
is

boosted to the required speed

for orbit.

go through the calculation here, but a KE of 60 MJ corresponds

to a speed of 1 1 .2 km/s whatever the mass involved. This is the
value of the escape speed from the surface of the earth."
If we give a payload any more energy than 60 MJ/kg at the surface of the earth or, equivalently, any more speed than 1 1 .2 km/s,
then, neglecting air resistance, the payload will escape from the
earth never to return. As it continues outward, its PE increases
and its KE decreases. Its speed becomes less and less, though it

more-advanced physics course you would learn how the value of escape
from any planet or any body, is given bv r = V IGMId where G is the
universal gravitational constant, M is the mass of the attracting body, and d is
the distance from its center. (At the surface of the body d would simply be the
In a

speed

v,

radius of the bodv.)

Interestingly enough, this might well be called the

maximum

Any object, however far from earth, released from rest and allowed
only under the influence of the earth's gravity would not exceed

falling speed.

to fall to earth

1 1

.2

km/s.

12.5

is

Escape Speed

171

never reduced to zero. The payload outruns the gravity of the

earth.

It

escapes.

The escape speeds of various bodies in the solar system are

shown in Table 12-1 Note that the escape speed from the sun is
620 km/s at the surface of the sun. Even at a distance equaling
.

that of the earth's orbit, the escape speed from the sun

The escape speed values

is

42.5 km/s.

in the table ignore the forces exerted

by

other bodies. A projectile fired from the earth at 11.2 km/s, for
example, escapes the earth but not necessarily the moon, and
certainly not the sun. Rather than recede forever, it will take up
an orbit around the sun.
The first probe to escape the solar system, Pioneer 10, was
launched from earth in 1972 with a speed of only 15 km/s. The
escape was accomplished by directing the probe into the path of
oncoming Jupiter. It was whipped about by Jupiter's great gravitational field, picking up speed in the process
just as the speed
of a ball encountering an oncoming bat is increased when it departs from the bat. Its speed of departure from Jupiter was increased enough to exceed the sun's escape speed at the distance
of Jupiter. Pioneer 10 passed the orbit of Pluto in 1984. Unless it
collides with another body, it will wander indefinitely through
interstellar space. Like a note in a bottle cast into the sea, Pioneer 10 contains information about the earth that might be of

interest to extra-terrestrials, in

hopes that

it

will

one day wash

up and be found on some distant "seashore."

Fig. 12-14

system

in

Pioneer 10, launched from earth in 1972, escaped from the solar
1984 and is wandering in interstellar space.

172

12

Satellite

Motion

Escape Speeds at the Surface of

Bodies in the Solar System

Table 12-1

Bodv

Sun
Sun

(earth masses)

330 000
(at

(earth radii)

(km/s)

620

109

a distance

of the earth's
orbit)

Jupiter

318

23 000

42.5

11

61.0

Saturn

95.2

37.0

Neptune
Uranus

17.3

3.4

25.4

14.5

3.7

22.4

Earth

1.00

1.00

11.2

Venus

0.82

0.96

10.4

Mars
Mercury

0.11

0.53

5.2

0.05

0.38

4.3

Moon

0.01

0.27

2.4

It is

important to point out that the escape speeds

for differ-

ent bodies refer to the initial speed given by a brief thrust, after

which there is no force to assist motion. One could escape the

earth at any sustained speed more than zero, given enough time.
Suppose a rocket is going to a destination such as the moon.
If the rocket engines burn out when still close to the earth, the
rocket needs a minimum speed of 1 1 .2 km/s. But if the rocket engines can be sustained for long periods of time, the rocket could
go to the moon without ever attaining 1 1.2 km/s.
It is interesting to note that the accuracy with which an unpiloted rocket reaches its destination is not accomplished by
staying on a preplanned path or by getting back on that path if it
strays off course. No attempt is made to return the rocket to its
original path. Instead, the control center in effect asks,

"Where

now with

respect to where it ought to go? What is the best

way to get there from here, given its present situation?" With the
aid of high-speed computers, the answers to these questions are
is it

used in finding a new path. Corrective thrusters put the rocket

on this new path. This process is repeated over and over again
all

way to the goal.

there a lesson to be learned here? Suppose you find that you

the
Is

are "off course."

You may,

like the rocket, find

it

more

fruitful to

take a course that leads to your goal as best plotted from your
present position and circumstances, rather than try to get back
on the course you plotted from a previous position and under,

perhaps, different circumstances.

173

Chapter Review

12

Chapter Review

12

Concept Summary
An

earth satellite

is

4.

a projectile that

enough horizontally so that

it

falls

moves fast
around the

through the orbit.

If something is launched from earth with a
great enough energy per kilogram of mass,
it will move so fast that it will never return
gies is constant all

a.

it.

The speed of a satellite in a circular orbit is

not changed by gravity.
The speed of a satellite in an elliptical orbit
increases as the distance from the earth decreases, and vice versa.
The sum of the kinetic and potential ener-

doesn't gravitational force change the

speed of a
5.

earth rather than into

Why

What

satellite in circular orbit? (12.2)

is

the period of a satellite in close

circular orbit about the earth?

b.

the period greater or less for greater

Is

distances from the earth? (12.2)

6.

What

7.

Why

8.

a.

At what part of an elliptical orbit

speed of a satellite maximum?

b.

Where

9.

an

ellipse? (12.3)

does gravitational force change the

speed of a satellite in elliptical orbit? (12.4)

to earth.

Important Terms

is

is it

minimum?

The sum of PE and

cular orbit

is

the

(12.4)

KE for a satellite

constant.

is

Is this

sum

in a cir-

also con-

stant for a satellite in an elliptical orbit?

apogee

(12.4)

(12.4)

ellipse (12.3)

escape speed (12.5)

10.

Why does

1 1

What

focus (12.3)

perigee (12.4)
period (12.2)

the force of gravity do no work on

a satellite in circular orbit, but does do work
on a satellite in an elliptical orbit? (12.4)
will

be the fate of a projectile that is

km/s? At 12 km/s? (12.5)

fired vertically at 8

Review Questions
1

If

12.

you simply drop a stone from a position of

rest,

how

far will

it

fall

vertically in the

2.

What do
have

to

still

the distances 8000

do with a

line

and

4.9

13.

How

the rocket

is

does a satellite

in relation to the

b.

tangent to the earth's

in circular orbit

move

surface of the earth? (12.2)

a.

How

fast

would a

particle have to be

ejected from the sun to leave the solar

surface? (12.1)
3.

when

close to the earth's surface? (12.5)

If

ways and drop

fall

does most of the work done in launch-

ing a rocket take place

first

you instead move your hand sideit (throw it), how far will it
vertically in the first second? (12.1)

second?

Why

14.

system?
What speed would be needed if it started
at a distance from the sun equal to the
earth's distance from the sun? (12.5)

What
(12.5)

is

the escape speed from the

moon?

174

15.

12

Why,

Although the escape speed from the surface of the earth

is

said to be

.2

lites

km/s,

Motion

communications

satel-

that "hover motionless" above the

same

spot on earth simply crash into the earth?

couldn't a rocket with enough fuel escape at

any speed? Defend your answer.

then, don't the

Satellite

(12.5)
5.

Would you expect

the speed of a satellite in

close circular orbit about the moon to be
less than,

equal

to,

or greater than 8 km/s?

Defend your answer.

Activity

Draw an ellipse with a loop of string, two tacks,

and a pen or pencil. Try different tack spacings

6.

Why

do you suppose that a space shuttle is

it in an easterly direction (the direction in which the earth
sent into orbit by firing

for a variety of ellipses.

spins)?

Think and Explain

1

7.

can orbit at a 5-km altitude above

the moon, but not at 5 km above the earth.
satellite

an astronaut in an orbiting space shuttle

wished to drop something to earth, how
could this be accomplished?
If

Why?
8.
2.

3.

Does the speed of a satellite around the

earth depend on its mass? Its distance from
the earth? The mass of the earth?
a cannonball

is fired

gravity changes

4.

from a

tall mountain,
along its trajectory. But if it is fired fast enough to go into
circular orbit, gravity does not change its
speed at all. Why is this so?
If

If

its

speed

by virtue of only the earth's gravity?

9.

Is

it

possible for a rocket to escape from earth

without ever traveling as

all

you stopped an earth satellite dead in its

it would simply crash into the earth.

tracks,

is the maximum possible speed of impact upon the surface of the earth for a faraway object initially at rest that falls to earth

What

fast as

1 1

.2

km/s?

Explain.
10.

the earth somehow became more massive,

would the escape speed be less than, equal
to, or more than 11.2 km/s? Defend your
If

answer.

on
Which moves

faster

Circular Motion

on a merry-go-round

side rail or a horse near the inside rail?

rotating carnival ride not

raised as

shown

rotations

and revolutions.

fall

out

when

a horse near the outWhy

do the riders of a

the rotating platform

is

Figure 13-1? If you swing a tin can at the end

of a string overhead in a circle, and the string breaks, does the
can fly directly outward or does it move tangent to the circle?
Why is it that astronauts orbiting in a space shuttle float in a
weightless condition, while those who orbit in a rotating facility
will experience normal earth gravity? These questions indicate
the flavor of this chapter. We begin by distinguishing between
in

Why do the occupants of this carnival ride not

fall out when it is tipped alFig. 13-1

most vertically?

175

176

13.1

13

Circular Motion

Rotations and Revolutions

The carnival ride shown in Figure 13-1 and an ice skater doing a
pirouette each turn around an axis, the straight line about which
rotation takes place.
the motion

is

When

this axis is located

called a rotation, or a spin.

carnival ride and the skater

is

within the body,

of both the

The motion

a rotation.

When an
motion

is

object turns about an external axis, the rotational

called a revolution. Although the carnival ride rotates,

the riders along

Fig. 13-2

ting at

its

its

outer edge revolve about the axis of the device.

The phonograph record

rotates

edge revolves around the same

around

its

axis while a ladybug

sit-

axis.

The earth undergoes both types of rotational motion: it revolves around the sun once every 364 days, and it rotates around
an axis through

13.2

its

geographical poles once every 24 hours.

Rotational Speed
We

began this chapter by asking which moves with the greater

speed on a merry-go-round a horse near the outside rail or a
horse near the inside rail? Similarly, which part of a phonograph record passes under the stylus faster a musical piece at
the beginning of the record or one near the end? You will get different answers if you ask people these questions. That's because
some people will think about linear speed and some will think
about rotational speed.
Linear speed is what we have been calling simply "speed"
the distance in meters or kilometers moved per unit of time.
A point on the outside of a merry-go-round or record moves a

13.2

177

Rotational Speed

greater distance in one complete rotation than a point on the

inside. The linear speed is greater on the outside of a rotating
object than closer to the axis.
Rotational speed (sometimes called angular speed) refers to
the number of rotations or revolutions per unit of time. All parts

and phonograph record circle the

same amount of time. All parts share the
or number of rotations or revolutions per

of the rigid merry-go-round

axis of rotation in the

same

rate of rotation,

unit of time. It is common to express rotational rates in revolutions per minute (RPM).* A common phonograph record, for ex-

ample, rotates at 33^ RPM. A ladybug sitting on the surface of

the record revolves at 33^ RPM.

The entire phonograph record rotates at the same rotational speed,

but ladybugs at different distances from the center travel at different linear
speeds. A ladybug sitting twice as far from the center moves twice as fast.

Fig. 13-3

Linear speed and rotational speed are related. Have you ever
ridden on a giant, rotating, round platform in an amusement
park? The faster it turns, the faster is your linear speed. Linear
speed is directly proportional to rotational speed.
Linear speed, unlike rotational speed, depends on the distance from the center. At the very center of the rotating platform, you have no speed at all; you merely rotate. But as you approach the edge of the platform, you find yourself moving faster
and faster. Linear speed is directly proportional to distance from
the center. Move out twice as far from the center, and you move
twice as fast. Move out three times as far, and you have three
times as much linear speed. If you find yourself in any rotating
system whatever, your linear speed depends on how far you are
from the axis of rotation. When a row of people locked arm in
arm at the skating rink makes a turn, the motion of "tail-end
Charlie" is evidence of this greater speed.

'

Physicists usually describe the rate of rotation in terms of the angle turned in a

unit of time.

Then the symbol

for rotational

speed

is a>

(the

Greek

letter

omega).

178

Circular Motion

13

Questions
1.

Which

correct
to say that a child on a merry-go-round
about the rotational axis of the merry-go-round,
or to say that the child re\>olves about this axis?
is

rotates

2.

On

a particular merry-go-round, horses along the outer

rail

are located three times as far from the axis of rota-

tion as horses along the inner rail. If a

boy

sitting

on an

inner horse has a rotational speed of 4 RPM and a linear

speed of 2 m/s, what will be the rotational and linear
speeds of his sister who sits on an outer horse?

13.3

Centripetal Force
you whirl a tin can on the end of a string, you find that you
must keep pulling on the string (Figure 13-4). You pull inward
on the string to keep the can revolving over your head in a circular path. A force of some kind is required for any kind of circular motion. Any force that causes an object to follow a circular
path is called a centripetal force. Centripetal means "centerseeking," or "toward the center." The force that holds the occuII

pants safely in the rotating carnival ride (Figure 1 3- 1 ) is a centerdirected force. Without it, the motion of the occupants would be
along a straight line they would not revolve.
Centripetal force is not a new kind of force. It is simply the
name given to any force that is directed at right angles to the
path of a moving object and that tends to produce circular motion. Gravitational and electrical forces act across empty space
as centripetal forces. Gravitational force directed toward the
center of the earth holds the moon in an almost circular orbit
about the earth. Electrons that revolve about the nucleus of the

Fig. 13-4
is

The onlj

force thai

exerted on the whirling can

(neglecting gravity

is

di-

ud

toward the center of

circular motion. It is called a
rec

centripetal force.
force

is

No outward

exerted on the can.

atom are held by an

electrical force that

is

directed toward the

central nucleus.

Answers
The child
same axis

around the axis, because it is external to the child. This

within the merry-go-round itself, so the merry-go-round rotates

revolves
is

about this axis.

2.

The rotational speed

of the sister

is

also 4

RPM; her

linear speed

is

6 m/s.

Since the merrv-go-round is rigid, all horses have the same rotational speed,
but the outer horse at three times the distance from the center has three times
the linear speed.

13.4

Centripetal and Centrifugal Forces

179

Centripetal force plays the main role in the operation of a cenA familiar example is the spinning tub in an automatic
washing machine. In its spin cycle, the tub rotates at high speed.
The tub's inner wall exerts a centripetal force on the wet clothes,
trifuge.

which are forced into a circular path. The tub exerts great force
on the clothes, but the holes in the tub prevent the tub from
exerting the same force on the water in the clothes. The water
therefore escapes. It is important to note that a force acts on the
clothes, not the water. No force causes the water to fly out. The
water will do that anyway, as it tends to move by inertia in a
straight-line path (Newton's first law) unless acted on by a centripetal or any other force. So interestingly enough, the clothes
are forced away from the water, and not the other way around.
When an automobile rounds a corner, the sideways friction
between the tires and the road provides the centripetal force
that holds the car on a curved path (Figure 13-6). If the force of
friction is not great enough, the car fails to make the curve and
the tires slide sideways. The car skids.

CENTER OF CURVATURE

CENTRIPETAL FORCE

Fig. 13-6

(Left)

For the car to go around a curve, there must be sufficient

friction to provide the required centripetal force. (Right) If the force of friction
is

not great enough, skidding occurs.

13.4

Centripetal

and Centrifugal Forces

In the preceding examples, circular motion

was described

as

caused by a center-directed force. Sometimes an outward force

is attributed to circular motion. This outward force is called centrifugal force.* Centrifugal means "center-fleeing," or "away from
the center." In the case of the whirling can,

it is

common

mis-

Both centrifugal and centripetal forces depend on the mass m, tangential

speed v, and radius of curvature rof the circularly moving object. If you take a
follow-up physics course, you'll learn how the exact relationship is F = mv 2 /r.
*

180

13

Circular Motion

conception to state that a centrifugal force pulls outward on the

can. If the string holding the whirling can breaks (Figure 13-7),
it is commonly stated that a centrifugal force pulls the can from
its circular path. But the fact is that when the string breaks, the
can goes off in a tangent straight-line path because no force acts
on it. We illustrate this further with another example.
Suppose you are the passenger of a car that suddenly stops
short. You pitch forward against the dashboard. When this happens, you don't say that something forced you forward. You know
that you pitched forward because of the absence of a force, which
a seat belt would have provided. Similarly, if you are in a car
that rounds a sharp corner to the left, you tend to pitch outward
to the right
not because of some outward or centrifugal force,
but because there is no centripetal force holding you in circular

Fig. 13-7

When

the string

breaks, the whirling can

moves

in a straight line, tan-

gent to

not outward from

the center of

path.

its

circular

motion (as a seat belt provides). The idea that a centrifugal force
bangs you against the car door is a misconception.
So when you swing a tin can in a circular path, there is no
force pulling the can outward. Only the pull by the string acts on
the can and pulls the can inward. The outward force is on the
string, not on the can.

-D
Fig. 13-8
ity) is

The only

force that

is

exerted on the whirling can (neglecting grav-

directed toward the center of circular motion. This

No outward

is

a centripetal force.

force acts on the can.

Now suppose there is a ladybug inside the whirling can (Figure 13-9). The can presses against the bug's feet and provides the
centripetal force that holds it in a circular path. The ladybug in
turn presses against the floor of the can, but (neglecting gravity)
the only force exerted on the ladybug is the force of the can on its
feet. From our outside stationary frame of reference, we see there
is no centrifugal force exerted on the ladybug, just as there was
no centrifugal force exerted on the person who lurched against
the car door. The "centrifugal-force effect" is attributed not to
any

real force but to inertia

the tendency of the moving body

to follow a straight-line path.

But

try telling that to the ladybug!

--^
13.5

Centrifugal Force in a Rotating Reference

Frame

CENTRIPETAL FORCE

181

The can provides

Fig. 13-9

the centripetal force neces-

sary to hold the ladybug in a

circular path.

13.5

Centrifugal Force in a Rotating Reference Frame

Our view of nature is very much influenced by the frame of reference from which we view it. When we sit on the seat of a fastmoving train, we have no speed at all relative to the train but an
appreciable speed relative to the reference frame of the stationary ground outside. We have just seen that in a nonrotating reference frame, the force that holds an object in circular motion is
a centripetal force. For the ladybug, the bottom of the can exerts
a force on its feet. No other force was acting on the ladybug.
But nature seen from the reference frame of the rotating system is different. In the rotating frame, in addition to the force of
the can on the ladybug's feet, there is a centrifugal force that is
exerted on the ladybug. Centrifugal force in a rotating reference
frame is a force in its own right, as real as the pull of gravity.
However, there is a fundamental difference: Gravitational force
is an interaction between one mass and another. The gravity we
experience is our interaction with the earth. But centrifugal force
in the rotating frame is not part of an interaction. It is like gravity, but with nothing pulling. Nothing produces it; it is a result of
rotation. For this reason, physicists rank it as a fictitious force
and not a real force like gravity, electromagnetism, and the nuclear forces. Nevertheless, to observers who are in a rotating system, centrifugal force seems to be, and is interpreted to be, a
very real force. Just as from the surface of the earth we find gravity ever present, so within a rotating system centrifugal force is
ever present.

CENTRIFU6AL FORCE

Fig. 13-10

From

the refer-

ence frame of the ladybug inside the whirling can, it is

being held to the bottom of
the can by a force that

rected

away from

of circular

bug

outward force a
which is as

centrifugal force,
it

di-

motion. The lady-

calls this

real to

is

the center

as gravity.

182

13

Circular Motion

Questions
1

A heavy

iron ball is attached by a spring to a rotating

platform, as shown in the sketch. Two observers, one in
the rotating frame and one on the ground at rest, observe

its motion. Which observer sees the ball being pulled

outward, stretching the spring? Which sees the spring

pulling the ball in a circle?

The spring

when

in the sketch is

the ball

is

observed to stretch 10

midway between

the rotational axis

the outer edge of the circular platform.

port

is

moved

so that the ball

will the spring be stretched

same

13.6

10

is

more

If

cm
and

the spring sup-

directly over the edge,

than, less than, or the

cm?

Simulated Gravity

Consider a colony of ladybugs living inside a bicycle tire the

old-fashioned balloon kind that has plenty of room inside. If we
toss the bicycle wheel through the air or drop it from an airplane high in the sky, the ladybugs will be in a weightless condition. They will float freely while the wheel is in free fall. Now
spin the wheel. The ladybugs will feel themselves pressed to the
outer part of the inner surface. If the wheel is spun not too fast
and not too slowly but just right, the ladybugs will be provided

Answer
1

The observer

frame of the rotating platform states that a cenball, which stretches the spring.
The observer in the rest frame states that a centripetal force supplied by the
stretched spring pulls the ball into a circle along with the rotating platform.
(The rest-frame observer can state in addition that the reaction to this centripetal force is the ball pulling outward on the spring. The rotating observer,
interestingly enough, can state no reaction counterpart to the centrifugal
in the reference

trifugal force pulls radially

outward on the

force.)
2.

have twice the linear speed at twice the distance from the
The greater the speed, the greater will be the centripetal/
centrifugal force (which will also be twice, and stretch the spring 20 cm). (We
will see in Chapter 18 that the stretch of a spring is directly proportional to

The

ball will

rotational axis.

the applied force.)

13.6

183

Simulated Gravity

with simulated gravity that will feel like the gravity to which
they are accustomed. Gravity is simulated by centrifugal force.
The "down" direction to the ladybugs will be what we would
call "radially outward," away from the center of the wheel.

Fig. 13-12

between the

The interaction
man and the

floor as seen at rest outside

the rotating system.

Fig. 13-11

If

the spinning wheel freely

falls,

the ladybugs inside will experi-

ence a centrifugal force that feels like gravity when the wheel spins at the appropriate rate. To the occupants, the direction "up" is toward the center of the
wheel and "down" is radially outward.

Today we all live on the outer surface of this spherical planet

and are held here by gravity. The planet has been the cradle of
humankind. But we will not stay in the cradle forever. We are
becoming a spacefaring people. Many people in the years ahead
will likely live in huge lazily-rotating habitats in space and will
be held to the inner surfaces by centrifugal force. The rotating
habitats will provide a simulated gravity so that the human body
can function normally.
Occupants of a space shuttle are weightless because they lack
a support force. Future space travelers need not be subject to
weightlessness, however. A rotating space habitat for humans,
like the rotating bicycle wheel for the ladybugs, can effectively
supply a support force and neatly simulate gravity.
Structures of small diameter would have to rotate at high
rates to provide a simulated gravitational acceleration of lg.
Sensitive and delicate organs in our middle ears sense rotation.
Although there appears to be no difficulty at a single revolution
per minute (RPM) or so, many people find difficulty in adjusting
to rates greater than 2 or 3 RPM (although some easily adapt
to 10 or so RPM). To simulate normal earth gravity at 1 RPM
requires a large structure
one 2 km in diameter. This is an
immense structure, compared to today's space shuttle vehicles.
Economics will probably dictate the size of the first inhabited
structures. They are likely to be small and not rotate at all. The
inhabitants will adjust to living in a weightless environment.

Larger, rotating habitats will likely follow later.

presses against the

tion)

The

man

floor
(ac-

and the man presses

back on the floor (reaction).

The only force exerted on the
man is by the floor. It is directed toward the center and
is

a centripetal force.

Fig. 13-13

As seen from

in-

side the rotating system, in

addition to the man-floor

interaction there
gal force exerted
at his center of

is

a centrifu-

on the

mass.

It

man
seems

as real as gravity. Yet, unlike

gravity, it has no reaction
counterpart
there is nothing
out there that he can pull
back on. Centrifugal force is

not part of an interaction, but

results

from rotation.

It is

therefore called a fictitious

force.

184

13

Fig. 13-14

A NASA depiction

Circular Motion

of a rotational space colony.

If the structure rotates so that inhabitants on the inside of the

outer edge experience lg, then half-way to the axis they would
experience 0.5g. At the axis itself they would experience weightlessness at Og. The variety of fractions of g possible from the rim
of a rotating space habitat holds promise for a most different
and (at this writing) yet unexperienced environment. We would
be able to perform ballet at 0.5g; diving and acrobatics at 0.2g
and lower-g states; three-dimensional soccer and new sports not
yet conceived in very-low-g states. People will be exploring possibilities never before available to them. This time of transition
from our earthly cradle to new vistas is an exciting time in which
to live
especially for those who will be prepared to play a role
in these new adventures.

At the risk of stating the obvious, this preparation can well begin by taking

your study of physics very seriously.

Chapter Review

13

13

185

Chapter Review

Concept Summary
An

object rotates

inside

itself; it

axis outside

5.

it

turns around an axis

when

it

turns around an

The rotational speed

is

the

number of rota-

made per

6.

unit of time.

A centripetal force pulls objects toward a center.

An object moving in a circle is acted on by
object

moves

in a circle, there

7.

8.

a rotating frame of reference,

which can simulate

an inward force or an outward force

on the clothes during the spin cycle
of an automatic washer? (13.3)
Is

it

When

a car makes a turn, do seat belts provide you with a centripetal force or a centrifugal force? (13.4)

there seems to be an outwardly directed centrifugal force,

the center? (13.3)

is

the circle.

From within

the direction of

that acts

no force pushing the object outward from

is

Does the force that holds the riders on the

carnival ride in Figure 13-1 act coward or

away from

a centripetal force.

When an

what

the force that acts on the can? (13.3)

itself.

tions or revolutions

whirl a can at the end of a string

in a circular path,

when

revolves

When you

gravity.
9.

If

the string breaks that holds a whirling can

in its circular path,

it

Important Terms

to

move

what kind of force causes

in a straight-line path

centrip-

What law

or no force?

etal, centrifugal,

of

physics supports your answer? (13.4)

axis (13.1)

centrifugal force (13.4)

centripetal force (13.3)

10.

Identify the action

and reaction

forces in the

linear speed (13.2)

interaction between the ladybug

revolution (13.1)
rotation (13.1)

whirling can of Figure 13-9.(13.5)

rotational speed (13.2)

1 1

Can you name

and the

and reaction forces

which one force is the
acting on an empty tin can

the action

in the interaction in

centrifugal force

Review Questions
1.

Distinguish between a rotation and a revo-

12.

Does a child on a merry-go-round revolve or

rotate around the merry-go-round's axis?

into a circle?

Why

centrifugal force in a rotating frame

is

not? (13.5)

called a "fictitious force?" (13.5)

lution. (13.1)

2.

Why or why

swung

13.

How can gravity be simulated

space station? (13.6)

in

an orbiting

(13.1)
14.
3.

Distinguish between linear speed and rota-

Why

will orbiting

late gravity likely

space stations that simube large structures? (13.6)

tional speed. (13.2)

15.
4.

What

is

the relationship of linear speed to

rotational speed? (13.2)

How will the values of g vary at different

distances from the hub of a rotating space
station? (13.6)

186

Activity
Hall

6.

and swing

a bucket of water

till

circular path overhead.

Why

it

makes

a turn on a level road,

between the

tires

ply the centripetal force.

doesn't the water

On

road in turn, pushes back on the car to produce a "normal force." Make a sketch and

show how

the edge of a

halfway between the axis and

phonograph record. What will

happen

linear speed

A ladybug
to

its

doubled?
and b?

(b)

It

sits at

if

(a) the

RPM rate

follow

if

you

fell off

A motorcyclist is able to ride on the vertical

wall of a bowl-shaped track, as shown in FigIs

that acts

it

of the

A space habitat

rotates so as to provide an
acceleration of \g for its inhabitants. If it is
rotated at a greater angular speed, how will

the acceleration for the inhabitants differ?

the edge of a rotating merry-go-round?

ure A.

component

the edge? (c) Both a

7.

What path would you

the horizontal

normal force decreases (or eliminates) the

friction force. (Can you see that if a curve is
banked at an angle to match a certain speed,
no friction occurs?)

sits

is

3.

a car

and the road supa banked road,

a car pushes into the road as it turns. The

Think and Explain

2.

When

friction

in

spill?

Circular Motion

13

8.

Explain why the faster the earth spins, the

less a person weighs, whereas the faster
a space colony spins, the more a person
weighs.

9.

Consider a too-small space habitat that consists of a rotating sphere of radius 4 m. If a

man standing inside is 2 m tall and his feet

centripetal or centrifugal force

on the motorcvcle?

what is the acceleration at the elevation of his head? (Do you see why rotating
are at lg,

space structures are designed to be large?)

Fig.

A
10.

4.

When

a gliding seagull suddenly turns in

flight,

what

is

force that acts

5.

its

Occupants
feel

the source of the centripetal

occupants

on

shuttle

it?

Does centripetal force do work on a rotating

object? Defend your answer.

in a single

space shuttle in orbit

weightless. Describe a

scheme whereby

in a pair of shuttles (or

even one

and a large massive object such as

an asteroid) would be able to use a long cable to continually experience a comfortable

normal earth

gravity.

Rotational Mechanics

Push on an object that

Some

objects will

is

free to

move, and you

move without

rotating,

set

some

it

in

motion.

rotate without

moving, and others do both. For example, a kicked football often

tumbles end over end. What determines whether an object will
rotate when a force acts on it? This chapter is about the factors

We will see that they underlie most of the

techniques used by gymnasts, ice skaters, and divers.
that affect rotation.

Torque

Every time you open a door, turn on a water faucet, or tighten a

nut with a wrench, you exert a turning force. This turning force
produces a torque (rhymes with fork). Torque is different from
force. If you want to make an object move, apply a force. Forces
tend to make things accelerate. If you want to make an object
turn or rotate, apply a torque. Torques produce rotation.

Fig. 14-1

A torque produces rotation.

187

188

14

FORCE
LEVER

ARM

Rotational Mechanics

A torque is produced when a force is applied with "leverage."

A doorknob is placed far away from the turning axis at its hinges
to provide more leverage when you push or pull on the doorknob. The direction of your applied force is important. In opening a door, you'd never push or pull the doorknob sideways to
make the door turn. You push perpendicular to the plane of the
door. Experience has taught you that a perpendicular push or
pull gives the greatest

amount

of rotation for the least

amount of

effort.

HIN6E

Fig. 14-2

When

dicular force
lever

arm

is

is

a perpen-

applied, the

the distance be-

tween the doorknob and the

edge with the hinges.

If you have ever used both a short wrench and a long wrench,
you also know that less effort and more leverage results with a
long handle. When the force is perpendicular, the distance from
the turning axis to the point of contact is called the lever arm.
If the force is not at a right angle to the lever arm, then only
the perpendicular component of the force, F x will contribute to
the torque. Torque is defined as:*
,

torque = force x x lever

arm

So the same torque can be produced by using a large force with

a short lever arm, or a small force with a long lever arm. Greater

torques are produced

when both

the force

and

lever

arm

are

large.

FORCE
O

Fig. 14-3

TORQUE

FORCE

The doorknobs on

these doors are placed at the

What do physics types

say about the practicality
center.

EVEN MORE TORQUE

of this?

Fig. 14-4

Although the magnitudes of the applied forces are the same in each
Only the component of forces perpendicular to

case, the torques are different.

the lever

arm

contributes to torque.

The units of torque are newton-meters. Work is also measured in newtonmeters (the same as joules), but work and torque are not the same. With work,
the force moves along (parallel to) the distance, while with torque the force
moves perpendicular to the distance.

^
Balanced Torques

14.2

189

Questions
1

a doorknob were placed in the center of a door rather

than at the edge, how much more force would be needed
to produce the same torque for opening the door?

If

you cannot exert enough torque to turn a stubborn

would more torque be produced if you fastened a
length of rope to the wrench handle as shown?

2. If

bolt,

[uJ

Balanced Torques

Torques are intuitively familiar to youngsters playing on a seesaw. Children can balance a seesaw even when their weights are
unequal. Weight alone does not produce rotation. Torque does,
and children soon learn that the distance they sit from the pivot
point is every bit as important as weight (Figure 14-5). The
heavier boy sits a short distance from the fulcrum (turning axis)
while the lighter girl sits farther away. Balance is achieved if the
torque that tends to produce clockwise rotation by the boy equals
the torque that tends to produce counterclockwise rotation by
the

girl.

Scale balances that work with sliding weights are based on

balanced torques, not balanced masses. The sliding weights are
adjusted until the counterclockwise torque just balances the
clockwise torque. Then the arm remains horizontal.

Fig. 14-5

A pair

can balance each

Answers
1

Twice as much, because the lever arm would be half as long in the center
of the door. Mathematically, 2F x (dll) = F x d, where F is the applied force,
d is the distance from the hinges to the edge of the door, and dll is the distance
to the center.

2.

No, because the lever arm is the same. To increase the lever arm, a better idea
would be to use a pipe that extends upward.

Fig. 14-6
This scale
balanced torques.

relies <>n

190

14

Rotational Mechanics

Computational Example
Suppose that a meter stick is supported at the center, and a
20-N block is hung at the 20-cm mark. Another block of unknown weight just balances the system when it is hung at
the 90-cm mark. What is the weight of the second block?

20

60

40

80

'M
20

100

?N

You can compute the unknown weight by applying the

The block of unknown weight
tends to rotate the system of blocks and stick clockwise,
and the 20-N block tends to rotate the system counterclockwise. The two torques are equal when the system is in
principle of balanced torques.

balance:

clockwise torque = counterclockwise torque

(FJh,

(FL d) cnun tcr clockwise

important to note that the lever arm for the unknown

is 40 cm, because the distance between the 90-cm
mark and the pivot point at the 50-cm is 40 cm. The lever
arm for the 20-N block is 30 cm, because its distance from
the midpoint of the stick is 30 cm. When known values are
substituted into the last equation, it becomes:
It is

weight

Fx

x (40 cm) = (20 N) x (30 cm)

The unknown weight

Fx

is

therefore

(20 N) x (30
(40

cm)

15

cm)

The answer makes sense. You can tell that the weight
must be less than 20 N because its lever arm is greater than
that of the block of

(40

known

cm) * (30 cm), or 4/3,

weight. In

that of the

fact, the lever

first

arm

is

block, so the weight

3/4 that of the first block. Anytime you use physics to compute something, be sure to consider whether or not your
answer makes sense. Computation without comprehension
is not conceptual physics!
is

1
14.3

191

Torque and Center of Gravity

you attempt to lean over and touch your toes while standing
with your back and heels to the wall, you soon find yourself rotating. This is because of a torque. Recall from Chapter 9 that if
there is no base of support beneath the center of gravity (CG), an
object will topple. When the area bounded by your feet is not
beneath your CG, there is a torque. Now you can see that the
cause of toppling is the presence of a torque.
If

V\

The L-shape bracket will topple because of a torque. Similarly, when

you stand with your back and heels to the wall and try to touch your toes, a
torque is produced when your CG extends beyond your feet.
Fig. 14-7

This chapter began by asking what determines whether or not

tumble end over end when kicked. The answer involves CG, force, and torque. You know that a force is required
to launch any projectile, whether it be a football or a Frisbee. If
a football will

the direction of the force is through the CG of the projectile, all

the force can do is move the object as a whole. There will be no

torque to turn the projectile around its own CG. If the force, inis directed "off center," then in addition to motion of the

stead,

CG, the projectile will rotate about its CG.

For example, when you throw a ball and impart spin to it, or
when you launch a Frisbee, a force must be applied to the edge
of the object. This produces a torque which adds rotation to the
projectile. If you wish to kick a football so that it sails through
the air without tumbling, kick it in the middle (Figure 14-8 top).
If you want it to tumble end over end in its trajectory, kick it
below the middle (Figure 14-8 bottom) to impart torque as well
as force to the ball.

Fig. 14-8

If

the football

is

kicked in line with the center

of gravity, it will move without rotating. If it is kicked

above or below
gravity,

it

its

center of

will also rotate

192

14

Rotational Mechanics

Rotational Inertia

14.4

Chapter 3 you learned about the law of inertia: an object at

and an object in motion tends to remain moving in a straight line. There is a similar law for rotation:
In

rest tends to stay at rest,

An object rotating about an

axis tends to keep on ro-

tating about that axis.

The resistance of an object to changes in its rotational state of

motion is called rotational inertia." Rotating objects tend to keep
rotating, while nonrotating objects tend to stay nonrotating.
Most

Fig. 14-9

of the

mass

of this industrial flywheel

is

concentrated along its edge so

that it has a greater rotational
inertia.

tation,

than

Once
it is

if its

it is

set into ro-

harder

to stop

mass were nearer

the axis.

Just as it takes a force to change the linear state of motion of

an object, a torque is required to change the rotational state of
motion of an object. In the absence of a net torque, a rotating top
keeps rotating, while a nonrotating top stays nonrotating.
Like inertia in the linear sense, rotational inertia depends on
the mass of the object. But unlike inertia, rotational inertia depends on the distribution of the mass. The greater the distance
between the bulk of the mass of an object and the axis about

which rotation takes place, the greater the rotational

Fig. 14-11

DIFFICULT
Fig. 14-10

TO ROTATE

By holding

a long pole, the tightrope walker increases his rotahim to resist rotation and gives him plenty of time to

tional inertia. This allows

re-adjust his CG.

Rotational inertia

depends on the distance of

mass from the axis

Rotational inertia

is

inertia.

sometimes called moment of inertia.

14.4

193

Rotational Inertia

A long pendulum has a greater rotational inertia than a short

It's "more lazy," so it swings to and fro more slowly
than a short pendulum. Hang a weight from a string. When the
string is short, the pendulum swings to and fro more frequently
than when it is long (Figure 14-12). Likewise with the legs of people and animals when their legs are allowed to swing freely. Longpendulum.

legged animals such as giraffes, horses, and ostriches normally

run with a slower gait than hippos, dachshunds and mice.
It is important to note that the rotational inertia of an object
is not necessarily a fixed quantity. It is greater when the mass
within the object is extended from the axis of rotation. You can
try this with your outstretched legs. Allow your outstretched leg
to swing to and fro from the hip like a pendulum. Now do the
same with your leg bent. In the bent position it swings to and fro
more frequently. To reduce the rotational inertia of your legs,
simply bend them. That is an important reason for running with
your legs bent. They are easier to swing to and fro.

Fig. 14-12

The short pen-

dulum will swing to and fro

more frequently than the long
pendulum.

r
Fig. 14-13

For similar mass

distributions, short legs have

less rotational inertia

than

long legs. Animals with short

Fig. 14-14

You bend vour

legs

when you run

to

reduce their rotational inertia.

Formulas for Rotational Inertia

When all the mass m of an object is

concentrated at the same

distance r from a rotational axis (as in a simple pendulum bob
swinging on a string about its pivot point, or a thin wheel turning about its center), then the rotational inertia / = mr 1 When
.

mass is more spread out, as in your leg, the rotational inertia

is less and the formula is different. Figure 14-15 compares rotational inertias for various shapes and axes. (It is not important
for you to learn these values, but you can see how they vary with
the shape and axis.)
the

legs more easily run with

quicker strides than animals
with long legs.

>

194

Rotational Mechanics

14

HOOP ABOUT NORMAL AXIS

SIMPLE

PENDULUM^.

ABOUT DIAMETER

..

SOLID
CYLINDER-

I=zm\- 7

STICK ABOUT

SOLID SPHERE
ABOUT CO

END

I=/3 mL
Fig. 14-15

l=

/5 rr\r

Rotational inertias of various objects each of mass m, about indi-

cated axes.

Rolling
Which will

A solid cylinder
down an incline faster

Fig. 14-16
rolls

than a hollow one, whether or

not they ha\e the
or diameter.

same mass

roll down an incline with greater acceleration

hollow or a solid cylinder of the same mass and radius? The answer is, the object with the smaller rotational inertia. Why? Because the object with the greater rotational inertia will require
more time to get rolling. Remember that inertia of any kind is a
measure of "laziness." Which has the greater rotational inertia
the hollow or the solid cylinder? The answer is, the one
with its mass concentrated farthest from the axis of rotation
the hollow cylinder. So a hollow cylinder has a greater rotational inertia than a solid cylinder of the same radius and mass
and will be more "lazy" in gaining speed. The solid cylinder will
roll with greater acceleration.
Interestingly enough, any solid cylinder will roll down an incline with greater acceleration than any hollow cylinder, regardless of mass or diameter. A hollow cylinder has more "laziness
per mass" than a solid cylinder.
Objects of the same shape but different sizes will accelerate equally when rolled down an incline. You should see this for
yourself and experiment. If started together, the smaller shape,
whether it be ball, disk, or hoop, will rotate more times than a
larger one, but both will reach the bottom of the incline in the

same

time.

14.5

Rotational Inertia and Gymnastics

195

Questions
1

Why

is

the rotational inertia of your leg swinging from

your hip
2.

A heavy

less

when your

iron cylinder

lar in shape, roll

leg is bent?

and a

down an

light

wooden cylinder, simiWhich will have the

incline.

greater acceleration?

Rotational Inertia

14.5

and Gymnastics

Consider the whole human body. You can rotate freely and stably about three principal axes of rotation (Figure 14-17). These
axes are each at right angles to the others (mutually perpendicular). Each axis coincides with a line of symmetry of the body
and passes through the center of gravity. The rotational inertia
of the body differs about each axis.

L0N6ITUDINAL AXIS

MEDIAN AXIS

Fig. 14-17

1.

The human body's principal axes

Answers
The rotational

inertia of

to the axis of rotation.

2.

The cylinders have

any object

Can you

is

less

of rotation.

when

different masses, but the

so both will accelerate equally

its

mass

is

concentrated closer

see that a bent leg satisfies this requirement?

down

same

rotational inertia per mass,

the incline. Their different masses make-

no difference, just as the different masses of objects

in free fall

acceleration. Similarly, different masses suspended from the

string

(pendulum bobs)

mass

will also roll

affect

of

masses down
equal accelerations. Similar round shapes

will accelerate equally. Sliding different

friction-free inclines will result in

of different

do not

same length

down

inclines with equal accelerations.

14

196
Rotational inertia
is

Rotational Mechanics

about the longitudinal axis, which

is because most of the mass
axis. A rotation of your body about

least

is

the vertical head-to-toe axis. This

is concentrated along this

your longitudinal axis is therefore the easiest rotation

to per-

form. An ice skater executes this type rotation when going into a
spin. Rotational inertia is increased by simply extending a leg or
the arms (Figure 14-18). The rotational inertia when both arms
are extended is found to be about three times greater than with
arms tucked in, so if you go into a spin with outstretched arms,

you will triple your spin rate when you draw your arms in. With
your leg extended as well, you can vary your spin rate by as much
as 6 times.

Rotations about the longitudinal axis. Rotational inertia in posiabout 5 or 6 times as great as in position a. so spinning in position d
and then changing to position a will increase the spin rate about 5 or 6 times.
Fig. 14-18

tion

is

You rotate about your transverse axis when you perform a

somersault or a flip. Figure 14-19 shows positions of different rotational inertia, starting with the least (when your arms and legs
are drawn inward in the tuck position) to the greatest (when
vour arms and legs are fully extended in a line). The relative
magnitudes of rotational inertia stated in the caption are with
respect to the body's center of gravity.

Fig. 14-19

Rotations about the transverse axis. Rotational inertia is least in a,

.5 times as great in b, 3 times as great in c, and
It is about

the tuck position.

5 times as great in d.

Rotational Inertia and Gymnastics

14.5

197

Rotational inertia is greater when the axis is through the

hands, such as when doing a somersault on the floor or swinging
from a horizontal bar with your body fully extended. The rotational inertia of a gymnast is up to 20 times as great when
swinging in fully extended position from a horizontal bar as
after dismount when she somersaults in the tuck position. Rotation transfers from one axis to another, from the bar to her
center of gravity, and she automatically increases her rate of rotation by up to 20 times. This is how she is able to complete two
or three somersaults before contact with the ground.

Fig. 14-20
axis.

When

The rotational inertia of a body is with respect to the rotational

gymnast pivots about the bar (a), she has a greater rotational

the

inertia than

when she

spins freely about her center of gravity

The third axis of rotation

back

axis, or

medial

for the

(b).

human body is the front-tocommon axis of rotation

axis. This is a less

and

is used in executing a cartwheel. Like rotations about the

other axes, rotational inertia can be varied with different body
configurations.

Question

Why is the rotational inertia greater for a gymnast when

she swings by her hands from a horizontal bar with her
body fully extended than when she executes a somersault
with her body fully extended?
Answer

Fig. 14-21

WhenJuliannc

When

McNamara

pivots around the

gymnast pivots about the bar, her mass is concentrated far from the
axis of rotation. Her rotational inertia is therefore greater. When she somei
saults. her mass is concentrated close to ihc axis because she is rotating about

full y

her center

inertia

the

ol gravity.

horizontal bar with her bod)

extended, her rotational

is

greatest.

198

14

^6~j Angular
14.6

Rotational Mechanics

Momentum
Anything that rotates, whether it be a colony in space, a cylinder
rolling down an incline, or an acrobat doing a somersault, keeps
on rotating until something stops it. A rotating object has an "inertia of rotation." Recall from Chapter 7 that all moving objects
have "inertia of motion," or momentum, which is the product of
mass and velocity. To be clear, let us call this kind of momentum
linear

momentum.

objects

is

Similarly, the "inertia of rotation" of rotating

called angular

momentum.

Like linear momentum, angular momentum is a vector quantity and has direction as well as a magnitude. When a direction
is assigned to rotational speed, we call it rotational velocity. Rotational velocity is a vector whose magnitude is the rotational
speed. (By convention, the rotational velocity vector, as well as

Fig. 14-22

A phonograph
more angular

turntable has

the angular momentum vector, have the same direction and

along the axis of rotation.)
Angular momentum is defined as the product of rotational
ertia

and rotational

momentum when

it is turning
than at 33; RPM.
It has even more angular momentum if a load is placed on

RPM

at 45

it

so

its

rotational inertia

angular
It is

momentum =

linear

in-

velocity.

rotational inertia x rotational velocity

the counterpart of linear

is

lie

momentum:

momentum = mass

x velocity

greater.

we won't treat the vector nature of momentum

even of torque, which also is a vector) except to acknowledge
the remarkable action of the gyroscope. The rotating bicycle
wheel in Figure 14-24 shows what happens when a torque by
In this book,

(or

Fig. 14-23

A gyroscope.

Low-friction swivels can be

turned in any direction without exerting a torque on the

whirling gyroscope. A- a result,

it

same

stays pointed

direction.

ir:

the

Angular momentum keeps the wheel axle horizontal when a torque

supplied by earth's gravity acts on it. Instead of toppling, the torque causes the
wheel to slowly precess about a vertical axis.
Fig. 14-24

Conservation of Angular

14.7

Momentum

199

4ir
change the direction of its angular moalong the wheel's axle). The pull of gravity
that acts to topple the wheel over and change its rotational axis
causes it instead to precess in a circular path about a vertical
axis. You must do this yourself to fully believe it. Full understanding will likely not come until a later time.
For the case of an object that is small compared to the radial
distance to its axis of rotation, such as a tin can swinging from
a long string or a planet orbiting around the sun, the angular
momentum is simply equal to the magnitude of its linear momentum, mv, multiplied by the radial distance, r. In shorthand
earth's gravity acts to

mentum (which

is

Fig. 14-25

An

centrated mass

-6m

object of con-

whirling in

a circular path of radius r

with a speed v has angular

notation,

angular

14.7

__-!-

momentum

momentum = mvr

Conservation of Angular

mvr.

Momentum

We know that an external net force is required to change the linear momentum of an object. The rotational counterpart is: an
is required to change the angular momenan object. We restate Newton's first law of inertia for rotating systems in terms of angular momentum:

external net torque

tum

of

An object or system of objects will maintain its angular

momentum

unless acted upon by an unbalanced ex-

ternal torque.

We

know that it is not easy to balance on a bicycle that is

The wheels have no angular momentum. If our center of
gravity is not above a point of support, a slight torque is produced
and we fall over. When the bicycle is moving, however, the wheels
have angular momentum. To tip the wheels requires a greater
all

at rest.

torque than before because the direction of the angular momentum would change. This makes the bicycle easy to balance when
it is

moving.
Fig. 14-26

Bicycles are easier

when their wheels

have angular momentum.
to balance

200

14

Rotational Mechanics

momentum

of any system is conserved if no

on the system, angular momentum is consystems in rotation. The law of conservation of an-

Just as the linear

net forces are acting

served for
gular momentum states:

no unbalanced external torque acts on a rotating

s\stem, the angular momentum of that system is
constant.
If

This

means

that the product of rotational inertia

velocity at one time will be the

same

and rotational

as at any other time.

illustrating angular momentum conFigure 14-27. The man stands on a lowli id ion turntable with weights extended. His rotational inertia,
with the help ol the extended weights, is relatively large in this
position. As he slowly turns, his angular momentum is the product of his rotational inertia and rotational velocity. When he pulls
the weights inward, the rotational inertia of himself and the
weights is considerably reduced. What is the result? His rotational speed increases! This example is best appreciated by the
turning person who feels changes in rotational speed that seem
to be mysterious. But it's straight physics! This procedure is used
by a figure skater who starts to whirl with his arms and perhaps
a leg extended, and then draws his arms and leg in to obtain a
greater rotational speed. Whenever a rotating body contracts,

An

interesting

servation

its

is

example

shown

in

rotational speed increases.

Fig. 14-27

Conservation

of

angular

momentum. When

and the whirling weights inward, he decreases

rotational speed correspondingly increases.

the

man

pulls his

his rotational inertia,

and

arms
his

Momentum

Conservation of Angular

14.7

201

Similarly, when a gymnast is spinning freely in the absence of

unbalanced torques on his body, his angular momentum does
not change. However, as we have already seen, he can change his
rotational speed by simply making variations in his rotational
inertia. He does this by moving some part of his body toward or
away from his axis of rotation.
Fig. 14-28

Rotational speed

controlled by variations in
the body's rotational inertia

is

as angular

momentum

is

conserved during a forward

somersault.

is held upside down and dropped, it is able to execute

and land upright even if it has no initial angular momentum. Zero-angular-momentum twists and turns are performed
by turning one part of the body against the other. While falling,
the cat rearranges its limbs and tail to change its rotational inertia. Repeated reorientations of the body configuration result in
the head and tail rotating one way and the feet the other, so that

If

a cat

a twist

the feet are

downward when

During

maneuver

zero.

this

the cat strikes the ground.

the total angular

When it is over, the cat

is

at rest. This

momentum

remains

manuever rotates

the

body through an angle, but it does not create continuing rotation. To do so would violate angular momentum conservation.

Humans can perform

similar twists without difficulty, though

not as fast as a cat can. Astronauts have learned to make zeroangular-momentum rotations about any principal axis. They do
these to orient their bodies in any preferred direction when floating freely in space.

The law of momentum conservation is seen in the planetary

motions and in the shape of the galaxies. It will be a fact of everyday life to inhabitants of rotating space habitats who will head
for these distant places.

Fig. 14-29
<jI

,i

Time-lapse photo

tailing cat.

202

14

Chapter Review

14

Concept Summary
Torque

for

4.

an object being turned

arm and

of the lever

the

is

the product

component

of the force

perpendicular to the lever arm.

When balanced torques act on an object,
there

is

How

do clockwise and counterclockwise

torques compare when a svstem is balanced?
(14.2)

5.

terms of center of gravity, support base, and

why can you not stand with heels
and back to a wall and then bend over to
touch your toes and return to your stand-up
In

torque,

no rotation.

When

the base of support is not under the

center of gravity, the gravitational force

position? (14.3)

produces a torque that causes toppling.

The resistance of an object to changes in its rotational state of motion is called rotational inertia.
The
greater the rotational inertia, the
harder it is to change the rotational speed
of an object.

Angular momentum, the "inertia of rotation," is

the product of rotational inertia and rotational

6.

Where must

a football be kicked so that it

won't topple end-over-end as it sails through
the air? (14.3)

7.

What

8.

What two

is

the law of inertia for rotation? (14.4)

quantities

make up

rotational in-

ertia? (14.4)

velocity.

Rotational Mechanics

Angular

momentum

is

conserved when no

external torque acts on a body.

9.

Why

does a short pendulum swing to and

than a long pendulum?

fro at a greater rate

(14.4)

Important Terms
angular momentum (14.6)
law of conservation of angular
lever

arm

linear

10.

momentum (14.7)

Why do your legs swing

idly

when

to

and

fro

more rap-

they are bent? (14.4)

(14.1)

momentum

11.

(14.6)

Which

will

have the greater acceleration

down an

rotational inertia (14.4)

rolling

rotational velocity (14.6)

small ball? (14.4)

incline

in

large ball or a

torque (14.1)
12.

Review Questions
1

Compare
object

What

down an

have the greater acceleration

incline
a hoop or a solid

the effects of a force exerted on an

and

a torque exerted

on an object.

13.

How

can a person vary his or her rotational

inertia? (14.5)

is

meanl bv

the lever

arm

of a force?

14.

Distinguish between rotational velocity and

rotational speed. (14.6)

(14.1)

3.

will

rolling

disk? (14.4)

(14.1)

2.

Which

In

what

to

produce maxii

directic

should a force be applied

im torque? (14.1

15.

Distinguish between linear

angular

momentum.

(14.6)

momentum and

203

14

Chapter Review

16.

What motion does

the torque produced by

impart to a vertically spinning bicycle wheel supported at only one
end of its axle? (14.6)

dle? Which is easier for turning stubborn

screws? Provide an explanation.

earth's gravity

2.

What

the

is

mass

of the rock

shown

in Fig-

ure A?
17.

What is the law

mentum? (14.7)

18.

What does it mean to say that angular momentum is conserved? (14.7)

of inertia for angular

Ocm

mo-

3.

What

15 c*

the

is

mass

5o|c

of the

meter

15 c*

lOCK**

stick

shown

in

Figure B?
19.

If

a skater

who

is

spinning pulls her arms in

0cm

so as to reduce her rotational inertia to half,

by how much

15cm

50cm

2.5cm

lOOcm

will her rate of spin increase?

(14.7)
Fig.

20.

What happens to a gymnast's angular momentum when he changes his body configu-

4.

ration during a somersault? His rotational

speed? (14.7)

You cannot stand with heels and back to the

wall and successfully lean over and touch
your toes without toppling. Would either
stronger legs or longer feet help you to do
this? Defend your answer.

Activities
1.

Ask a friend

5
to stand facing a wall

against the wall. Then ask your friend to

stand on the balls of the feet without toppling backward. Your friend won't be able to
do it. Now you explain why it can't be done.
2

with toes

Place the ends of a uniform horizontal board

on a pair of weighing scales. You'll note that
each scale reads half the weight of the board.
If a toy truck with a heavy brick in it is placed
on the center of the board, the reading on
each scale will increase by half the weight of
the loaded truck. Why? How would you go
about figuring how much weight is supported by each scale when the load is not in
the middle? Make predictions for various
places before you check by experiment. (Try
to do this before you measure torques in the
lab part of your course.)

6.

Which

7.

have the greater acceleration

incline
a bowling ball or a
volleyball? Defend your answer.
rolling

will

down an

The most popular gyroscope around

Frisbee.

8.

Why and how

that

it

do you throw a

spins about

Which is easier for prying open a stuck

cover from a can of paint a screwdriver
with a thick handle or one with a long han-

the

its

long axis

its

some-

football so

when

travel-

ing through the air?

9.

a trapeze artist rotates twice each second

while sailing through the air, and contracts
to reduce her rotational inertia to one third,
how many rotations per second will result?
If

You

sit in

the middle of a large, freely rotat-

ing turntable at an

Think and Explain

is

What function, besides being a place

for gripping and catching, does

what thicker curved rim serve?

10.

1.

walk along the top of a fence why does

holding your arms out help you to balance?
If you

amusement

park.

If

you

crawl toward the outer rim, does the rotation speed increase, decrease, or remain unchanged? What law of physics supports your

Special Relativity

Space and Time

Everyone knows that we move in time, at the rate of 24 hours

per day. And everyone knows that we can move through space, at
rates ranging from a snail's pace to those of supersonic aircraft
and the space shuttles. But relatively few people know that motion through space is related to motion in time.
The first person to understand the relationship between space
and time was Albert Einstein. Einstein went beyond common
sense when he stated in 1905 that in moving through space we
also change our rate of proceeding into the future
time itself is
altered. This view was introduced to the world in his special theory of relativity. This theory describes how time is affected by
motion in space at constant velocity, and how mass and energy
are related. Ten years later Einstein announced a similar theory,
called the general theory of relativity, which encompasses accelerated motion as well. These theories have enormously changed
the way scientists view the workings of the universe. This book
discusses only the special theory and leaves the general theory
for followup study later in your education.
This chapter will serve merely to acquaint you with the basic
ideas of special relativity as they relate to space and time. Chapter 16 will continue with the relationship between mass and
energy. These ideas, for the most part, are not common to your
everyday experience. As a result, they don't agree with common
sense. So please be patient with yourself if you find that you do
not understand them. Perhaps your children or grandchildren
will find them very much a part of their everyday experience. If
so, they should find an understanding of relativity considerably

less difficult.

204

15.1

205

Spacetime

Spacetime
Newton and others before Einstein thought of space as an inexpanse in which all things exist. We are in space, and we
move about in space. It was never clear whether the universe exfinite

ists in

space, or space exists within the universe. Is there space

Or is space only within

same question could be raised for time. Does
outside the universe?

the universe?

The

the universe exist

does time exist only within the universe? Was there

time before the universe came to be? Will there be time if the
universe ceases to exist? Einstein's answer to these questions is
that both space and time exist only within the universe. There is
no time or space "outside."
in time, or

The universe does not exist in a certain part of infinite space, nor
does it exist during a certain era in time. It is the other way around: space and
time exist within the universe.
Fig. 15-1

Einstein reasoned that space and time are two parts of one
whole called spacetime. To begin to understand this, consider
your present knowledge that you are moving through time at
the rate of 24 hours per day. This is only half the story. To get the
other half, convert your thinking from "moving through time"
to "moving through spacetime." From the viewpoint of special
relativity, you travel through a combination of space and time
spacetime at a constant speed. If you stand still, then all your
traveling is through time. If you move a bit, then some of your
travel is through space and most is still through time. What hap-

206

15

pens to time

<#*

When you stand

you are traveling at the

Fig. 15-2

maximum
the

rate in time: 24

maximum

If

you travel

at

rate through

space (the speed of

time stands still.

light),

Motion

15.2

is

to travel

through space

at the

speed of

that all your traveling

timeless.

-.

hours per day.

you were

Space and Time

would be through
space, with no travel through time! You would be as ageless
as light, for light travels through space only (not time) and is
light?

still,

if

The answer

Special Relativity

Motion in space affects motion in time. Whenever we move

through space, we to some degree alter our rate of moving into
the future. This is time dilation, a stretching of time that occurs
ever so slightly for everyday speeds, but significantly for speeds
approaching the speed of light. In the high-speed spacecrafts of
tomorrow, people will be able to travel noticeably in time. They
will be able to jump centuries ahead, just as today people can
jump from the earth to the moon. To understand time dilation
and how this can be, you first need to understand several ideas:
the relativity of motion and the postulates of special relativity.

Is Relative

we discuss motion, we must

which the motion is being observed
and measured. For example, you may walk along the aisle of a
moving bus at a speed of 1 km/h relative to your seat, but at
100 km/h relative to the road outside. Speed is a relative quantity. Its value depends upon the place
the frame of reference
where it is observed and measured. An object may have different
Recall from Chapter 2 that whenever

specify the position from

speeds relative to different frames of reference.

Fig. 15-3

The bag of

gro-

ceries has an appreciable

speed in the frame of reference of the building, but in

the frame of reference of the
freely-falling elevator

no speed at

it

has

Suppose your friend always pitches a baseball at the same

speed of 60 km/h. Neglecting air resistance and other small effects, the ball is moving at 60 km/h when you catch it. Now suppose your friend pitches the ball to you from the flatbed of a
truck that moves toward you at 40 km/h. How fast does the ball
meet you? You'll have to be sure to wear a catcher's mitt, because the speed of the ball will be 100 km/h (the 60 km/h relative to the truck plus the 40 km/h relative to the ground). Speed

all.

is

Your speed is
km/h relative to your seat,
and 100 km/h relative to the
Fig. 15-4
1

road.

relative.

15.3

The Speed of Light

Is

207

Constant

Suppose the truck moves away from you

40 km/h and your

you need no glove
at all, for the ball reaches you at a speed of 20 km/h (since 60 km/h
minus 40 km/h is 20 km/h). This is not surprising, for you expect
that the ball will be traveling faster when the truck approaches
you and that the ball will be traveling more slowly when the
at

friend again pitches the ball to you. This time

truck recedes.

TRUCK

AT REST

TRUCK MOVES TOWARD YOU

TRUCK MOVES AWAY FROM YOU

The ball is always pitched at 60 km/h relative to the truck, (a) When
both you and the truck are at relative rest the ball is traveling at 60 km/h when
you catch it. (b) When the truck moves toward you at 40 km/h, the ball is traveling at 100 km/h when you catch it. (c) When the truck moves away from you
at the same speed, the ball is traveling at 20 km/h when you catch it.
Fig. 15-5

The idea that speed is a relative quantity goes back to Galileo

and was known long before the time of Einstein. As you will learn
in this chapter, Einstein expanded the relativity of speed to include the relativity of things that seem unchangeable.

15.3

The Speed of Light Is Constant

Suppose you actually caught baseballs thrown off a moving

truck out in a parking lot and found that no matter what the
speed or direction of the truck, the ball always got to you at only
one speed 60 km/h. That is to say, if the truck zooms toward

208

15

Special Relativity

Space and Time

at 50 km/h and your friend pitches the ball at his speed of

60 km/h, you catch the ball with the same speed as if the truck
were not moving at all. Furthermore, if the truck moves away
from you at whatever speed, the ball still gets to you at 60 km/h.
This all seems quite impossible, for it is contrary to common experience. And if you did experience this, you would have to reevaluate your whole notion of reality. To put it mildly, you would
be quite confused.

you

Baseballs do not behave this way. But

it

turns out that light

measurement of the speed of light in empty space

same value of 300 000 km/s, regardless of the speed of

does! Every
gives the

We do not ordinarily nobecause light travels so incredibly fast.

The fact that light has only one speed in empty space was discovered at the end of the last century.* * Light from an approaching source reaches an observer at the same speed as light from a
receding source. And the speed of light is the same whether we
move toward or away from a light source. How did the physics
community regard this finding? They were as perplexed as you
would be if you caught baseballs at only one speed no matter
how they were thrown. Experiments were done and redone, and
always the results were the same. Nothing could vary the speed
of light. Various interpretations were proposed, but none were
satisfactory. The foundations of physics were on shaky ground.
Albert Einstein looked at the speed of light in terms of the definition of speed. What is speed? It is the amount of space traveled
compared to the time of travel. Einstein recognized that the
classical ideas of space and time were suspect. He concluded
that space and time were a part of a single entity
spacetime.
The constancy of the speed of light, Einstein reasoned, unifies
space and time.
The special theory of relativity that Einstein developed rests
on two fundamental assumptions, or postulates.
the source or the speed of the receiver."'
tice this

The speed of
found to be the same
frames of reference.
Fig. 15-6

light

is

in all

The presently accepted value for the speed of light is 299 792 km/s, which
off to 300 000 km/s. This corresponds to 186 000 mi/s.

we round

In 1887 two American physicists, A. A. Michelson and E. W. Morley, performed an experiment to determine differences in the speed of light in different
directions. They thought that the motion of the earth in its orbit about the sun
would cause shifts in the speed of light. The speed should have been faster
when light traveled in the same direction as the earth, and slower when it
traveled at right angles to the earth. Using a device called an interferometer,
they found that the speed was the same in all directions. For Michelson s many
experiments on the speed of light, he was the first American honored with a
Nobel Prize.

15.4

The

15.4

209

First Postulate of Special Relativity

The

First Postulate

of Special Relativity

Einstein reasoned that there is no stationary hitching post in the

universe relative to which motion should be measured. Instead,
all motion is relative and all frames of reference are arbitrary. A
spaceship cannot measure its speed relative to empty space, but
only relative to other objects. If, for example, spaceship A drifts
past spaceship B in empty space, spaceman A and spacewoman
B will each observe only the relative motion. From this observation each will be unable to
at rest,

if

determine

who

is

moving and who

is

E'

either.

Spaceman A considers himself at rest and sees spacewoman B pass

But spacewoman B considers herself at rest and sees spaceman A pass by.
Who is moving and who is at rest?
Fig. 15-7
by.

This is a familiar experience to a passenger in a car at rest

waiting for the traffic light to change. If you look out the window
and see the car in the neighboring lane start to move backward,
you may find to your surprise that the car you observe is really
at rest and the car you are in is moving forward. Or vice versa. If
you could not see out the windows, there would be no way to
determine whether your car was moving with constant velocity
or was at rest.
In a high-speed jetliner we flip a coin and catch it just as we
would if the plane were at rest. Coffee pours from the flight attendant's coffee pot as it does when the plane is standing on the
runway. If we swing a pendulum, it moves no differently when
the plane is moving uniformly (constant velocity) than when not
moving at all. There is no physical experiment we can perform
to determine our state of uniform motion. Of course, we can look
outside and see the earth whizzing by, or send a radar signal out.
However, no experiment confined within the cabin itself can determine whether or not there is uniform motion. The laws of
physics within the uniformly-moving cabin are the same as those
in a stationary laboratory.

Fig. 15-8

A person playing

pool on a smooth and

moving ocean
have

to

liner

fast-

does not

make adjustments

to

compensate lor the speed ol

the ship. The laws ol physic S
are the same for the ship
whether it is moving uniformly or

is

al resl

210

15

Special Relativity

These examples illustrate one

eial relativity.

All the

It is

laws

Einstein's
of

moving frames

of

first

Space and Time

the two building blocks of spepostulate of special relativity:

nature are the same

in all

uniformly

of reference.

Any number

of experiments can be devised to detect accelerated

motion, but none can be devised, according to Einstein, to detect the state of uniform motion.

15.5

The Second Postulate of Special

Relativity

One

of the questions that Einstein as a youth asked his schoolteacher was, "What would a light beam look like if you traveled
along beside it?" According to classical physics, the beam would
be at rest to such an observer. The more Einstein thought about
this, the more convinced he became of its impossibility. He came
to the conclusion that i/ an observer could travel close to the

speed of light, he would measure the light as moving away from

him at 300 000 km/s.
This is the idea that makes up Einstein's second postulate of
special relativity:

The speed of light in empty space will always have the

same value regardless of the motion of the source or
the motion of the observer.
of light in all reference frames is always the same.
Consider, for example, a spaceship departing from the space
station shown in Figure 15-9. A flash of light is emitted from the
station at 300 000 km/s, which we call simply c. No matter what
the speed of the spaceship relative to the space station, an ob-

The speed

server on the spaceship will measure the speed of the flash of

light passing her as the same speed c. If she sends a flash of her
own to the space station, observers on the station will measure
the speed of these flashes as c. The speed of the flashes will be no
different

if

the spaceship stops or turns around

observers
value c.

All

The speed of a
Hash emitted by either
the spaceship or the space
station is measured as c by
observers on the ship or the
space station. Evt -one who
light
measures the spci

who measure

and approaches.
it has the same

the speed of light find

Fig. 15-9
light

will net the

same

-#

fC3>
(T-VZ

15.6

Time

211

Dilation

The constancy of the speed of light is what unifies space and

And for any observation of motion through space, there is
a corresponding passage of time. The ratio of space to time for
light is the same for all who measure it. The speed of light is a
time.

constant.

SPACE -C

SPACE
TIME

TIME

space and
time measurements of light

Fig. 15-10

All

are unified bv

Special relativity turns around some of our conceptions about

We think that speed is relative, that it depends on the
speeds of the source and the observer. Yet, the speed of light is
absolute independent of the speeds of the source or observer.

c.

the world.

Time, on the other hand, is thought of as absolute. It seems to

pass at the same rate regardless of what is happening. Yet, Einstein did not accept this, and proposed that time depends on the
motion between the observer and the event being observed.
We measure time with a clock. A clock can be any device that
measures periodic intervals, such as the swings of a pendulum,
the oscillations of a balance wheel, or the vibrations of a quartz
crystal. We are going to consider a "light clock," a rather impractical device, but one that will help to describe time dilation.
Imagine an empty tube with a mirror at each end (Figure
15-11). A flash of light bounces back and forth between the parallel mirrors. The mirrors are perfect reflectors, so the flash
bounces indefinitely. If the tube is 300 000 km in length, each
bounce will take 1 s in the frame of reference of the light clock. If
the tube is 3 km long, each bounce will take 0.00001 s.
Suppose we view the light clock as it whizzes past us in a highspeed spaceship (Figure 1 5- 12). We see the light flash bouncing up
and down along a longer diagonal path.

<P

Fig. 15-12

(a)

An observer moving with

the spaceship observes the lighi Hash

between the mirrors of the light clock, (b) An observer who

passed by the moving ship observes the flash moving along a diagonal path.

moving

vertically

is

MIRROR

'*- LI6HT
FLASH

MIRROR
A stationary light
bounces between
mirrors and "ticks

Fig. 15-11

clock. Light

parallel
off"

equal intervals of time.

212

15

Special Relativity

Space and Time

But remember the second postulate of relativity: the speed

measured by any observer as c. Since the speed of light
will not increase, we must measure more time between bounces!
The light clock is a measure of time for the moving spaceship.
We would measure more time for bounces in the light clock and
for all the seconds and minutes experienced by the inhabitants
of the spaceship. We would observe time in the spaceship running more slowly than it does where we are.
will be

Fig. 15-13

The longer

dis-

tance taken b\ the light Hash

following the diagonal
path must be di\ ided In a
corresponding!) longer time
in ten al to \ ield an un\ ai \ ing
value for the speed ut light.
in

57

"*

\-y

r
-_

DISTANCE

TIME

TIME

The slowing of time is not peculiar to the light clock. It is time

the moving frame of reference, as viewed from our frame
of reference, that slows. The heartbeats of the spaceship occupants will have a slower rhythm. All events on the moving ship
will be observed by us as slower. It is time itself that is dilated.
How do the occupants on the spaceship view their own time?
Do they perceive themselves moving in slow motion? Do they
itself in

experience longer lives as a result of time dilation? As

it

turns

none of these things. Time for them is the same

as when they do not appear to us to be moving at all. Recall Ein-

out, they notice

laws of nature are the same in all uniformly moving frames of reference. There is no way they can tell
uniform motion from rest. They have no clues that events on
board are seen to be dilated when viewed from other frames of
stein's first postulate: all

reference.
Fig. 15-14

Mathematical

PATH OF LI6HT AS SEEN

FROM A POSITION OF REST

detail of Figure 15-13.

MIRRORS
POSITION

AT
1

MIRRORS

AT

POSITION 2

MIRRORS AT
POSITION 3

"

15.6

Time

213

Dilation

The Time Dilation Equation'

Figure 15-14 shows three successive positions of the
it moves to the right at constant speed v. The
diagonal lines represent the path of the light flash as it
starts from the lower mirror at position 1, moves to the
upper mirror at position 2, and then back to the lower mirlight clock as

ror at position

3.

The symbol t represents the time it takes for the flash to

move between the mirrors as measured from a frame of reference fixed to the light clock. This is the time for straight
up or down motion. Since the speed of light is always c, the
light flash is observed to move a vertical distance ct in the
frame of reference of the light clock. This is the distance between mirrors and is at right angles to the horizontal motion of the light clock. This vertical distance is the same in
both reference frames.
The symbol t represents the time it takes the flash to move
from one mirror to the other as measured from a frame of
reference in which the light clock moves to the right with
speed v. Since the speed of the flash is c and the time to go
from position 1 to position 2 is /, the diagonal distance traveled is ct During this time t, the clock (which travels horizontally at speed v) moves a horizontal distance vt from
position 1 to position 2.
These three distances make up a right triangle in the figure, in which ct is the hypotenuse, and ct and vt are legs. A
.

well-known theorem of geometry (the Pythagorean theorem) states that the square of the hypotenuse is equal to the
sum of the squares of the other two sides. If we apply this
to the figure,

we

obtain:
c2 t2

c2 t2
t"[\

--

v2 t2

~(v '/c

= c tl +
= cH\
2

v2 t2

tl

)]
2
t

tl

=
1

The mathematical derivation

- (v 2 /c 2
)

Vl -

(v 2 lc

of this equation for time dilation

is

included here mainiv to show that it involves only a bit ol geometry and
elementary algebra. It is not expected that you master it! (If you take a
follow-up physics course, you can master

it

then.)

l*w

214

15

Special Relativity

Space and Time

Questions
1.

Does time dilation mean that time

slowly in moving systems or that
more slowly?

it

really passes more

only seems to pass

you are moving in a spaceship at a high speed relative

would you notice a difference in your pulse
rate? In the pulse rate of the people back on earth?

2. If

to the earth,

3.

A and B agree on measurements of time if

A moves at half the speed of light relative to B? If both A
and B move together at 0.5c relative to the earth?
Will observers

How

do occupants on the spaceship view our time? Do they

The answer is no motion is relative, and from their frame of reference it appears that we are the
ones who are moving. They see our time running slow, just as we
see their time running slow. Is there a contradiction here? Not at

see our time as speeded up?

all. It is

physically impossible for observers in different frames

and the same realm of spacetime. The

measurements in one frame of reference need not agree with the
measurements made in another reference frame. There is only
one measurement they will always agree on: the speed of light.
of reference to refer to one

15.7

The Twin Trip

A dramatic

illustration of time dilation

identical twins,
trip

2.

3.

is

the classic case of the

who takes a high-speed roundthe other stays home on earth. When the trav-

one an astronaut

journey while

Answers
The slowing of time in moving systems is not merely an illusion resulting
from motion. Time really does pass more slowly in a moving system compared
to one at relative rest, as we shall see in the next section. Read on!
There would be no relative speed between you and your own pulse, so no relativistic effects would be noticed. There would be a relativistic effect between
you and people back on earth. You would find their pulse rate slower than normal (and they would find your pulse rate slower than normal). Relativity effects are always attributed to "the other guy."

When A and B have

motions relative to each other, each will observe

frame of reference of the other. So they will not agree

different

a slowing of time in the

on measurements of time.

same frame

of reference

When

and

they are moving in unison, they share the

will agree

on measurements of time. They will

and they will each see events on

see each other's time as passing normally,

earth in the

same slow motion.

15.7

The Twin Trip

215

is younger than the stay-at-home twin.

younger depends on the relative speeds involved. If
the traveling twin maintains a speed of 50% the speed of light
for one year (according to clocks aboard the spaceship), 1.15
years will have elapsed on earth. If the traveling twin maintains
a speed of 87% the speed of light for a year, then 2 years will
have elapsed on earth. At 99.5% the speed of light, 10 earth years
would pass in one spaceship year. At this speed the traveling
twin would age a single year while the stay-at-home twin ages

eling twin returns, he

How much

10 years.

r
C.'J

The question

arises, since

as well the other

way around

motion

is

relative,

why

isn't it just

why wouldn't the traveling twin

return to find his stay-at-home twin younger than himself? We

show that from the frames of reference of both the earth-

will

it is the earthbound twin who

ages more.
First, consider a spaceship hovering at rest relative to a distant planet. Suppose the spaceship sends regularly-spaced brief
flashes of light to the planet (Figure 15-16). Sometime will elapse
before the flashes get to the planet, just as 8 minutes elapses before sunlight gets to the earth. The light flashes will encounter
the receiver on the planet at speed c. Since there is no relative
motion between the sender and receiver, successive flashes will
be received as frequently as they are sent. For example, if a flash
is sent from the ship every 6 minutes, then after some initial delay, the receiver will receive a flash every 6 minutes. With no motion involved, there is nothing unusual about this.

bound twin and traveling twin,

Fig. 15-15

The traveling

twin does not age as fast as

the stay-at-home twin.

216

Special Relativity

15

ROC<ET

SHIP AT REST"
RELATIVE TO EARTH

(
C*>?'J

,*

Space and Time

*
I

y SENDS
,

/!

'
i

FLASH EVERY

6 MINUTES

SEES FLASH

EVERY

6 MINUTES

Fig. 15-16

When no motion

is

involved, the light flashes are received as fre-

quently as the spaceship sends them.

When motion

is

involved, the situation

quite different.

is

It is

important to note that the speed of the flashes will still be c, no

matter how the ship or receiver may move. How frequently the
flashes are seen, however, very much depends on the relative
motion involved. When the ship travels toward the receiver, the
receiver sees the flashes more frequently. This happens not only
because time is altered due to motion, but mainly because each
succeeding flash has less distance to travel as the ship gets closer
to the receiver. If the spaceship emits a flash every 6 minutes, the
flashes will be seen at intervals of less than 6 minutes. Suppose
the ship is traveling fast enough for the flashes to be seen twice
as frequently. Then thev are seen at intervals of 3 minutes (Figure 15-17).
\

<
,

SENDS FLASH EVERY

6 MINUTES

,-'

Fig. 15-17

more

When

the sender

moves toward

r/

SEES FLASH

'

'

EVERY
3 MINUTES
,

the receiver, the flashes are seen

frequently.

the ship recedes from the receiver at the same speed and
emits flashes at 6-min intervals, these flashes will be seen
half as frequently by the receiver, that is, at 12-min intervals
(Figure 15-18). This is mainly because each succeeding flash has
a longer distance to travel as the ship gets farther away from the
If

still

receiver.

The

effect of

moving away

is

just the opposite of

moving closer

IT217

The Twin Trip

15.7

\
\

\
1

\
1

SENDS 'FLASH EVEKyV

6^MINUTES
\

\
\

12

EVERY
MINUTES

"

SEES FLASH '

[/

.V,
/

When the sender moves away from the receiver, the flashes are
spaced farther apart and are seen less frequently.
Fig. 15-18

to the receiver.

when

So

if

the flashes are received twice as frequently

is approaching (6-min flash intervals are

seen every 3 min), they are received half as frequently when it is
receding (6-min flash intervals are seen every 12 min)."
The light flashes make up a light clock. In the frame of reference of the receiver, events that take 6 min in the spaceship are
seen to take 12 min when the spaceship recedes, and only 3 min
when the ship is approaching.

the spaceship

Questions
Here's a simple arithmetic question: If a spaceship travone hour and emits a flash every 6 min, how many

1.

els for

flashes will be emitted?

A spaceship sends equally spaced flashes while approach-

2.

ing a receiver at constant speed. Will the flashes be equally

spaced when they encounter the receiver?

A spaceship emits

3.

flashes every 6

min

for

one hour.

If

the

receiver sees these flashes at 3-min intervals, how much

time will occur between the first and the last flash (in the

frame of reference of the receiver)?

1.

Answers
Ten flashes, since (60 min)/(6 min/flash) = 10

flashes.

as long as the ship moves at constant speed, the equally-spaced flashes

be seen equally spaced but more frequently. (If the ship accelerated while
sending flashes, then they would not be seen at equally spaced intervals.)

2. Yes,

will

3. All

10 flashes will be seen in 30

The frequencies
That

is,

for

min

approach and

[(10 flashes) x (3 min/flash)

= 30 min].

each other.
approach are seen i as

for recession are reciprocals of

flashes that are seen 2 times as frequently for

frequently for recession. For higher speeds, il seen 3 times as frequently for
approach, the flashes are seen \ as frequentlj foi recession; or 4 times for approach, ; for recession, and so on.

218

Time

Special Relativity Space and

15

apply this doubling and halving of flash intervals to the

Suppose the traveling twin recedes from the earthbound
twin at the same high speed for one hour, then quickly turns
around and returns in one hour. The traveling twin takes a round
trip of two hours, according to all clocks aboard the spaceship.
This trip will twt be seen to take two hours from the earth frame
Let's

twins.

however. We can see

from the ship's light clock.

of reference,

this

with the help of the flashes

^5>

^
/

SENDS FLASH EVERY

6 MINUTES
'

STILL RECEIVIN6 FLASHES AT 12 -MINUTE

INTERVALS FROM RECEDING SHIP

r
i

ak*

i
\

ON WAY BACK

SHIP

'

*
-

\js

STILL SENDIN6 FLASH

EVERY 6 MINUTES

i'ii/

/-^

* # * * * * A-

'

,
'

11/

STILL SENDIN6 FLASH

EVERY 6 MINUTES

/
I

'

SEES FLASHES
\
.

FROM

APPROACHING SHIP
/
EVERY 3 MINUTES ^*'

'

'

/
'

The spaceship emits flashes each 6 min during a two-hour trip.

During the first hour, it recedes from the earth. During the second hour, it approaches the earth.
Fig. 15-19

As the ship recedes from the earth, it emits a flash of light every
These flashes are received on earth every 12 min. During
the hour of going away from the earth, a total of 10 flashes are
emitted. If the ship departs from the earth at noon, clocks aboard
the ship read 1 p.m. when the tenth flash is emitted. What time
will it be on earth when this tenth flash reaches the earth? The
answer is 2 p.m. Why? Because the time it takes the earth to
receive 10 flashes at 12-min intervals is (10 flashes) x (12 min/
flash), or 120 min (= 2 h).
6 min.

^15.7

219

The Twin Trip

Suppose the spaceship is somehow able to suddenly turn

around in a negligibly short time and return at the same high
speed. During the hour of return it emits 10 more flashes at
6-min intervals. These flashes are received every 3 min on earth,
so all 10 come in 30 min. A clock on earth will read 2:30 p.m.
when the spaceship completes its 2-hour trip. We see that the
earthbound twin has aged a half hour more than the twin aboard
the spaceship!

~EARTH FRAME OF REFERENCE

10 FLASHES
10 FLASHES

12 AAIN

3 MIN

120 MIN

= 30 MIN
150 MIN

2^2 HOURS

SPACESHIP FRAME OF REFERENCE:

20 FLASHES 6 MIN = 120 MIN
2 HOURS
Fig. 15-20

The

trip that takes 2 h in the

frame of reference of the spaceship

takes 2.5 h in the earth's frame of reference.

The

result

is

same

the

same from

either frame of reference. Con-

time with flashes emitted

from the earth at regularly spaced 6-min intervals in earth time.
From the frame of reference of the receding spaceship, these
flashes are received at 12-min intervals (Figure 15-21a). This
means that 5 flashes are seen by the spaceship during the hour of
receding from earth. During the spaceship's hour of approaching, the light flashes are seen at 3-min intervals (Figure 15-2 lb),
so 20 flashes will be seen.
sider the

trip again, only this

SENDS FLASH EVERY

6 MINUTES

SENDS FLASH EVERY

6 MINUTES

SEES FLASH EVERY

'

'

SEES FLASH EVERY

3 MINUTES,
,

12 MINUTES

B
Flashes sen! from earth at 6-min intervals are seen at 12-min intervals by the ship when it recedes, and at 3-min intei vals when n approaches.
Fig. 15-21

fe*

T
/

>

220

Special Relativity

15

Space and Time

Hence, the spaceship receives a total of 25 flashes during its

2-hour trip. According to clocks on the earth, however, the time
it took to emit the 25 Hashes at 6-min intervals was (25 flashes) x
(6 min Hash), or 150 min (= 2.5 h). This is shown in Figure 15-22.

EARTH FRAME OF REFERENCE:

Z5 FLASHES

6 MIN = 150 MIN

Z\ HOURS
====

'SPACESHIP'

FRAME OF REFERENCE

5 FLASHES
20 FLASHES

@
@

12 MIN = 60 MIN
3 MIN = 60 MIN

120MIN

2 HOURS
Fig. 15-22

Frame

A time interval

of 2.5

h on earth

is

seen to be 2 h in the spaceship's

of reference.

So both twins agree on the same results, with no dispute as

who ages more than the other. While the stay-at-home twin
remains in a single reference frame, the traveling twin has experienced two different frames of reference, separated by the acceleration of the spaceship in turning around. The spaceship has in
effect experienced two different realms of time, while the earth
has experienced a still different but single realm of time. The
twins can meet again at the same place in space only at the exto

pense of time.

15.8

Space and Time Travel

Before the theory of special relativity was introduced,

it

was

ar-

gued that humans would never be able to venture to the stars. It

was thought that our life span is too short to cover such great
distances
at least for the distant stars. Alpha Centauri is the
nearest star to earth, after the sun, and it is 4 light years away.*
It was therefore thought that a round trip even at the speed of

A light year is the distance that light travels in one year.

equals 9.45 x 10 IJ km.

One

light

year

15.8

Space and Time Travel

221

would require 8 years. The center of our galaxy is some

30 000 light years away, so it was reasoned that a person traveling even at the speed of light would require a lifetime of 30 000
years to make such a voyage. But these arguments fail to take
into account time dilation. Time for a person on earth and time
for a person in a high-speed spaceship are not the same.
A person's heart beats to the rhythm of the realm of time it is
in. One realm of time seems the same as any other realm of time
to the person, but not to an observer who is located outside the
person's frame of reference
for she sees the difference. As an example, astronauts traveling at 99% the speed of light could go to
the star Procyon (10.4 light years distant) and back in 21 years
in earth time. It would take light itself 20.8 years in earth time
to make the same round trip. Because of time dilation, it would
seem that only 3 years had gone by for the astronauts. All their
clocks would indicate this, and biologically they would be only 3
years older. It would be the space officials greeting them on their
return who would be 21 years older.
At higher speeds the results are even more impressive. At a
speed of 99.90% the speed of light, travelers could travel slightly
more than 70 light years in a single year of their own time. At
99.99% the speed of light, this distance would be pushed appreciably further than 200 years. A 5-year trip for them would take
them farther than light travels in 1000 earth-time years.
Such journeys seem impossible to us today. The amounts of
energy involved to propel spaceships to such relativistic speeds
are billions of times the energy used to put the space shuttles
into orbit. The problems of shielding radiation induced by these
high speeds seems formidable. The practicality of such space
journeys are prohibitive, so far.
If and when these problems are overcome and space travel
becomes routine, people will have the option of taking a trip and
returning in future centuries of their choosing. For example, one
might depart from earth in a high-speed ship in the year 2050,
travel for 5 years of so, and return in the year 2500. One might
live among earthlings of that period for a while and depart again
to try out the year 3000 for style. People could keep jumping into
the future with some expense of their own time, but they could
not trip into the past. They could never return to the same era on
earth that they bid farewell to.
Time, as we know it, travels only one way forward. Here
on earth we constantly move into the future at the steady rate of
24 hours per day. An astronaut leaving on a deep-space voyage
must live with the fact that, upon her return, much more time
will have elapsed on earth than she has experienced on her voyage. Star travelers will not bid "so long, see you later" to those
they leave behind but, rather, a permanent "goodby."
light

Fig. 15-23

From

the earth

frame of reference, light takes

30 000 years to travel from
the center of the Milky Way
galaxy to our solar system.
From the frame of reference
of a high-speed spaceship, the
trip takes less time.

From

the

frame of reference of light itself, the trip takes no time.

There is no time in a speed-oflight frame of reference.

222

Special Relativity

15

Chapter Review

Concept Summary

6.

According to Einstein's special theory of relativity, time is affected by motion in space at constant velocity.

Time appears to pass more slowly in a

frame of reference that is moving relative
7.

laws of nature are the same in all uniformly moving frames of reference.
No experiment can be devised
to detect
whether the observer is moving at con-

The speed of a ball you catch that is thrown

from a moving truck depends on the speed
and direction of the truck. Does the speed of
light caught from a moving source similarly
depend on the speed and direction of the
source? Explain. (15.3)

to the observer.

What does
light

8.

mean

9.

10.

the

is

first

postulate of special rela-

What

is

the second postulate of special rela-

The
any

ratio of circumference to diameter for

size circle

is jr.

11.

The path
in a

Important Terms

12.

special theory of relativity (15.1)

time dilation (15.1)

2.

Can you

is

is

Why, then, does the

be moving faster? (15.6)

we view

seen to be longer
light not

appear

a passing spaceship and see that

is running slow, how

do they see our time running? (15.6)

the inhabitants' time

13.

Review Questions
What

If

the

(15.5)

a stationary frame of ref-

erence.
to

is

of light in a vertical "light clock"

high-speed spaceship

when viewed from

first postulate of special relativity (15.4)
postulate (15.3)
second postulate of special relativity (15.5)

what
waves?

Similarly,

ratio of space to time for light

motion of the observer.

spacetime (15.1)

speed of

tivity? (15.5)

The speed of

1.

to say that the

tivity? (15.4)

stant velocity.
light in empty space is the same in
frames of reference.
The speed of light has the same value regardless of the motion of the source or the

What

it

a constant? (15.3)

is

All the

all

Space and Time

When

a flashing light source approaches

you, does the speed of light, or the frequency

spacetime? (15.1)

of arrival of light flashes, or both, increase?

(15.7)

in

3.

travel while

remaining

in

one place

space? Explain. (15.1)

14.

light travel through space? Through

time? Through both space and time? (15.1)

a.

How many

b.

How many

Does

frames of reference does the

stay-at-home twin experience in the twin
trip?

frames of reference does the

traveling twin experience? (15.7)

4.

What

5.

What does

is

tim

dilation? (15.1)
15.

it

relative? (15.2

mean

to say that

motion

is

Is it possible for a person with a 70-year life

span to travel farther than light travels in 70

years? Explain. (15.8)

~"^"

Chapter Review

15

223

Think and Explain

and yet

travel that distance in a

10-year

trip? Explain.
1.

you were in a smooth-riding train with no

windows, could you sense the difference between uniform motion and rest? Between
accelerated motion and rest? Explain how
you could do this with a bowl filled with
If

6.

Can you

7.

Explain

water.

2.

When you change your

position and

move

8.

Suppose you're playing catch with a friend

in a moving train. When you toss the ball in
the direction of the moving train, how does
the speed of the ball appear to an observer

9.

server were riding on the train?)

Suppose you're shining a light to a friend on

moving train. When you shine the light in
the direction of the moving train, strictly
speaking, how would the speed of light appear to an observer standing at rest outside
the train? (Does it increase or appear the
same as if the observer were riding with the

5.

Light travels a certain distance in, say, 10000

it possible that an astronaut could
travel more slowly than the speed of light

years. Is

we

that

when we

look out into

see into the past.

One

of the fads of the future might be "century hopping," where occupants of high-

Consider a high-speed spaceship equipped

with a flashing light source. If the spaceship
approaches you so that the frequency of
flashes is doubled, how will your measurements of the time between flashes (the pe-

from similar measurements by

occupants of the spaceship? Is this period
constant for a constant relative speed? For
accelerated motion? Defend your answer.
riod) differ

train?)

it is

speed spaceships would depart from the

earth for a few years and return centuries
later. What are the present-day obstacles to
such a practice?

standing at rest outside the train? (Does it

increase or appear the same as if the ob-

4.

why

the universe,

through space, what else are you moving

through?
3.

get younger by traveling at speeds

near the speed of light?

10.

you were in a high-speed spaceship travelaway from the earth at a speed close to
that of light, would you measure your normal pulse to be slower, the same, or faster?
How would your measurements be of the
pulse of friends back on earth if you could
monitor them from your ship? Explain.
If

ing

224

Biography:

Albert Einstein

IP
Biography:

225

Albert Einstein

Biography: Albert Einstein

(1879-1955)
which he won the 1921 Nobel prize in
The second paper was on the statistical
aspects of molecular theory and Brownian mo-

was born in Ulm, Germany, on

March 14, 1879. According to popular legend, he
was a slow child and learned to speak at a much

effect, for

later age than average; his parents feared for a

while that he might be mentally retarded. Yet
his elementary school records show that he was

tion, a

Albert Einstein

remarkably gifted

in

mathematics, physics, and

playing the violin. He rebelled, however, at the

practice of education by regimentation and rote,
and was expelled just as he was preparing to
drop out at the age of 15.
Largely because of business reasons, his family moved to Italy. Young Einstein renounced his
German citizenship and went to live with family
friends in Switzerland. There he was allowed to
take the entrance examinations for the

renowned

Swiss Federal Institute of Technology in Zurich,

two years younger than normal age. But because
of difficulties with the French language he did
not pass the examination. He spent a year at a
Swiss preparatory school in Aarau, where he
was "promoted with protest in French." He tried
the entrance exam again at Zurich and passed.
As a student he cut many lectures, preferring
to study on his own, and in 1900 he succeeded in
passing his examinations by cramming with the
help of a friend's meticulous notes. He said later
of this, "... after I had passed the final examination, I found the consideration of any scientific
problem distasteful to me for an entire year."
It was not until two years after graduation
that he got a steady job, as a patent examiner at
the Swiss Patent Office in Berne. Einstein held
this position for over seven years. He found the

work rather

interesting,

sometimes stimulating
mainly freeing

his scientific imagination, but

him

of financial worries while providing free

time to think about problems in physics.

With no academic connections whatsoever,
and with essentially no contact with other physicists, he laid out the main lines along which
twentieth-century theoretical physics has developed. In 1905, at the age of 26, he earned his Ph.D.
in physics and published three major papers.
The first was on the quantum theory of light,
including an explanation of the photoelectric

physics.

proof for the existence of atoms. His third

and most famous paper was on special relativity.

In 1915 he published a paper on the theory of
general relativity which presented a new theory
of gravitation that included Newton's theory as a
special case. These trailblazing papers have
greatly affected the course of

modern

physics.

were not limited to physBerlin during World War I and

Einstein's concerns
ics.

He

lived in

denounced the German militarism of

his time.

He

publicly expressed his deeply felt conviction

that warfare should be abolished and an inter-

national organization founded to govern disputes between nations. In 1933, while Einstein
visiting the United States, Hitler came to
power. Einstein spoke out against Hitler's racial
and political policies and resigned his position
at the University of Berlin. Hitler responded by
putting a price on his head. Einstein then accepted a research position at the Institute for Ad-

was

vanced Study in Princeton, New Jersey.

In 1939, one year before Einstein became an
American citizen, and after German scientists
fissioned the uranium atom, he was urged by
several prominent American scientists to write
the famous letter to President Roosevelt pointing out the scientific possibilities of a nuclear

bomb. Einstein was a

pacifist,

Hitler developing such a

but the thought of

bomb prompted

his ac-

The outcome was the development of the

first nuclear bomb, which, ironically, was detonated on Japan after the fall of Germany.
tion.

Einstein believed that the universe

ferent to the

human

is

indif-

condition, and stated that

if humanity were to continue, it must create a

moral order. He intensely advocated world peace
through nuclear disarmament. Nuclear bombs,
Einstein remarked, had changed everything but
our way of thinking.

He was a man oi unpretentious disposition

with a deep concern for the welfare of his fellow
beings. Albert Einstein was much more than a
great scientist
he was a great human being.

Special Relativity
Length, Mass, and

Energy

The speed of light is the speed limit for all matter. Suppose that
two spaceships are each traveling at nearly the speed of light
and are moving directly toward each other. The realms of spacetime for each ship are so distorted that the relative speed of approach is still less than the speed of light! For example, if both
ships are traveling toward each other at 80% the speed of light
with respect to the earth, each would measure their speed of approach as 98% the speed of light. There are no circumstances
where the relative speeds of any material objects surpass the
speed of light.
Why is the speed of light the universal speed limit? To understand this, we must know how motion through space affects the
length and mass of

moving

objects.

Length Contraction
For moving objects, space as well as time undergoes changes.
outside observer, moving objects appear
to contract along the direction of motion. The amount of contraction is related to the amount of time dilation. For everyday
speeds, the amount of contraction is much too small to be measured. For relativistic speeds, the contraction would be noticeable. A meter stick aboard a spaceship whizzing past you at 87%
the speed of light, for example, would appear to you to be only
0.5 meter long. If it whizzed past at 99.5% the speed of light, it
would appear to you to be contracted to one-tenth its original
length. As relative speed gets closer and closer to the speed of

When viewed by an

Fig. 16-1

A meter stick trav-

eling at 87 r

the speed of light

relative to an observer

would

be measured as only half as

lone as normal.

light, the

to zero.

226

measured lengths

of objects contract closer

and closer

227

Length Contraction

16.1

Do people aboard the spaceship also see their meter sticks

and everything else in their environment contracted? The answer is no. People in the spaceship see nothing at all unusual

about the lengths of things in their

did,

it

would

own

reference frame.

If

they

violate the first postulate of relativity. Recall that

the laws of physics are the same in all uniformly moving reference frames. Besides, there is no relative speed between themselves and the events they observe in their own reference frame.
There is a relative speed between themselves and our frame of
reference, however, so they will see our meter sticks contracted
to half size
and us as well.
all

In the frame of reference of the meter stick, its length is one meter.
Observers from this frame see our meter sticks contracted. The effects of relativity are always attributed to "the other guy."
Fig. 16-2

The contraction of speeding objects is the contraction of space

Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion
are the same in the two frames of reference. So if an object is
moving horizontally, no contraction takes place vertically (Figitself.

ure 16-3).

u = 0.87c

u-=0.995c

(j=0.999c

a=c('?)

As relative speed increases, contraction in the direction of motion

increases. Lengths in the perpendicular direction do not change.
Fig. 16-3

228

Special Relativity

16

Length, Mass, and Energy

Relativistic length contraction can be expressed

mathemati-

cal lv as:

L = L\
In this equation, v

is

(t-/c

the speed of the object relative to the ob-

sen er, c is the speed of light, L is the length of the moving object
as measured by the observer, and L is the measured length of
the object at rest.

Suppose that an object

stituted for
It

\'

in the

was stated

is

earlier that

if

= 0. When is subL - L, as we would expect.

an object were moving at 87% the

at rest, so that v

equation,

we

find

to half its length. When 0.87c is

substituted tor v in the equation, we find L = 0.5L,,. Or when
0.995c is substituted for v, we find L = 0.1L, as stated earlier. If
the object moves at c, its length would contract to zero. This is

speed of

light,

it

would contract

one of the reasons that the speed of

the speed of any material object.

light is the

upper

limit for

Question

A spacewoman

travels by a spherical planet so fast that

appears to her to be an ellipsoid (egg shaped). If she sees
the short diameter as half the long diameter, what is her
speed relative to the planet?
it

16.2

The Increase of Mass with Speed

we push an object that is free to move, it will accelerate. If we
maintain a steady push, it will accelerate to higher and higher
speeds. If we push with a greater and greater force, the acceleration in turn will increase. It might seem that the speed should
increase without limit, but there is a speed limit in the universe
the speed of light. In fact, we cannot accelerate a material object
enough to reach the speed of light, let alone surpass it.
We can understand this from Newton's second law. Recall that
the acceleration of an object depends not only on the applied
force, but on the mass as well: a = F/m. Einstein stated that
If

Answer
The spacewoman passes the spherical planet

at

87^

the speed of light.

This equation (and those that follow) is simply stated as a "guide to thinking" about the ideas of special relativity. The equations are given here without

any explanation as

to

how

they are derived.

16.2

The Increase of Mass with Speed

when work

done

is

to increase the

increases as well. This increase of

229

speed of an object,

mass means

its

mass

that a constant

applied force produces less and less acceleration as the object's

speed increases. The relationship between mass and speed is expressed mathematically by:

m
Vl -

(v7c 2 )

Here m is the relativistic mass of the moving object its mass as

measured by an observer. Again, v is the speed of the object relative to the observer. The symbol m represents the rest mass, the
mass the object would have at rest. When an object is given
kinetic energy,

The
which

its

mass

is

greater than
pushed, the

its

rest mass.

more

its mass increases,

and less response to the applied force. As
v approaches c, the denominator of the equation approaches
zero. This means that the mass m approaches infinity! An object
pushed to the speed of light would have infinite mass and would
require an infinite force, which is clearly impossible. Therefore

faster

an object

is

results in less

we

find another reason for saying that nothing made of matter

can be accelerated to the speed of light.
Subatomic particles have been accelerated to nearly the speed
of light, however. The masses of particles accelerated beyond 99%
the speed of light increase thousands of times, as evidenced when
a beam of electrons is directed into a magnetic field. Charged
particles moving in a magnetic field experience a force that

them from their normal paths. The amount of deflection

known to depend on the mass the greater the mass, the less

deflects
is

the deflection.

The

particles are found to deflect less than pre-

dicted unless the relativistic mass increase is taken into account (Figure 16-4). Only when the mass increase is taken into
account do the particles strike their predicted targets. Physicists
working with high-speed subatomic particles in atomic accelerators find the increase of mass with speed an everyday fact
of

life.

ELECTROMAGNETS

ELECTRON BEAM
If the masses ol the electrons did not increase with speed, the beam
would follow the dashed line. But because ol the increased inertia, the highspeed electrons in the beam are not deflected as much.

Fig. 16-4

230

16

Special Relativity

Length, Mass, and Energy

The Mass-Energy Equivalence

The most remarkable insight of Einstein's special theory of relativity is his conclusion that mass and energy are equivalent. As
stated earlier, the energy pumped into atomic particles in an accelerator increases their mass. This is simply a consequence of
the fact that energy and mass are equivalent to each other.
Consider the energy that goes into running particle accelerWe all know that the energy that goes into accelerating
the particles comes from a power plant someplace. The accelerator facility must pay its electric bill like other consumers of
ators.

energy.
Let us follow this further. Energy generated at the

most

likely

comes from

power plant

either the chemical combustion or nu-

clear reactions of certain fuels.

If

the process

is

chemical com-

bustion, the masses of hydrocarbon molecules produced by

combustion are reduced by about one part in a billion. If the

process is nuclear fission, the masses of the fission fragments
after reaction are reduced by about one part in a thousand. That
is, the masses of the fission fragments are about a thousandth
less than the mass of the initial uranium atoms before reaction.
In either the chemical or the nuclear case, mass is given off when
energy is given off. Now comes the interesting question: how
much mass is gained by the accelerator particles? If we neglect
the inefficiencies of
just as
Fig. 16-5

Saying that a

power plant delivers 90 milmegajoules of energy to

is equivalent to
saying that it delivers one
gram of energy to its consumers. This is because mass and
energy are equivalent.
lion
its

consumers

much mass

power transmission,

the fuel fragments lose

as the particles at the accelerator gain!

The Mass-Energy Equivalence

16.3

231

mass every bit as much as it deis the same as saying it delivers the other. This is because mass and energy are in a practical
sense one and the same. When something gains energy, it gains
mass. When something loses energy, it loses mass.
Einstein realized that anything with mass even if it is not
moving has energy. Conversely, anything with energy even if

So a power company

livers energy.

To say

it

delivers

delivers one

not matter (such as light or microwaves, for example) has

mass! The amount of energy E is related to the amount of mass
m by the most celebrated equation of the twentieth century:
it is

E = mc

The c 2 is the conversion factor for energy units and mass units.*
Because of the large magnitude of c, the speed of light, a small
mass corresponds to a huge amount of energy. For example, the
energy equivalent of a single gram of matter is greater than the
energy used daily by the populations of our largest cities.
The change in mass for energy changes is so slight that it has
not been detected until recent times. When we strike a match,
for example, a chemical reaction occurs. Phosphorus atoms in
the match head rearrange themselves and combine with oxygen
in the air to form new molecules. The resulting molecules have
very slightly less mass than the separate phosphorus and oxygen
molecules. From a mass standpoint, the whole is slightly less
than the sum of its parts, but not by very much by only about
one part in 10 9
According to Einstein, the missing mass has not been destroyed. It has been carried off in the guise of radiant energy and

E is the amount of energy given off, the "missElc 2 For all chemical reactions that give off energy, there is a corresponding decrease in mass.
It is important to realize that the total amount of mass is the
same before and after the reaction ifyou remember to include the
mass equivalent of the energy. Similarly, the total energy is the
same before and after the reaction if you include the energy
equivalent of the rest mass (the mass of matter at rest) involved.
Mass is conserved and energy is conserved. Some people say
that mass is converted to energy (or vice versa). It is really more
accurate to say that rest mass is converted to "pure" energy (energy that is not due to rest mass). The total amount of mass does
not change, nor does the total amount of energy.
kinetic energy. If

ing"

mass

is

just

meters per second and m is in kilograms, then E will be in

mass and energy had been understood long ago when
physics concepts were first being formulated, there would likely be no separateunits for mass and energy. Furthermore, with a redefinition of space and time
units, c could equal
and = mc 2 would simply be E = m.

When

joules.

If

is in

the equivalence of

232

16

In

nuclear reactions, the decrease

more than

Fig. 16-6

[none second, 4.5

mass are

billion tonsol rest

converted to radiant energy in

the sun. The sun is so massive,

however, that in a million

years onl} one ten-millionth ol
the sun's rest mass will have
been converted to radiant
energy.

Special Relativity

Length, Mass, and Energy

in rest

mass

is

considerably

chemical reactions about one part in 10\ The

decrease ol rest mass in the sun and other stars by the process of
thermonuclear fusion bathes the solar system with radiant energy and nourishes life. The present stage of thermonuclear fusion in the sun has been going on for the past 5 billion years, and
there is sufficient hydrogen fuel for fusion to last another 5 billion years. It is nice to have such a big sun!
The equation E = mc 2 is not restricted to chemical and nuclear
reactions. Any change in energy corresponds to a change in mass.
The increased kinetic energy of a baseball is accompanied by a
slight increase in mass. The filament of a light bulb energized
with electricity has more mass than when it is turned off. A hot
cup of tea has more mass than the same cup of tea when cold. A
wound-up spring clock has more mass than the same clock when
unwound. But these examples involve incredibly small changes
in mass
too small to be measured by conventional methods. No
wonder the fundamental relationship between mass and energy
in

was not discovered until this century.

The equation E = mc is more than a formula for the conversion of rest mass into pure energy, or vice versa. It states that
energy and mass are the same. Mass is simply congealed energy.
If you want to know how much energy is in a system, measure
2

its

mass. For an object at rest,

that is hard to shake.

its

energy

is its

mass.

It is

energy

itself

Question
equation E = mc 2 another
say that matter transforms into pure energy when
eling at the speed of light squared?

Can we look

16.4

The Correspondence
If

new theory

the old theory.

is

at the

way and
it is

trav-

Principle
valid,

it

must account

for the verified results of

New theory and old must overlap and agree in the

region where the results of the old theory have been fully verified.
Answer
No, no, no! There are several things wrong with that statement. As matter is
its mass increases rather than decreases. In fact, its mass approaches infinity. At the same time, its energy approaches infinity. It has more
mass and more energy. Matter cannot be made to move at the speed of light, let
2
alone the speed of light squared (which is not a speed!). The equation E = mc
propelled faster,

simplv means that energy and mass are "two sides of the same coin."


16.4

The Correspondence

233

Principle

This requirement is known as the correspondence principle.

If the equations of special relativity are valid, they must correspond to those of the mechanics of Newton classical mechanics when speeds much less than the speed of light are
considered.
The relativity equations for time, length and mass are:

t=

<

Vl = U Vl

Vl -

{vie) 2

-{vIcY
2

(v/c)

We

can see that these equations each reduce to Newtonian values for speeds that are very small compared to c. Then, the ratio
2
(v/c) is very small, and for everyday speeds may be taken to be
zero. The relativity equations become
*0

L =

Vl L Vl -0

m
,

Vl -

= L

everyday speeds, the length, mass, and time of moving obunchanged. The equations of special relativity hold for all speeds, although they are significant only for
speeds near the speed of light.

So

for

jects are essentially

Einstein's theory of relativity has raised

many

philosophical

time? Can we say that it is nature's

way of seeing to it that everything does not all happen at once?
And why does time seem to move in one direction? Has it always
moved forward? Are there other parts of the universe where time
moves backward? Perhaps these unanswered questions will be
answered by the physicists of tomorrow. How exciting!
questions. What, exactly,

is

234

16

Special Relativity

Length, Mass, and Energy

Chapter Review

16

Concept Summary

How do the lengths of objects

frame appear? (16.1)

out (dilated).

When
moves at very high speed relative to an observer, it is measured as contracted
in the direction ol motion.

in that

an object

2.

When an

object moves at very high speed rean observer, its mass is measured as
greater than the value when it is not moving.

How

long would a meter stick appear if it

were traveling like a properly thrown spear
at 99.5% the speed of light? (16.1)

lative to

Einstein realized that

alent

3.

long would a meter stick appear if it

were traveling at 99.5% the speed of light,

but with

mass and energy are equiv-

anything with mass also has energy, and

anything with energy also has mass, according

to the equation E = mc 1
Only when the release of energy is very

How

its

length perpendicular to

(Why

rection of motion?
this

and the

last

its di-

are your answers to

question different?) (16.1)

great

is

the release ot

mass

large

enough

4.

to

be detected.
During any reaction, the total amount of
mass remains the same if the mass equivalent of the energy released is taken into

(16.1)

5.

account.

During anv reaction, the total amount of

energy remains the same if the energy
equivalent of the rest mass involved is
taken into account.

you were traveling in a high-speed spacewould meter sticks on board appear to

vou to be contracted? Defend vour answer.
If

ship,

6.

What happens

a.

What

rest

b.

If

rest

a.

Important Terms
b.

rest

mass

mass

If

16.2)

we

wit

moving

p.

mass?
mass of

mass be

it

is

kg, will

greater, less, or

accelerated to a high

mass of an object if
speed of light? (16.2)

the

to the

it

What

is meant by the equivalence of mass

and energy?
What does the equation E = mc mean?
2

(16.3)

(16.2)
9.

>s

events in a frame of reference

time appears to be stretched

us.

Does the equation

E = mc

apply only to
reactions that involve the atomic nucleus?
(16.3)

Review Questions
1.

relativistic

What would be

correspondence principle (16.4)

relativistic

is meant by
an object has a

were pushed
8.

of an object that

to higher speeds? (16.2)

the same, if
speed? (16.2)

7.

mass

pushed

its

According to the correspondence principle, the

equations for the special effects due to motion at
high speed match the old, well tested equations
governing motion at everyday speeds when small
values of the speed are substituted.

to the

is

10.

If

an object

kind, does

it

supplied with energy of any

then have more mass? (16.3)

is

235

Chapter Review

16

The masses

3.

utility that services the accelerator?

Explain. (16.3)
12.

In the preceding question, the

power

practically equal to

+ v2

v,

Pretend that the spaceship of the first question is somehow traveling at speed c with
respect to you, and it fires a rocket at speed c
with respect to itself. Use the equation to
show that the speed of the rocket with respect to you

utility

energy from the mass of fuel, whether

coal, oil, or atomic nuclei. If we neglect all
gets

is

mass when they are

accelerated to speeds near that of light. Does

this mass increase consume energy from the

power

day speeds V

of particles in research acceler-

ators gain appreciable

is still c!

its

4.

power plant and in the

and at the accelerator,

inefficiencies at the

transmission lines

If

high-speed

tracted to half
tivistic

spaceship

its

length,

mass compare

appears

how

con-

will its rela-

to its rest

mass?

how

does the mass increase of the accelerated particles compare with the decrease in
mass of the fuel at the power plant? Ex-

5.

Where does

14.

What is

travel past the earth at relaspeeds in a spaceship and earth observers tell you that your ship appears to be
contracted. Comment on the idea of checking their observations by putting a meter
tivistic

plain. (16.3)

13.

Suppose you

solar energy originate? (16.3)

measuring rulers
your spaceship.

stick or finer

the correspondence principle? (16.4)

How

does the relativistic mass of a car moving at ordinary speeds compare to its rest

mass?

6.

to parts of

why we say there is a speed

Give two reasons

limit for particles in the universe.

(16.4)
7.

The two-mile-long

linear accelerator at Stan-

ford University in California "appears" to be

less

Think and Explain

1.

You observe a spaceship moving away from

you at speed v,. A rocket is fired straight

8.

speed v 2 relative to the spaceship. It

so happens that the speed of the rocket relative to you is not v, + v 2 For motion in a

is

sum V

of electrons that are accelerated

would you notice an increase in their mass?

In the mass of the target they are about to

and v

The masses

Explain.

Stanford accelerator become thousands of times as great as the rest mass by

the time the electrons reach the end of their
trip. In theory, if you could travel with them,

fired at

straight line, the relativistic

it.

in the

ahead from the spaceship, so that it also

moves away from you. Suppose that it is

locities v,

than a meter long to the electrons that

travel in

hit? Explain.

of ve-

given by

v,

9.

v2

The electrons

that illuminate your

TV screen

travel at about one-fourth the speed of light

and have an increased mass of nearly

Does this relativistic effect tend
crease or decrease your electric bill?
cent.

Suppose
light,

v,

and v 2 are each half the speed

or 0.5c.

Show

that the speed

rocket relative to you

is

V of

of

Substitute small values of

v,

10.

and

to in-

the

0.8c.

Since there

is

an upper limit on the speed of

a particle, does
2.

3 per-

v, into the

preceding equation and show that for every-

it

follow that there

is

there-

an upper limit on its kinetic energy or

momentum? Defend your answer.
fore

-*-

II

Properties of Matter

People say tme balloon floats because it's lighter than

helium 6as inside is composed of lighter
atoms than the air outside that helium has less density
than air. but does this explain why the balloon floats?
isn't the explanation that the bottom of the balloon is
"deeper" in our "ocean of air" than the top, and that the
atmospheric pressure up against the bottom is slightly
greater than the pressure down against the top? and
air--- that the

difference in upward and downward pressures

a buoyant force that exceeds the weight
of the balloon? you'll find a lot

this

of "everyday physics"

in

unit 2

produces

237

Suppose you were to break apart a large boulder with a heavy

sledgehammer. You break it into rocks. Then you break the rocks
into stones, and the stones into gravel. You keep going and break
the gravel into sand, and the sand into a powder. The powder
is made of fine crystals, which are particles of the minerals that
make up the rock. If the crystals are broken down, you have
atoms, the building blocks of
atoms.

17.1

all

matter. Everything

is

made

of

Elements

material things that exist shoes, ships, mice, people, the

are made of atoms. Interestingly enough, the incredible
number of different things that exist are made of a surprisingly
small number of kinds of atoms. Just as only three colors of dots
of light combine to form almost every conceivable color on a
television screen, only about 100 distinct kinds of atoms combine to form all the materials we know about. Each of these kinds
All

stars

atoms belongs to a different element.

To date ( 1 985), there are 1 09 known elements. Of these, only 90
are found in nature. The others are made in the laboratory with
high-energy atomic accelerators and nuclear reactors. These elements are too unstable (radioactive) to occur naturally in any
appreciable amounts.
All matter, however complex, living or nonliving, is some
combination of the elements. From a pantry having about 100
of

bins,
rials

each containing a different element, we have all the mateneeded to make up any substance occurring in the known

universe.

Common

materials are composed of not more than about 14

of these elements, for the majority of elements are relatively rare.

238

17.2

239

Atoms Are Ageless

Living things, for example, are composed primarily of four elements: carbon (C), hydrogen (H), oxygen (0), and nitrogen (N).
The letters in parenthesis represent the chemical symbols for
these elements. Table 17-1 lists the 14 most common elements.
Table 17-1

The 14 Most Common Elements

hydrogen (H)
nitrogen (N)
sodium (Na)

aluminum

The

magnesium (Mg)

(Al)

phosphorus

carbon (C)
oxygen (0)

(P)

silicon (Si)

sulfur (S)

chlorine (CI)

potassium (K)

calcium (Ca)

iron (Fe)

atoms of all are hydrogen. Hydrogen atoms are

most abundant. They comprise over 90 percent of the
atoms in the known universe. More complex and heavier atoms
are simply the result of hydrogen atoms squeezed together in the
deep interiors of stars under enormous temperatures and pressures. Nearly all the elements that occur in nature are remnants
of stars that exploded long before the solar system came into
lightest

also the

being.

Atoms Are Ageless

17.2
I

Atoms are much older than the materials they compose. The age
many atoms goes back to the origin of the universe and the
age of most atoms is more than the age of the sun and earth.

of

Fig. 17-1

are

made

Both you and she

of Stardust

in the

sense that the carbon, oxygen,

nitrogen,
that

and other atoms

make up your body

origi-

nated in the deep interior of

ancient stars that have long
since exploded.

240

The Atomic Nature of Matter

17

Atoms in your body have been around long before the solar
system came into existence. They cycle and recycle among innumerable forms, both living and nonliving. Every time you
breathe, lor example, only part of the atoms that you inhale are
exhaled in your next breath. The remaining atoms are taken into
your body to become part of you, and most leave your body
sooner or later.
Strictly speaking, you don't "own" the atoms that make up
your body you borrow them. We all share from the same atom
pool, as atoms migrate around, within, and throughout us. So
some of the atoms in the ear you scratch today may have been
part of your neighbor's breath yesterday!
Most people know we are all made of the same kinds of atoms.
But what most people don't know is that we are made of the same
atoms atoms that cycle from person to person as we breathe,

sweat, and vaporize.

Atoms Are Small

17.3

There are about 10 2 atoms in a gram of water (a thimbleful).

The number 10 2< is an enormous number. It is roughly the number of drops of water in the Atlantic Ocean. So there are as many
atoms in a drop of water as there are drops of water in the Atlantic Ocean! No wonder a cupful of DDT or any material thrown
into the ocean spreads around and is later found in every part of
*

the world's oceans.

The same

is

true of materials released into

the atmosphere.

so small that there are about as many atoms in the

your lungs at any moment as there are breathfuls of air in
the atmosphere of the whole world. It takes about 6 years for one
of your exhaled breaths to become evenly mixed in the atmosphere. At that point, every person in the world inhales an average of one of your exhaled atoms in a single breath. And this occurs for each breath you exhale! When you take into account the
many thousands of breaths that people exhale, there are many
atoms in your lungs at any moment that were once in the lungs
of every person who ever lived. We are literally breathing each

Atoms are

air in

*o \i3">

Fig. 17-2

atoms

in a

There are as many

normal breath of

air as there are breathfuls of

air in the

world.

atmosphere of the

other's breaths.

Atoms are too small to be seen at least with visible light.

You could connect microscope to microscope and never "see" an
atom. This is because light travels in waves, and atoms are
smaller than the wavelengths of visible light. The size of a particle visible under the highest magnification must be larger than
the wavelength of light.

241

Evidence for Atoms

17.4

Question
Is

your brain made of atoms that were once part of Albert

Einstein?

Evidence for Atoms

17.4

The first somewhat direct evidence for the existence of atoms was
unknowingly discovered in 1827. A Scottish botanist, Robert
Brown, was studying the spores of pollen under a microscope.
He noticed that the spores were in a constant state of agitation,
always jiggling about. At first, Brown thought that the spores
were some sort of moving life forms. Later, he found that inanimate dust particles and grains of soot also showed this kind of
motion. The perpetual jiggling of particles that are just large
enough to be seen is called Brownian motion. Brownian motion
is now known to result from the motion of neighboring atoms
too small to be seen.

More direct evidence for the existence of atoms is available

A photograph of individual atoms is shown in Figure 17-3.
The photograph was made not with visible light but with an electron beam. A familiar example of an electron beam is the one
today.

that sprays the picture

electron

beam

properties.

It

on your television screen. Although an

a stream of tiny particles, electrons,

so happens that a high-energy electron
is

it

has wave

beam has

wavelength more than a thousand times smaller than the wavelength of visible light. With such a beam, atomic detail can be
seen. The historic (1970) photograph in Figure 17-3 was taken
with a powerful and very thin electron beam in a scanning electron microscope. It is the first photograph of clearly distinguishable atoms.
More recently, IBM researchers have developed an electron
microscope small enough to be held in your hand; it is called a
scanning tunneling microscope. An image of graphite taken with
this remarkable instrument in 1985 is shown in Figure 17-4. The
gray "hill top" areas indicate the location of individual carbon
atoms in the graphite layer.

Fig. 17-3

The

strings of

dots are chains of thorium

atoms taken with a scanning

electron microscope by re-

searchers at the University

of Chicago's Enrico Fermi

Answer
and of Charlie Chaplin too. However, these atoms are combined differently than they were before. The next time you have one of those days when you
Yes,

feel like you'll

that the

atoms

on earth

who

never amount to anything significant, take comfort in the thought

that now compose you will be part of the bodies of all the people-

are yet to be!

Institute.

242

17

The Atomic Nature of Matter

An image ol graphobtained using the small

scanning tunneling mit roFig. 17-4
ite

"bumps"

scope. The

indicate

the location ol individual car-

bon atoms.

Molecules
Sometimes, atoms combine to form larger particles called moleexample, two atoms of hydrogen (H) are combined
with a single atom of oxygen (O) in a water molecule (FLO). The
gases nitrogen and oxygen, which make up most of the atmosphere, are each made of simple two-atom molecules (N, and O,).
In contrast, the double helix of deoxyribonucleic acid (DNA), the
cules. For

basic building block of

life, is

composed

of millions of atoms.

Models of simple molecules. The atoms that compose a molecule are

mixed together, but are connected in a well-defined way.

Fig. 17-5

not just

Not

all

matter

als (including

is

made

common

of molecules. Metals

table salt) are

joined in molecules. Matter that is a

perature tends to be made of molecules.
Matter made of molecules can contain

Fig. 17-6

An

elec

croscope photo
molecules.

ol

and rock miner-

made of atoms that are not

gas or liquid at room tem-

all the same kind of

molecule or it can be a mixture of different kinds of molecules.
Pure water contains only FLO molecules, whereas "clean" air
contains molecules of several different gases.
Like atoms, molecules are too small to be seen with optical microscopes. More direct evidence of molecules is seen in electronmicroscope photographs. The photograph in Figure 17-6 is of
virus molecules composed of thousands of atoms. These giant

17.7

The Atomic Nucleus

243

too small to be seen with visible light. So we

an atom or a molecule that is invisible to light
may be visible to a shorter-wavelength electron beam.
We are able to detect molecules through our sense of smell.
Noxious gases such as sulphur dioxide, ammonia, and ether are
clearly sensed by the organs in our nose. The smell of perfume is
the result of molecules that rapidly evaporate from the liquid
and jostle around completely haphazardly in the air until some
of them accidentally enter our noses. The perfume molecules are
certainly not attracted to our noses! They are just a few of the
billions of aimlessly jostling molecules that have moved out in
all directions from the liquid perfume.

molecules are

still

see again that

Compounds

17.6

A compound is a substance made of atoms of different elements

combined in a fixed proportion. The chemical formula of the
compound tells the proportions of each kind of atom. For exgas carbon dioxide, the formula C0 indicates that
carbon (C) atom there are two oxygen (O) atoms. Water,
table salt, and carbon dioxide are all compounds. Air, wood, and
salty water are not compounds.
A compound may or may not be made of molecules. Water
and carbon dioxide are made of molecules. On the other hand,
table salt (NaCl) is made of different kinds of atoms arranged in
a regular pattern. Every chlorine atom is surrounded by six so-

ample,

in the

for every

dium atoms (see Figure 17-7). In turn, every sodium atom is surrounded by six chlorine atoms. As a whole, there is one sodium
atom for each chlorine atom, but there are no separate sodiumchlorine groups that can be labeled molecules.
Compounds have different properties from the elements from
which they are made. At ordinary temperatures, water is a liquid, whereas hydrogen and oxygen are both gases. Salt is an edible solid, whereas chlorine is a poisonous gas.

The Atomic Nucleus

7.7

An atom

is

mostly empty space. Almost

all its

into the central region called the nucleus.

The

mass

New

is packed
Zealander

physicist Ernest Rutherford discovered this in 1909 in his

nowfamous gold-foil experiment. Rutherford's group directed a beam

of electrically-charged particles (alpha particles) from a radio-

Table salt (NaCl) is

that is not made
of molecules. The sodium and
Fig. 17-7

compound

chlorine atoms are arranged

in a repeating pattern.

Each

atom

six

is

surrounded by

atoms of the other kind.

244

17

RADIUM IN
LEAD BLOCK

ZINC
,

SULFIDE.

SCREEN

The Atomic Nature of Matter

active source through a very thin gold

foil.

They measured the

angles at which the particles were deflected from their straightline path as they emerged. This was accomplished by noting
spots of light on a zinc-sulphide screen that nearly surrounded
the gold foil (Figure 17-8). Most particles continued in a more or
less straight-line path through the thin foil. But, surprisingly,
particles were widely deflected. Some were even scattered
back along their incident paths. It was like firing bullets at a
piece of tissue paper and finding some bullets bouncing backward.
Rutherford reasoned that the particles that were undeflected
traveled through regions of the gold foil that were empty space.
The few particles that were deflected were repelled from the
massive centers of gold atoms that themselves were electrically
charged. Rutherford had discovered the atomic nucleus.
Although the mass of an atom is primarily concentrated in
the nucleus, the nucleus occupies only a few quadrillionths of
the volume of an atom. Atomic nuclei (plural of nucleus) are
extremely compact, or equivalently, extremely dense. If bare
atomic nuclei could be packed against each other into a lump
cm in diameter (about the size of a large pea), it would weigh
133 000 000 tons!
Huge electrical forces of repulsion prevent such close packing of atomic nuclei. This is because each nucleus is electrically
charged and repels other nuclei. Only under special circumstances are the nuclei of two or more atoms squashed into contact. When this happens, the violent reaction known as nuclear
fusion takes place. Fusion occurs in the core of the sun and other
stars and in a hydrogen bomb.
The principal building block of the nucleus is the nucleon."
When the nucleon is in its electrically neutral state, it is a neutron. When it is in its electrically charged state, it is a proton. All
neutrons are identical; they are copies of each other. Similarly,
all protons are identical.
Atoms with the same number of protons all belong to the same
element. There may be a difference in the number of neutrons,
however, in which case we speak of different isotopes. The nucleus of the most common hydrogen atom contains only a single
proton. When one proton is accompanied by a single neutron,
we have deuterium an isotope of hydrogen. When there are two
neutrons in the hydrogen nucleus, we have the isotope tritium.
Every element has a variety of isotopes. The lighter nuclei have
roughly equal numbers of protons and neutrons. The more massive nuclei have more neutrons than protons.

some

Fig. 17-8

The occasional

large-angle scattering of

alpha particles from the gold

atoms led Rutherford to the
discovery of the small, very
massive nucleus at their
centers.

Nucleons are composed of

discussed in Chapter 39.

still

smaller particles called quarks, which are

17.8

Electrons in the

Atom

245

Atoms are classified by their atomic number, which is the

same as the number of protons in the nucleus. Since the nucleus
of a hydrogen atom has a single proton, its atomic number is 1
Helium has two protons, so its atomic number is 2. Lithium has
three protons and its atomic number is 3, and so on, in sequence
to naturally occurring uranium with atomic number 92.
Electric charge comes in two kinds, positive and negative.
You will learn more about electric charge later. For now it is sufficient to know that like kinds of charge repel one another and
unlike kinds attract one another. Protons have a positive electric

charge and thus repel other protons. Protons are held together
within a nucleus, in spite of their mutual repulsion, by the very
strong nuclear force, which acts only across tiny distances. (The
strong nuclear force is discussed in Chapter 38.)

17.8

Electrons in the

Atom

The number of protons in the nucleus is normally electrically

balanced by an equal number of electrons outside the nucleus.
These are the electrons that make up the flow of electricity in
electric circuits. Electrons have only about jm the mass of a
nucleon and contribute very little mass to the atom. The negatively

charged electrons are attracted

to the positive nucleus,

but repel other electrons.

When an atom

it normally does not atatoms. But when atoms are close together,
the negative electrons on one atom may at times be closer to the
positive nucleus of a neighboring atom, which results in a net
attraction between the atoms. This is how atoms combine to
form molecules.
When the number of electrons do not equal the number of
protons in an atom, the atom is electrically charged and is said
to be an ion. When electrons are knocked off neutral atoms by
any means, positive ions are produced. This is because the net
charge is positive. When neutral atoms gain electrons by any
means, negative ions are produced. Compounds that are not
made of molecules are made of ions. In salt, for example, the so-

is

electrically neutral,

tract or repel other

dium atoms are positive ions. The chlorine atoms are negative
ions. The compound is held together by the forces between the
positive and negative ions.
Just as our solar system is mostly empty space, the atom is
mostly empty space. The nucleus and surrounding electrons occupy only a tiny fraction of the atomic volume. If it were not for
the electric forces of repulsion between the electrons of neigh-

The classic model

atom consists ot a tinv

Fig. 17-9

of the

nucleus surrounded by orbiting electrons.

246

17

The Atomic Nature of Matter

boring atoms, solid matter would be much more dense than it is.
We and the solid floor are mostly empty space, because the atoms
making up these and all materials are themselves mostly empty
space. But we don't fall through the floor. The electric forces of
repulsion keep atoms from caving in on each other under pressure. Atoms too close will repel (if they don't combine to form
molecules), but when atoms are several atomic diameters apart,
the electric forces on each other are negligible.
To explain how atoms of different elements interact to form
compounds, scientists have produced the shell model of the
atom. Electrons are pictured as being in spherical shells around
the nucleus. In the innermost shell, there are at most two elec-

second shell, there are at most eight. In the third

most 18. In the fourth shell, there are at most
32. Consider aluminum as an example. The neutral atom has 13
electrons. The first shell has two, the second shell has eight, and
the remaining three are in the third shell.
trons. In the

shell, there are at

HYDROGEN - ONE ELECTRON

IN ONE SHELL

ALUMINUM

THIRTEEN

ELECTRONS IN
THREE SHELLS

LITHIUM- THREE

ELECTRONS

TWO

IN

SHELLS

HELIUM - TWO
ELECTRONS IN
ONE SHELL
Fig. 17-10

The

shell

model of the atom pictures the electrons

in concentric,

spherical shells around the nucleus.

can never contain more than

with 19 electrons, has 2, 8, 8, and 1 in
The third shell cannot have 9 electrons when it is the

The outermost

shell of electrons

eight. Thus, potassium,

four shells.
last shell.

The arrangement of electrons about the atomic nucleus dicwhether and how atoms join to become molecules, melting
and freezing temperatures, electrical conductivity, as well as the
taste, texture, appearance, and color of substances. The electron
arrangement quite literally gives life and color to the world.
The periodic table is a chart that lists atoms by their atomic
number and by their electron arrangements. (See Figure 17-11.)
tates

17.8

ElectronsJn the

S*
ct>LL

Atom

*<f;

-OI3

247

^ a>|s

S*

to

in
co

CD

coflc,

71

Lu

Luletium

70

Yb

Ytterbium

174.967

103

Lr

173.04

102

No

Lawrencium

(260)

Nobelium

(259)

u-

eoQ

*P
0OQ_

520HS

69
j

Tm

Thulium

168.93

HP

CUJ-TD

""Wis

00

68 Er

CD

Erbium

167

100

Fm

(257)
Fermium

ti

<oO II

26

><2

S20.

_C'~

Is

oC

-r

a -

93

67

0OQ_

Ho

Holmium

164

99

Es

(254)
Einsteinium

mQQ

0^5

=<

CD

<oQ as

oC
.

si

65

Tb

Tefbium

158.93

98 Cf

97

Bk

(251)
Californium

,-fi

Berkelium

(247)

25

co-o;

r-O

k<;

64 Gd

r-Q.

63

fcJSS:

62 Sm

Eu

157

96 Cm

151.96

95

Am

150.36

94

Pu

Gadolinium

Europium

Samarium

Curium

Americium

(247)

(243)

(244)
Plutonium

E--

E
CO 0)

?r=

-.:

048

S Q.|-

r!

93

Neptunium

Np

237

oTJl'

Z|I

ao

<m:~ |S

029
Uranium

238

Ois *'
-.a

92

036

wQ. If

91 Pa

Proctactimum

231

T>
12

co. <U 1

58 Ce

*-o|g

140

57

La

Lanthanum

139

Q u H

038

90

Th

Thorium

232

91

3t32

fN

Cerium

028

89

Ac

Actinium

227

<N(/)

eCBi!

Sw

<0

3-

\p)

IS

""(Dig

Wis

a, ra
00

DC is

oU.

jg

C Cll)

248

17

The Atomic Nature of Matter

As you icad across from left to right, each element has one more
proton than the preceding element. As you go down, each element has one more electron shell than the one above.
Elements in the same column have similar chemical properties. That is, they form compounds with the same elements according to similar formulas. Elements in the same column are
said to belong to the same family of elements. Elements in the
same familv have the same number of electrons in the outer shell.

17.9

The

States of Matter
Matter exists in four states. You are familiar with the solid, liqand gaseous states. In the plasma state, matter consists of
bare atomic nuclei and free electrons. The plasma state exists
only at high temperatures. Although the plasma state is less common to our everyday experience, it is the predominant state of
matter in the universe. The sun and other stars as well as much of
the intergalactic matter are in the plasma state. Closer to home,
the glowing gas in a fluorescent lamp is a plasma.
In all states of matter, the atoms are constantly in motion. In
the solid state the atoms and molecules vibrate about fixed positions. If the rate of molecular vibration is increased enough, molecules will shake apart and wander throughout the material,
vibrating in non-fixed positions. The shape of the material is no
longer fixed but takes the shape of its container. This is the liquid state. If more energy is put into the material and the molecules vibrate at even greater rates, they may break away from
one another and assume the gaseous state.
All substances can be transformed from one state to another.
We often observe this changing of state in the compound H,0.
When solid, it is ice. If we heat the ice, the increased molecular
motion jiggles the molecules out of their fixed positions, and we
have water. If we heat the water, we can reach a stage where continued increase in molecular vibration results in a separation
between water molecules, and we have steam. Continued heating causes the molecules to separate into atoms. If we heat these
to temperatures exceeding 2000C, the atoms themselves will be
shaken apart, making a gas of free electrons and bare atomic
nuclei. Then we have a plasma.
uid,

The following three chapters treat the solid, liquid, and gaseous states in turn.

Chapter Review

17

249

Chapter Review

17

Concept Summary

element

matter is made from only about 100 different

kinds of atoms.
Each kind of atom belongs to a different
element.
Most atoms have been recycling through
matter even before the solar system came
All

into being.

Atoms are too small

with visible
light but can be photographed with an
electron microscope.

A compound

is

a substance

ments combined

to see

made of molecules,
made of atoms joined

are

which are particles

ion (17.8)

isotope (17.7)

molecule (17.5)
neutron (17.7)
nucleon (17.7)
nucleus (17.7)
periodic table (17.8)

plasma

(17.9)

proton (17.7)
shell

model of the atom

(17.8)

made of different ele-

in a fixed proportion.

Some compounds

(17.1)

family (17.8)

Review Questions
1

How many elements are known today? (17.1)

2.

Which element has the

together.

Other compounds are made of different

kinds of atoms arranged in a regular pattern.
is

mostly empty space.

almost entirely in
The nucleus

made

is

of protons

and neu-

4.

What

trons.

don't

The number of protons determines the element to which the atom belongs.
An electrically neutral atom has electrons

body?

outside the nucleus equal in

protons inside the nucleus.

mass

nucleus.

its
is

Its

The

shell

atoms? (17.1)

3. How does the age of most atoms compare

^-^ with the age of the solar system? (17.2)

The atom

lightest

model

of the

atom

number

5.

is meant by the statement that you

"own" the atoms that make up your

(17.2)

How does

number of atoms
your lungs compare to the number of breaths of air in the atmosphere of the
whole world? (17.3)
the approximate

in the air in

to the

pictures elec-

trons in spherical shells around the nucleus.

The periodic

table

ranged according
ture

is

a chart of elements ar-

to similar

6.

How

7.

What causes

atomic struc-

do the sizes of atoms compare to the

wavelengths of visible light? (17.3)

and similar properties.

dust particles to

Brownian motion?

Important Terms
I)

atom

(17.1)

atomic number (17.7)

Brownian motion (17.4)
chemical formula (17.6)

compound

(17.6)

Individual atoms cannot be seen with visible

light; yet there is a photograph of individual

atoms

9.J

move with

(17.4)

in

Figure 17-3. Explain. (17.4)

Distinguish between an
cule.(17.5)

atom and

a mole-

250

10/

17

a.
b.

How many elements compose pure water?

How many individual atoms arc there in
a water molecule?

11. Ja.

made
b.

example

Cite an

17.5)

of a substance that

is

hydrogen atom? The lithium atom? The

minum atom?

&

is

not

made

Think and Explain

1.

'

b.

What

f 2. jwhic h

compound?

Cite the chemical formulas for at least

three

14.

is

compounds.

did Rutherford discover when he bombarded a thin foil of gold with subatomic-

to the

mass

of the

whole atom?

How does the size of an atomic nucleus compare to the size of the whole atom? (17.7)

4.

18.

a.

b.

0,

are older, the atoms in the body of an

body of a

smell almost immediately.

From an atomic

point of view, exactly what

is

happening?

In what way does the number of protons in

an atomic nucleus dictate the chemical properties of the element?

A
\

What

3/ Suppose that your brother comes into the

room wearing shaving lotion, which you

(17.7)

17.

H H

does the mass of an atomic nucleus

compare

16.

of the following chemical

baby?

(17.6)

What

How

which

elderly person, or those in the

particles? (17.7)

15.

Identify

formulas represent pure elements:

He.Na, NaCl, Au, U.

(17.5)

are the four states of matter? (17.9)

of

True or false: We smell things because certain molecules are attracted to our noses.

/l3./a.

What

alu-

(17.8)

molecules.

ol

Cite a substance that

molecules. (17.5)
12.

The Atomic Nature of Matter

toms are mostly empty space, and strucytures such as a floor are composed of atoms

and are therefore also empty space. So why

don't you fall through the floor?

are the two kinds of nucleons? (17.7)

What

is an isotope?
Give two examples. (17.7)

6.

What element

results

if

you add a proton to

the nucleus of carbon? (See periodic table.)

19.

How

number of an element
number of protons in its
number of electrons that

does the atomic

compare

to the

nucleus? To the

7.

What element

8.

You could swallow a capsule of the element

germanium without harm. But if a proton
were added to each of the germanium nu-

results if two protons and two

neutrons are ejected from a uranium nucleus?

normally surround the nucleus? (17.7)

20.

How

2\\

a.

does the mass of an electron compare

to the mass of a nucleon? (17.8)

clei,

b.

What

an ion?
Give two examples. (17.8)

sule.

is

9.

22. At the

atomic

block of iron
mostly empty space. Explain. (17.8)

23.

What is

24.

According

level, a solid

the periodic table of elements?

is

how manv

model

to

swallow the cap-

Why?

Assuming that all the atoms stay in the atmosphere, what are the chances that at least one
of the atoms you exhaled in your very first
breath will be inhaled in vour next breath?

17.8)
10./

the shell

you would not want

of the atom,

iron shells are there in the

What does

the addition or subtraction of

heat have to do with whether or not a substance is a solid, liquid, gas, or plasma?

Solids

Humans have been

classifying solids since the Stone Ages, when

distinguished between rocks for shelter and rocks for
tools. Since then, our knowledge of solids has grown without
pause. Before the turn of the century, it was thought that the content of a solid was what determined its characteristics
what
made diamonds hard, lead soft, iron magnetic, and copper electrically conducting. It was believed that to change a substance,
one merely varied the contents. We have since found that the
characteristics of a solid are due to its structure
that is, the
arrangement of atoms that make up the material.
This shift in emphasis from the study of the content to the
study of the structure of solid materials has changed the role of
investigators from being finders and assemblers of materials to
actual makers of materials. In today's laboratories people are
continually creating new synthetic (humanmade) materials.

they

first

Crystal Structure
When we look carefully at samples of rock minerals, such as quartz,
mica, or galena, we see many smooth, flat surfaces. These flat sur-

'

The mineral
samples are made of crystals, or regular geome tric shapes Each
sample is made of many crystals, assembled in various directions. The samples themselves may have very irregular shapes,
as if they were tiny cubes or other small units glued together to
faces are at angles to each other within the mineral.

make

a freeform solid sculpture.

Not
in

all

many

crystals are evident to the naked eye. Their existence

was not discovered until the advent of X-ray beams

The X-ray pattern caused by the crystal strucof common table salt (sodium chloride) is shown in Figsolids

early in the century.

ture

251

18

252
lire 18-1

Solids

Rays from the X-ray tube are blocked by a lead screen

except for a narrow beam that hits the crystal of sodium chloride. The radiation that penetrates the crystal produces the pattern

shown on

white spot
ol

the photographic film beyond the crystal. The

center is caused by the main unscattered beam
size and arrangement of the other spots indicate

in the

rays. The
arrangement

ol sodium and chlorine atoms in the crystal.

Ever) crystalline structure has its own unique X-ray pattern. All
cr> stals of sodium chloride will always produce this same design.
The patterns made by X rays on photographic film show that
the atoms in a crystal are in an orderly arrangement. For example, in a sodium chloride crystal, the atoms are arranged like
a three-dimensional chess board or a child's jungle gym (Fig-

the

ure 18-2).

X-rav pattern
caused b\ the crystal strucFig. 18-1

ture ol

common

tabic salt

(sodium chloride).

Fig. 18-2

A model of a sodium chloride crystal. The large spheres represent

The small ones represent sodium atoms.

chlorine atoms.

Fig. 18-3

Crystal structure

quite evident on galvanized

(zinc-coated) metals.

is

Metals such as iron, copper, and gold have relatively simple

crystal structures. Tin and cobalt are only a little more complex.
Metal crystals can be seen if you look carefully at a metal surface that has been cleaned (etched) with acid. You can also see
them on the surface of galvanized iron that has been exposed to
the weather, or on brass doorknobs that have been etched by the
perspiration of hands.

253

Density

18.2

Density

18.2

J
One
is

of the properties of solids, as well as liquids

and even

gases,

the measure of the compactness of the material: density

We

think of density as the "lightness" or "heaviness" of materials.

Den sity is a mea^urejaf-hQW. much matter is squeezed into a given
space it is the amount of mass per unit volume:
;

density

mass
volume

What happens

to the density of a chocolate bar when you break

two? The answer is, nothing. Each piece may have half the
mass, but each piece also has half the volume. Density is not
mass and it is not volume. Density is a ratio; it is the amount
of mass per unit volume. A pure iron nail has the same density as
a pure iron frying pan. The frying pan may have 100 times as
many iron atoms and have 100 times as much mass, but they'll
take up 100 times as much space. The mass per unit volume for
the iron nail and the iron frying pan is the same.
it

in

Fig. 18-4

When

the loaf of bread

is

squeezed,

its

volume decreases and

its

density increases.

Both the masses of atoms and the spacing between atoms determine the density of materials. Osmium, a hard bluish-white
metallic element, is the densest substance on earth, even though
the individual osmium atom is less massive than individual
atoms of gold, mercury, lead, and uranium. The close spacing ol
osmium atoms in an osmium crystal gives it the greatest density.
More atoms of osmium fit into a cubic centimeter than other
more massive and more widely spaced atoms.

254

18

Solids

The densities of a few materials, in units of grams per cubic

centimeter, are given in Table 18-1 Densities vary somewhat with
.

temperature and pressure, so except for water, densities are given

at 0C and atmospheric pressure. Note that water at 4C has a
1 .00 g/cm\ The gram is now defined as the mass of a
cubic centimeter of water at a temperature of 4C. Gold, which
has a density of 19.3 g/cm\ is 19.3 times more massive than an
equal volume of water.

density of

Densities of a few substances

Table 18-1

Densitv

Densitv

Liquids

Solids

Osmium

22.6

Mercury

Platinum
Gold

21.4

Glycerin

19.3

Sea water

1.03

Uranium

19.0

1.00

0.90
0.81

Lead

11.3

Water at 4C
Benzene

Silver

10.5

Ethvl alcohol

Copper

8.9

Brass

8.6

Iron

7.8

Steel

7.8

Tin

7.3

Diamond
Aluminum

3.5

13.6

1.26

2.7

Graphite

2.25

Ice

0.92

Pine wood
Balsa wood

0.50
0.12

A quantity known as weight density can be expressed by

amount of weight a body has compared to its volum e:

weight densitv =
.

Weight density

is

the

weight

volume
r-2

commonly used when

discussing liquid pres-

sure (see next chapter)."

Weight density is common to British units, where one cubic foot of fresh water
(almost 7.5 gallons) weighs 62.4 pounds. So fresh water has a weight density
of 62.4 lb/ft 3 Salt
.

water

is

a bit denser, 64

lb/ft'.

255

Elasticity

18.3

Questions
1.

Which has

the greater density

kg of water or 10 kg of

water?
2.

Which has the greater

aluminum?

3.

Which has

density

5 kg of lead or 10 kg of

the greater density

g of uranium or the

whole earth?

The Value of a Simple Computation

One of the reasons gold was used as money is that it is
nearly the most dense of all substances and could therefore
be easily identified. A merchant who suspected that gold
was diluted with a less valuable substance had only to compute its density by measuring its mass and dividing by its
volume. The merchant would then compare this value to
the density of gold, 19.3 g/cm\
Consider a gold nugget with a mass of 57.9 g that is measured to have a volume of exactly 3 cm 3 Is the nugget pure
gold? We compute its density:
.

density
Its

mass
volume

57.9 g
3

cm

19.3 g/cm'

density matches that of gold, so the nugget can be pregold. (It is possible to get the same density

sumed to be pure

by mixing gold with a platinum alloy, but this is unlikely

since platinum is several times more valuable than gold.)

Elasticity
When you hang

a weight on a spring, the spring stretches. When

you add additional weights, the spring stretches still more. When
you remove the weights, the spring returns to its original length.

We
1

say the spring

Answers
The density

of

is

elastic.

any amount of water

(at 4C) is

.00g/cm\

2.

Any amount of lead always has a greater density than any amount
minum. The amount of material is irrelevant.

3.

Any amount
is

of

uranium

actually 5.5 g/cm\

is

much

more dense than

less

the earth.

The density

than the density of uranium (19.0

of alu-

ol the

earth

g/crrT).

256

Solids

18

When a batter hits a baseball, he temporarily changes its

When an archer shoots an arrow, she first bends the bow,

shape.

which springs back to its original form when the arrow is released. The spring, the baseball, and the bow are examples of
elastic objects. Elas ticit y is that prope rty of a body by which it
experiences a change in shape when a deforming force acts on it
and by which it returns to its orig inal shape when the deformingforce

re moved

is

materials return to their original shape when a deforming force is applied and then removed. Materials that do
not r esume their original shape after being distorted are said to"
be inelastic. Clay, putty, and dough are inelastic materials. Lead
\'oi

is

Fig. 18-5

When

The bou

is

removed,

it

returns to

its

is

also inelastic, since

it is

easy to distort

it

permanently.

By hanging a weight on a spring, you are applying a

clastic.

the deforming Force

all

found that the stretch or compression

proportional to the applied force (Figure 18-6).
the spring.

It is

is

force to
directly

original shape.

Fig. 18-6
If

of the spring is directly proportional to the applied force.

doubled, the spring stretches twice as much.

The stretch

the weight

is

This relationship was noted by the British physicist Robert

Hooke, a contemporary of Isaac Newton, in the midseventeenth
century.

It is

pression, x,

called
is

Hooke 's

law. The

amount

of stretch or

com-

directly proportional to the applied for ce F. In

shorthand notation,

is stretched or compressed beyond a ceramount, it will not return to its original state. Instead, it
will remain distorted. The distance bevond which perman ent
distortion occurs is called the elastic limit. Hooke's law holds
only as long as the force does not stretch or compress the material bevond its elastic limit.

If

tain

an elastic material

18.4

257

Compression and Stretching

Questions
1

When

is hung from the end of a tree branch,

observed to sag a distance of 10 cm. If,
instead, a 40-kg load is hung from the same place, by
how much will the branch sag? How about if a 60-kg
load were hung from the same place? (Assume that none
of these loads makes the branch sag beyond its elastic

a 20-kg load

the branch

is

limit.)
2. If

a force of 10

much

stretches a certain spring 4 cm,

stretch will occur for an' applied force of 15

how

N?

Compression and Stretching

Steel is an excellent elastic material. It can be stretched and it

can be compressed. Because of its strength and elastic properties, it is used to make not only springs but construction girders
as well. Vertical girders of steel used in the construction of tall
buildings undergo only slight compression. A typical 25-meterlong vertical girder used in high-rise construction is compressed
about a millimeter when it carries a 10-ton load. Most deformation occurs when girders are used horizontally, where the tendency is to sag under heavy loads.
A horizontal beam supported at one or both ends is under
stress from the load it supports, including its own weight. It
undergoes a stress of both compression and stretching. Consider
the beam supported at one end in Figure 18-7. It sags because of
its own weight and because of the load it carries at its end.

2.

Answers
A 40-kg load has twice the weight of a 20-kg load. In accord with Hooke's law,
F ~ x, two times the applied force will result in two times the stretch, so the
branch should sag 20 cm. The weight of the 60-kg load will make the branch
sag 3 times as much, or 30 cm. (When the elastic limit is exceeded, then the
amount of sag cannot be predicted with the information given.)
The spring

will stretch 6

cm. By

ratio

ION
4 cm

and proportion,

15N
x

read "10 N is to 4 cm as 15 N is to x." Solving for x gives x = (15 N)x

(4 cm)/(10 N) = 6 cm. In lab you will learn that the ratio ol force to stretch is
called the spring constant k (in this case k = 2.5 N/cm), and Hooke's law is
expressed as the equation F = kx.

which

is

18

258

The top part

ol the

What happens

in the

Fig. 18-7

pressed.

Solids

beam is stretched and the bottom pari is commiddle portion, between top and bottom?

Can you see that the top part of the beam tends to be stretched?
Molecules tend to be pulled apart. The top part is slightly longer
because of its deformation. And can you see that the bottom part
of the beam is compressed? Molecules there are squeezed. The
bottom part is slightly shorter because of the way it is bent. So
is stretched, and the bottom part of the beam is comA little thought will show that somewhere in between
the top and bottom, there will be a region where the two effects
overlap, where there is neither compression nor stretching. This

the top part

pressed.

is

the neutral layer.

Consider the beam shown in Figure 18-8. It is supported at

both ends, and carrying a load in the middle. This time the top
of the beam is compressed and the bottom is stretched. Again,
there is a neutral layer along the middle portion of the length of
the

beam.

Fig. 18-8

The top part of the beam

stretched.

Where

is

is

compressed and the bottom part is

is not under stress due

the neutral layer (the part that

to

compression or stretching)?

An I-beam is like a
bar with omeofthe
steel scooped from its middle
where it is need
least. The
Fig. 18-9

solid

beam

is

therefoi

nearlv the

same

ighter For
s

_'th.

Have you ever wondered why the cross-section of steel girders

has the form of the letter / (Figure 18-9)? Most of the material
in these I-beams is concentrated in the top and bottom bars,
whereas the piece joining the bars is of thinner construction.
Why is this the shape?
The answer is that the stress is predominantly in the top and
bottom bars when the beam is used horizontally in construe-

18.5

259

Scaling

tion. One bar tends to be stretched while the other tends to be

compressed. Between the top and bottom bars is a stress-free
region that acts principally to connect the top and bottom bars
together. This is the neutral layer, where comparatively little
material is needed. An I-beam is nearly as strong as if it were a
solid bar, and its weight is considerably less.

Question
If you had to make a hole horizontally through the tree
branch shown, in a location that would weaken it the least,
would you bore it through the top, the middle, or the bottom?

18.5

Scaling

Did you ever notice how strong an ant is for its size? An ant can
carry the weight of several ants on its back, whereas a strong elephant couldn't even carry one elephant on its back. How strong
would an ant be if it were scaled up to the size of an elephant?
Would this "super ant" be several times stronger than an elephant? Surprisingly, the answer is no. Such an ant would not be
able to lift its own weight off the ground. Its legs would be too
thin for its greater weight and would likely break.
Ants have thin legs and elephants have thick legs for a reason.
The proportions of things in nature are in accord with their size.
The study of how size affects the relationship between weight,
sTrength, and surface area is known as scaling. As the size of

Answer
It would be

best to drill the hole in the middle, through the neutral layer.

Wood

branch are being stretched, and if you drilled the holethere, that part of the branch may pull apart. Fibers in the lower part are being
compressed, and if you drilled the hole there, that part ol the branch might
crush under compression. In between, in the neutral layer, the hole will not affect the strength of the branch because fibers there are being neither stretched
nor compressed.
fibers in the top part of the

260

18

Solids

it grows heavier much faster than it grows

You can support a toothpick horizontally at its ends
and you'll notice no sag. But support a tree of the same kind of
wood horizontally at its ends and you'll see a noticeable sag. The
tree is much heavier compared to its strength than the toothpick.
Weight depends on volume, and strength comes from the area
ol the cross-section. To understand this weight-strength relationship, consider a very simple case
a solid cube of matter,
centimeter on a side.
A -cubic centimeter cube has a cross section of square centimeter. That is, if we sliced through the cube parallel to one of
its laces, the sliced area would be
square centimeter. Compare
this to a cube that has double the linear dimensions, a cube 2
centimeters on each side. Its cross-sectional area will be 2 x 2
(or 4) square centimeters and its volume will be 2 x 2 x 2 (or 8)
cubic centimeters. If it has the same density it would be 8 times
heavier. Careful investigation of Figure 18-10 shows that for increases of linear dimensions the cross-sectional area (as well as
the total area) grows as the square of the increase, whereas volume and weight grow as the cube of the increase.

a thing increases,

stronger.

LENGTH OF SIDE = 2 cm
CROSS- SECTION AREA = 4

>

VOLUME

MASS =

(2*2*2) = 8 cm*
8 GRAMS

<M^>
V NT
LENGTH OF SIDE= 1cm
CROSS-SECTION AREA=1cm

VOLUME

(U1*1)

MASS =

s=

1cm 5

GRAM

LENGTH OF SIDE = 4 cm

CROSS- SECTION AREA = 16cm

3
=
VOLUME (4*4 *4) 64 cm

"^\
LENGTH OF SIDE = 3 cm
CROSS- SECTION AREA= 9 cm 2
VOLUME (3*3*3) = 27 cm 3
MASS =

27

MASS = 64 GRAMS

GRAMS
Fig. 18-10

II

the linear dimensions of an object are multiplied by

some num-

grows bv the square of the number, and the volume (and

hence the weight) grows by the cube of the number. If the linear dimensions of
2
the cube are increased by 2. the area grows bv 2 = 4, and the volume grows by
2* = 8. If the linear dimensions are increased bv 3. the area grows bv 3' = 9,
and the volume grows by 3 3 = 27.
ber, then the area

261

Scaling

18.5

The volume (and weight) multiplies much more than the corresponding increase of cross-sectional area. Although the figure
demonstrates the simple example of a cube, the principle applies to an object of any shape. Consider an athlete who can lift
his weight with one arm. Suppose he could somehow be scaled
up to twice his size that is, twice as tall, twice as broad, his
bones twice as thick and every linear dimension increased by a
factor of 2. Would he be twice as strong? Would he be able to lift
himself with twice the ease? The answer to both questions is
no. Since his twice-as-thick arms would have 4 times the crosssectional area, he would be 4 times as strong. At the same time,
his volume would be 8 times as great, so he would be 8 times as
heavy. Thus, he could lift only half his weight. In relation to his
weight, he would be weaker than before.
The fact that volume (and weight) grow as the cube of the
increase, while strength (and area) grow as the square of the
increase is evident in the disproportionate thick legs of large
animals compared to small animals. Consider the different legs
of an elephant and a deer; or a tarantula and a daddy longlegs.

Questions
1

Suppose a cube 1 cm long on each side were scaled up

to a cube 10 cm long on each edge. What would be the
volume of the scaled-up cube? What would be its crosssectional surface area? Its total surface area?

an athlete were somehow scaled up proportionally to

twice size, would he be stronger or weaker?

2. If

to King Kong and other

cannot be taken seriously. The fact that the consequences of scaling are conveniently omitted is one of the differences between science and science fiction.

So the great strengths attributed

fictional giants

Answers
The volume

of the scaled-up cube would be (length of side) = (10 cm)*, or

1000 cm*. Its cross-sectional surface area would be (length of side)- = (10 cm)',
or 100 cm 2 Its total surface area = 6 sides x (length of side)' = 600 cm-.
1

2.

The scaled-up athlete would be 4 times as strong, because the

area of his twice-as-thick bones and muscles increases by a
could

lift

8 times as

maximum
much

load 4 times as heavj as before. But his

as before, so he

would be weaker

cross-sectional
factor ol 4

Ik-

own weight

in relation to his

is

weight.

8 times the weight gives him

former value. This means that if
weight before, he could now Mil only hall his new

Having 4 times the strength while carrying

a strength- to- weight ratio ol only hall

its

he could just lilt his own

weight. In sum, while his actual strength would increase, his strength-toueight ratio would decrease

262

Solids

18

THE SURFACE AREA OF A

cm 5 VOLUME (SHOWN OPENED
z
THE RATIO OF
UP) IS 6cm
1

SURFACE AREA

VOLUME

"

6_
1

WHEN THE VOLUME OF A CUBE

WHEN

(27Crn

VOLUME OF A CU6E
IS 2*2*2 (8 cmv THE SURFACE
AREA IS 24 cm z THE RATIO OF
THE

THE SURFACE AREA

IS

IS

33*3

54 cm z

SURFACE AREA
rue
THE dat.A
F
M1{0 rsc
VOLUME

SURFACE AREA _ 24 _ 3

VOLUME

),

"
8

54 = 2
27

Fig. 18-1

increase in

As an object grows proportionally

volume than in surface area. As a

in all directions, there is a greater

result, the ratio of surface

area to

volume decreases.

Fig. 18-12

phant has

The African
less surface

ele-

area

to its u eight than

other animals. It compensates
for this with its large ears,

compared

which significanth increase

its radiating surface irea and
promote cooling.

Important also is the comparison of total surface area to volume. A study of Figure 18-11 shows that as the linear size of an
object increases, the volume grows faster than the total surface
area. (Volume grows as the cube of the increase, and both crosssectional area and total surface area grow as the square of the
increase.) So as an object grows, its surface area and volume
grow at different rates, with the result that the surface area to
volume ratio decreases. In other words, both the surface area
and the volume of a growing object increase, but the growth of
surface area compared to the growth of volume decreases. Relatively few people really understand this idea. The following examples may be helpful.
The big ears of elephants are not for better hearing, but for
cooling. They are nature's way of making up for the small ratio
of surface area to volume for these large animals. The heat that
an animal radiates is proportional to its surface area. If an elephant did not have large ears, it would not have enough surface
area to cool its huge mass. The large ears of the African elephant
greatly increase overall surface area, and enable it to cool off in
hot climates.
At the biological level, living cells must contend with the fact
that the growth of volume is faster than the growth of surface
area. Cells obtain

nourishment by diffusion through their

sur-

263

Scaling

18.5

As cells grow, surface area increases, but not fast enough

keep up with volume. For example, if the surface area increases by four, the corresponding volume increases by eight.
Eight times as much mass must be sustained by only four times
as much nourishment. This puts a limit on the growth of a living
faces.

to

So cells divide, and there is life as we know it. That's nice.

Not so nice is the fate of large animals when they fall. The statement "the bigger they are, the harder they fall" holds true and is
a consequence of the small ratio of surface area to weight. Air
resistance to movement through the air is proportional to the
surface area of the moving object. If you fall out of a tree, even in
the presence of air resistance, your speed increases at the rate of
very nearly lg. You don't have enough surface area compared to
your weight unless you wear a parachute. Small animals, on
the other hand, need no parachute. They have plenty of surface
area compared to their small weights. An insect can fall from
the top of a tree to the ground below without harm. The surface
area per weight ratio is in the insect's favor in a sense, the incell.

own

parachute.
The fact that small things have more surface area compared
to volume, mass, or weight is evident in the kitchen. An experienced cook knows that more skin results when peeling 5 kg of
small potatoes than when peeling 5 kg of large potatoes. Smaller
objects have more surface area per kilogram. Crushed ice will
cool a drink much faster than a single ice cube of the same mass,
because crushed ice presents more surface area to the beverage.
The rusting of iron is also a surface phenomenon. Iron rusts
when exposed to air, but it rusts much faster and is soon eaten
away if it is in the form of small filings.
Chunks of coal burn, while coal dust explodes when ignited.
Thin french fries cook faster in oil than fat fries. Flat hamburgers
cook faster than meatballs of the same mass. Large raindrops
fall faster than small raindrops, and large fish swim faster than
small fish. These are all consequences of the fact that differences
in volume and differences in area are not in the same proportion
to each other.
It is interesting to note that the rate of heartbeat in a mammal is proportional to the size of the mammal. The heart of a
tiny shrew beats about twenty times as fast as the heart of an
elephant. In general, small mammals live fast and die young;
larger animals live at a leisurely pace and live longer. Don't feel
bad about a pet hamster that doesn't live as long as a dog. All
warmblooded animals have about the same lifespan not in
terms of years, but in the average number of heartbeats (about
800 million). Humans are the exception: we live two to three
times longer than other mammals of our size.
sect

is its

264

Solids

18

Chapter Review

Concept Summary

What evidence can you

2~J

scopic crystal nature

Man)

made

solids are

The atoms

an ordcrh

arrangement.

[3j What evidence can you

v^
is

density, the

amount

4.)

A second property

(18.2)
ol solids is elasticity.

Which has

6.)

Elastic materials return to their original

shape when a deforming force is applied

and removed, as long as they are not deformed beyond their elastic limit.
According to Hooke's law, the amount of
stretch or compression is proportional to

remain distorted
removed.

Inelastic materials

the force

Scaling

is

is

the study of

how

What

8\

after

is

the difference between

and weight density?

What

9?) a.

and

a heavy bar
( 1

8.2)

Does the mass of a loaf of bread change when

you squeeze it? Does its volume change? Does
its density change? (18.2)

7.)

is

size affects the rela-

tionship between weight, strength,

the greater density

of pure gold or a pure gold ring?

the applied force (within the elastic limit.)

What happens to the density of a uniform

piece of wood when you cut it in half? (18.2)

Uranium is the heaviest atom. Why is uraf\ Urs

V/ niu m metal not the most dense material?

unit volume.

cite for the visible

related both to the masses of the

Density

atoms and the spacing between atoms.

Weight density is th-.- amount of weight per

micro-

crystal nature of certain solids? (18.1)

of

is

cite for the

certain solids? (18.1)

ol crystals.

in a crystal are in

One property ol solids

mass per unit volume.

ol

steel

surface.

is

the evidence for the claim that

elastic?

That putty

b.

mass density

(18.2)

is

inelastic? (18.3)

Important Terms

(M What

is

Hooke's law? (18.3)

crystal (18.1)

(Tl.

What

is

an

elastic limit? (18.3)

density (18.2)
elastic (18.3)
elastic limit (18.3)

\12.J

If

the weight of a 2-kg

by

elasticity (18.3)

cm, how much

mass stretches a spring

will the spring stretch

when it supports 6 kg? (Assume the spring

has not reached its elastic limit.) (18.3)

Hooke's law (18.3)

inelastic (18.3)

scaling (18.5)

weight density (18.2)

Review Questions

^b

How

does the arrangement of atoms differ in

and noncrystalline substance?

a crystalline
(18.1)

a steel beam slightly shorter

stands vertically? (18.4)

Is

\14/\What

is

the neutral layer in a

^-^ supports a

load ?

( 1

when

beam

it

that

8 .4)

445 J Why are the cross sections of metal beams

I-shaped, not rectangular? (18.4)

Chapter Review

18

What

16.J

265
,4.lSuppose the spring in the preceding question is placed next to an identical spring so
that both side-by-side springs support the
8-kg load. By how much will each spring

the weight-strength relationship in

is

scaling? (18.5)

17J

the linear dimensions of an object are

doubled, by how much does the overall
area increase?
Bv how much does the volume increase?

a.

If

b.

&
1

stretch?

Compression and stretching

(18.5)

beam

in a

load

As the volume of an object is

surface area also increases,
but the ratio of its surface area to volume
decreases. Explain. (18.5)

True or

false:

increased,

/ 19.)

Which

V_^

ice

will cool a drink faster

its

own

beam

carrying

is

how

if

the

Show by means

weight).

of a simple sketch

its

cube or 10 grams of crushed

is

stresses occur

that supports loads (even

a horizontal load-

stretched at the top and

compressed at the bottom. Then show a case

where the opposite occurs: compression at
the top and stretching at the bottom.

10-gram

ice? (18.5)

Me
letal beams are not "solid" like wooden
\_ beams, but are "cut out" in the middle so
that their cross section has an I-shape. What
are the advantages of this shape?

fc.

20. /a.

b.

Which has more

mouse?
Which has more

an elephant or a
skin per body weight

skin

an elephant or a mouse?

(18.5)
([

Consider a model

"M

steel

bridge that

1/100

is

the exact scale of the real bridge that

^j7

is

to

be

built.
a.

model bridge weighs 50 N, what

weigh?
If the model bridge doesn't appear to sag
under its own weight, is this evidence
If

the

will the real bridge

Activity

b.

you nail 4 sticks together to form a rectangle,

they can be deformed into a parallelogram without too much effort. But if you nail 3 sticks together to form a triangle, the shape cannot be
changed without actually breaking the sticks or
If

the nails.

The

triangle

is

the strongest of all the

that the real bridge,

you use a batch of cake batter for cupinstead of a cake and bake them for
the time suggested for baking a cake, what
will be the result?

8 A If

\_/ cakes

geometrical figures. Try it and see, and then look

at the triangles used in strengthening structures
of

many

built exactly to

if

scale, will not sag either? Explain.

kinds.
\

Explain, in terms of scaling,

9.

V^vantage

why

it

is

an ad-

that natives of the hot African des-

ert tend to

be relatively

tall

and slender, and

natives of the Arctic region tend to be short

Think and Explain

T*

Which has
of

2).

and

Which has
or a

liter

the greater weight

liter of ice

10.

it

supports.

A piece

of wire will cool

stretched out than

when

rolled

ball.)

Nourishment

is

obtained from food through

the inner surface area of the intestines.

of water?

A certain spring stretches

gram

when

up into a

is it

3.

stout. (Hint:

faster

volume a kilogram
lead or a kilogram of aluminum?
the greater

If

reached, how far will

ports a load of 8 kg?

cm

for each kilo-

the elastic limit

it

stretch

when

is
it

not
sup-

that a small organism, such as a

Whv

worm,

has a simple and relatively straight intestinal tract, while a large organism, such as
a human being, has a complex and manyfolded intestinal tract?

Liquids

We

live on the only planet in the solar system covered predominantly by a liquid. The earth's oceans are made of H 2 in the liquid state. If the earth were a little closer to the sun, the oceans
would turn to vapor. If the earth were a little farther, its surface
would be solid ice. It's nice that the earth is where it is.
In the liquid state, molecules can flow. They freely move from
position to position by sliding over one another. The shape of a

liquid takes the shape of

19.1

its

container.

Pressure in a Liquid
A liquid in a container exerts forces against the walls and bottom of the container. To investigate the interaction between the
liquid and the walls, it is useful to discuss the concept of pressure. Recall from Chapter 4 that pressure is defined as the force
per area on which the force acts
.

=
pressure
r

force

area

The pressure that a block exerts against a table is simply the

weight of the block divided by its area of contact. Similarly, for a
Fig. 19-1

The liquid exerts a

liquid in a container, the pressure the liquid exerts against the

pressure against the bottom

of its container, just as the
block exerts a pressure against
the table.

*
Pressure may be measured in any unit of force divided by any unit of area.
The standard international (SI) unit of pressure, the newton per square meter,
is called the pascal (Pa), named after the seventeenth century theologian and
Pa is very small, approximately the presscientist Blaise Pascal. A pressure of
sure exerted by a dollar bill resting flat on a table. Science types more often
use kilopascals (1 kPa = 1000 Pa).
1

266

19.1

267

Pressure in a Liquid

bottom of the container

is the weight of the liquid divided by the

area of the container bottom.
How much a liquid weighs, and hence how much pressure it

depends on its density. Consider two identical containers,

one filled with mercury and the other filled to the same depth
with water. For the same depth, the denser liquid exerts the
greater pressure. Mercury is 13.6 times as dense as water. So for
the same volume of liquid, the weight of mercury is 13.6 times
the weight of water. Thus, the pressure of the mercury on the
bottom is 13.6 times the pressure of the water.
For liquids of the same density, the pressure will be greater at
the bottom of the deeper liquid. Consider the two containers in
Figure 19-2. If the liquid in the first container is twice as deep as
the liquid in the second container, then like two blocks one atop
exerts,

the other, the pressure of the liquid at the

bottom

of the first con-

tainer will be twice that of the second container.

The two blocks exert twice as much pressure as one block against

Fig. 19-2

first container is twice as deep, so the

twice that exerted by the liquid in the sec-

the table. Similarly, the liquid in the

pressure

it

exerts on the

bottom

is

ond container.
It turns out that the pressure of a liquid at rest depends only
on the density and the depth of the liquid. Liquids are practically incompressible, so except for changes in temperature, the
density of a liquid is normally the same at all depths. The pres-

sure of a liquid*

is

pressure = weight density x depth.

This relationship is derived from the definitions of pressure and density.

Consider an area at the bottom of a container of liquid. Pressure is produced
by the weight of the column of liquid directly above this area. From the definition of weight density as weight divided by volume, this weight of liquid can
be expressed as weight density times volume. The volume of the column is

simply the area multiplied by the depth. Then we get

pressure =

force

area

_ weight _ weight density x volume

area

area

x ^ifea-x depth
.
...
densitv
, _..
*z
b
= weight densitv _,
x depth
= weight
.

268

Liquids

19

At a given depth, a given liquid exerts the same pressure

the bottom or sides of its container, or even
against any surface
the surface of an object submerged in the liquid to that depth.
The pressure a liquid exerts depends only on the density and

depth of the liquid and on the atmospheric pressure. At the top of

the liquid, the pressure equals the atmospheric pressure. Lower

down, the pressure against any surface is greater the greater the
depth. If one liquid is more dense than a second liquid, then at the
same depth the pressure is greater in the more dense liquid.
Interestingly enough, the pressure does not depend on the
amount of liquid. Neither the volume nor even the total weight
of liquid matters. For example, if you sampled water pressure at
one meter beneath a large lake surface and one meter beneath a
small pool surface, the pressures would be the same." The dam
that must withstand the greater pressure is the dam with the
deepest water behind it, not the most water (Figure 19-3).

LARGE BUT SHALLOW

LAKE
,

The water pressure is greater at the bottom of the deeper pond, not
pond with the most water. The dam holding back water twice
as deep must withstand greater average water pressure, regardless of the total
volume of water.
Fig. 19-3

necessarily the

The

ume

fact that

water pressure depends on depth and not on

vol-

nicely illustrated with "Pascal's vases" (Figure 19-4).

is

Note that the water surface in each of the connected vases is at

the same level. This occurs because the pressures at equal depths

The density

of fresh

water

is

gram per cubic centimeter, which

is

the

same

as 1000 kilograms per cubic meter. Since the weight (mg) of 1000 kilograms

is

(1000 kg) x (9.8 N/kg) = 9800 N, the weight density of water is 9800 newtons
per cubic meter (9800 N/m J). (Seawater is slightly denser than fresh water; its
weight density is about 10 000 N/'m\) Water pressure in a pool or lake is simply
equal to the atmospheric pressure plus the product of the weight densitv of
fresh water and the depth in meters. For example, at a depth of 1 m the water
pressure is 101 300 N/nr + 9 800 Him 1 or 10 100 Him 1 In SI units, pressure
2
is measured in pascals (1 Pa = 1 N/m ), so this would be
10 100 Pa; or in kilo,

pascals,

10.1 kPa.

19.1

269

Pressure in a Liquid

beneath the surfaces are the same. At the bottom of each vase,
for example, the pressures are equal. If they were not, the liquid
would flow until the pressures were equalized. This is the reason
that water seeks its own level.

Fig. 19-4

The pressure

of the liquid

is

the

same

at

any given depth below the

surface, regardless of the shape of the container.

Questions
there more pressure at the bottom of a bathtub of water
30 cm deep or at the bottom of a pitcher of water 35 cm
deep?

Is

2.

brick mason wishes to mark the back of a building at

the exact height of bricks already layed at the front of

the building. How can he measure the same height using

only a transparent garden hose and water?

At any point within a liquid, the forces that produce pressure

when you are

swimming under water, no matter which way you tilt your head,
you feel the same amount of water pressure on your ears.
are exerted equally in all directions. For example,

When

the liquid

is

pressing against a surface, there

is

a net

force directed perpendicular to the surface (Figure 19-5 top).

If

there

is

a hole in the surface, the liquid initially

moves

per-

pendicular to the surface (Fig. 19-5 bottom). Gravity, of course,

causes the path of the liquid to curve downward. At greater
depths, the net force

escaping liquid

2.

is

greater and the horizontal velocity of the

greater.

is

Answers
is more pressure at the bottom of
in it. The fact that there is more water

the pitcher, because the water

in the

is

deeper

bathtub does not matter.

To measure the same height, the brick mason can extend a garden hose from
the front to the back of the house, and fill it with water until the water level
level of

water

ol

in the

bricks

in

other end

(Top) The forces in

a liquid that produce pres-

sure against a surface add up

to a net force that

is

perpen-

dicular to the surface. (Bot-

tom) Liquid escaping through

moves perpen-

a hole initially

dicular to the surface.

There

reaches the height

Fig. 19-5

the front. Since water seeks

ol

the hose will be the same!

its

own

level, the

270

19

Liquids

If you have ever lilted a submerged object out of water, you are
familiar with buoyancy, the apparent loss of weight of objects

when submerged

in a liquid.

It

is

a lot easier to

lift

a boulder

submerged on the bottom of a river bed than to lift it above the

water surface. The reason is that when the boulder is submerged,
the water exerts an upward force that is opposite in direction to
gravity. This upward force is called the buoyant force.
To understand where the buoyant force comes from, look at
Figure 19-6. The arrows represent the forces by the liquid that
produce pressure against the submerged boulder. The forces are
greater at greater depth. The forces acting horizontally against
the sides cancel out, so the boulder is not nudged sideways. But
the forces acting upward against the bottom are greater than
Fig. 19-6

The upward forces

against the bottom of a sub-

those acting downward against the top. This is simply because

the bottom of the boulder is deeper within the water. The differ-

ence
If

merged object are greater

than the

downward

forces

against the top. There

upward

force, the

is

a net

buoyant

upward and downward forces is the buoyant force.

submerged object is greater than the buoyant

in

the weight of a

is equal to the buoyant

remain at any level, like a fish.
than the buoyant force, the object will rise to

force, the object will sink. If the

force, the
If

submerged object

the weight

force

the surface

is

and

less

weight

will

float.

To further understand buoyancy,

about what happens when an object

it

is

helps to think some more

placed in water. If a stone

is placed in a container of water, the water level will rise (Figure 19-7). Water is said to be displaced, or moved elsewhere,
by the stone. A little thought will tell us that the volume that
is, the amount of space taken up or the number of cubic centimeters of water displaced is equal to the volume of the stone.
A completely submerged object always displaces a volume of liquid
equal to its own volume.

THIS VOLUME OF
LIQUID = VOLUME
OF THE STONE

Fig. 19-7
to the

When an

volume

object

of the object

is

submerged,

itself.

it

displaces a volume of water equal

271

Archimedes' Principle

19.3

way to determine the volume of an irregushaped object. Simply submerge it in water in a measuring
cup and note the increase in volume of the water. That increase
This gives us a good

larly

in

volume

is

volume

also the

of the

submerged

object.

THIS VOLUME OF
LIQUID = VOLUME
OF THE STONE

Fig. 19-8
full,

the

When an

volume

object

is

submerged

of water that overflows

in a container that
is

is

initially

brim

equal to the volume of the object

itself.

Archimedes' Principle
The relationship between buoyancy and displaced liquid was
discovered in ancient times by the Greek philosopher Archimedes (third century B.C.). It is stated as follows:

An immersed

object

is

the weight of the fluid

This relationship

is

buoyed up by a force equal

it

to

displaces.

called Archimedes' principle.

It is

true of

and gases, which are both fluids.

The word immersed means either completely or partially submerged. For example, if we immerse a sealed one-liter container
halfway into water, it will displace a half liter of water and be
buoyed up by the weight of a half liter of water. If we immerse it
all the way (submerge it), it will be buoyed up by the weight of
liquids

full liter

of water (9.8 newtons). Unless the completely sub-

merged container

compressed, the buoyant force will equal

is because it
will displace the same volume of water, and hence same weight
of water, at all depths. The weight of this displaced water (not
the weight of the submerged object!) is the buoyant force.
A 300-gram block weighs about 3 N in air. Suppose the block
displaces 2 N of water when it is submerged (Figure 19-10). The
buoyant force on the submerged block will also equal 2 N. The
block will seem to weigh less under water than above water. In
the water, its apparent weight will be 3 N minus the buoyant
force of 2 N, or only
N. The apparent weight of a submerged
object is its weight in air minus the buoyant force.
is

the weight of one liter of water at any depth. This

Fig. 19-9

liter of

water

occupies 1000 cubic centimeters, has a mass of one

kilogram, and weighs 9.8 N.
Any object with a volume of

one liter will experience a

buoyant force of 9.8 N when
fully

submerged

in water.


272

19

Liquids

Fig. 19-10
A block weighs less in water than in air. The buoyant force that acts
on the submerged block is equal to the weight of the water displaced. So the
block appears lighter under water by an amount equal to the weight of water
(2 N) that has spilled into the smaller container. The apparent weight of the
block under water equals its weight in air minus the buoyant force (3 N - 2 N

N).

Questions
1.

2.

A block

1 -liter (L) container completely filled with mercury

has a mass of 13.6 kg and weighs 133.3 N. If it is submerged in water, what is the buoyant force acting on it?

held suspended beneath the water in the three

and C, shown in Figure 19-11. At what
position is the buoyant force on it greatest?
is

positions, A, B,

3.

1.

A stone

is thrown into a deep lake. As it sinks deeper and

deeper into the water, does the buoyant force on it increase? Decrease? Remain unchanged?

Answers
The buoyant force is equal to the weight of 1 L of water about 10 N because the volume of water displaced is
L. The mass or weight ot mercury is
irrelevant; 1 L of anything submerged in water will displace
L of water and
be buoyed upward with a force of 1C 4. Or look at it this way: when the con-

tainer

is

put

The water,

in the water,

it

pushes

of water out of the

way by

its

presence.

and reaction for n, pushes back on the container with a

force equal to the weight of
L of water. If the container had twice the volume, it would displace 2 L of water and be buoyed up with a force equal to the
weight of 2 L of water. (The concepts involved with buoyant force are best
conceptual physics'.)
dealt with if you visualize what is going on
in action

2.

The buoyant

amount

force will be the

will equal the

The difference in
the upward and downward
force acting on the submerged block is the same at
any depth.

B,

Fig. 19-11

3.

and

same

Why? Because the

each case. The buoyant force

at all three positions.

of water displaced will be the

same

in

weight of the water displaced, which

is

the

same

at positions A,

C.

Like the previous question, the volume of water displaced

is practically incompressible, so its density

depth. Water

is
is

the
the

same at any
same at all

depths, and equal volumes of water will weigh the same. The buoyant force on
the sinking stone will be the

same

at all depths.

19.4

The

Effect of Density

273

on Submerged Objects

Perhaps your teacher will summarize Archimedes' principle

by way of a numerical example to show that the difference between the upward acting and the downward acting forces due to
water pressure on a submerged block is numerically identical to
the weight of liquid displaced. It makes no difference how deep
the block is submerged, for although the pressures are greater
with increasing depth, the difference in pressures on the bottom and top of the block is the same at any depth (Figure 19-11).
Whatever the shape of a submerged object, the buoyant force is
equal to the weight of liquid displaced.

The

19.4

Effect of Density

on Submerged Objects

You have learned that the buoyant force that acts on a submerged
object depends on the volume of the object. Small objects displace small amounts of water and are acted on by small buoyant
forces. Large objects displace large amounts of water and are
acted on by larger buoyant forces. It is the volume of the submerged object not its weight that determines buoyant force.

misunderstanding of this idea is at the root of a lot of confusion that you or your friends may have about buoyancy!)
Thus far, the weight of the submerged object has not been
(A

taken into account. Now we will consider its role.

Whether an object will sink or float in a liquid has to do with
how great the buoyant force is compared to the object's weight.
Careful thought will show that if the buoyant force is exactly
equal to the weight of a completely submerged object, then the
weight of the object must be the same as the weight