Daniel Kleppner
of

Professor

Associate

Physics

Massach usetts Institute

of Technology
J.

Robert

Formerly

Associate

Kolenkow

Professor

of Physics,Massachusetts
Institute

of

AN

INTRODUCTION

Technology)

TO

MECHANICS)

II)
Boston,

Massachusetts

Burr

Ridge,

Illinois

New
Dubuque, Iowa Madison,Wisconsin
San Francisco, California St. Louis,Missouri)))

York,

New

York



McGraw-Hill)
A

AN

Copyright
All

INTRODUCTION

TO

rights

of The McGraw.HiU Companies

Division

@ 1973 by McGraw-Hill, Inc.
reserved. Printed in the United States

of America. Except

as permitted

under the Copyright Act of 1976, no part of this publication
may be reproduced or
in any form or by any means, or stored in a data base or retrieval
distributed
without the prior written

of the publisher.)
system,
permission

MECHANICS)
Printed

and

by Book-mart

bound
20

BKMBKM

Press, Inc.)

998)

This book was set in News Gothic by The Maple Press Company.
The editors were Jack L. Farnsworth
and J. W. Maisel;
the
was Edward A. Butler;
designer
and the production supervisor was
Sally
Ellyson.
The drawings were done by Felix Cooper.)


Library

of Congress

Cataloging

in

Publication

Data)

Kleppner, Daniel.
An

introduction

to mechanics.)

1. Mechanics.
QA805.K62
ISBN 0-07-035048-5)

I.

Kolenkow,

531


Robert,

joint

72-11770)

author.

II

Title.)))


To

our

parents

Beatrice and Otto
Katherine

and

John)))



OF EXAMPLES
xv


LIST

xi

PREFACE

CONTENTS)

TO THE

1

1.1

VECTORS

AND

KINEMATICS

-A

FEW

TEACHER

xix)

2


INTRODUCTION

1.2 VECTORS 2
of a Vector, The Algebra
Definition

1.3

OF

COMPONENTS

MATHEMATICAL

1.4

BASE VECTORS

PRELIMINARIES)

1.5

DISPLACEMENT

1.6

VELOCITY

Motion in


10

AND

THE POSITION
13
ACCELERATION
AND

FORMAL

1.8

MORE

1.9

MOTION

Dimension, 14; Motion in Several
14; A Word
Dimensions,
Units, 18.
SOLUTION
OF KINEMATICAL EQUATIONS 9
THE DERIVATIVE OF A VECTOR
23
ABOUT
IN PLANE POLAR COORDINATES 27


27; Velocity in
Coordinates,
in Polar Coordinates,

Acceleration

The

1.1

Series,

References

PROBLEMS

2

NEWTON'S LAWS 53

MECHANICS)

First

Law,

MOMENTUM)

55;


Second

Newton's

Law,

56;

Newton's

Third Law, 59.

2.3 STANDARDSAND UNITS
64
67.
The Fundamental
64; Systems of Units,
Standards,
2.4 SOME APPLICATIONSOF NEWTON'S
68
LAWS
2.5 THE EVERYDAY
OF PHYSICS 79
FORCES
80; The Electrostatic Force, 86;
Field,
Weight, and the Gravitational
Gravity,
and Atomic

Contact
Force of a String,
87; Tension-The
87; Tension
Forces,
Forces,91;The Normal Force, 92; Friction, 92; Viscosity, 95; The Linear Restoring
97.
Force: Hooke's Law, the Spring, and Simple Harmonic
Motion,
Note 2.1 THE GRAVITATIONAL
OF A SPHERICAL
ATTRACTION
101
SHELL

PROBLEMS

3

52

INTRODUCTION

2.2

NEWTONIAN

dr/dt, 31;

to Calculus Texts, 47.


2.1

Newton's

27; Evaluating

Coordinates,

36.

47)

NEWTON'S

OF

Polar

APPROXIMATION METHODS 39
45.
41; Taylor's Series,42;Differentials,

LAWS-THE
FOUNDATIONS

about

MATHEMATICAL


Binomial

Some

VECTOR 11

and

1.7

Note

8

One

Dimensions

Polar

3.

of Vectors,

A VECTOR

3.1

3.2


112
INTRODUCTION
DYNAM
ICS OF A SYSTEM

Center of
3.3

103)

IMPULSE

RELATION

3.5

PARTICLES

OF MOMENTUM

CONSERVATION

Center of Mass Coordinates,
3.4

OF

113

Mass, 116.

AND

122

127.

A RESTATEMENT

OF THE

MOMENTUM

130

MOMENTUMAND

THE

FLOW OF

MASS

133)))


CONTENTS)

viii)

3.6

Note

MOMENTUM TRANSPORT 139
3.1
CENTER OF MASS 145
147)

PROBLEMS

WORK

4

AND
ENERGY)

4.1

INTRODUCTION

4.2

INTEGRATING

4.4
DI

THE

EQUATION


OF MOTION

IN

ONE

153

DI M ENSION

4.3

152

THE

THEOREM

WORK-ENERGY

INTEGRATING THE
M ENS IONS 158

EQUATION

IN

OF


156
DIMENSION
ONE
MOTION IN SEVERAL

4.5

THE
WORK-ENERGY
160
THEOREM
4.6 APPLYING THE WORK-ENERGY
162
THEOREM
4.7 POTENTJAL ENERGY
168
Illustrations
of Potential Energy, 170.
4.8 WHAT POTENTIAL
ENERGY TELLS US ABOUT FORCE
Stability, 174.
4.9 ENERGY DIAGRAMS 176
4.10 SMALL OSCILLATIONS
IN A BOUND SYSTEM 178
4.11 NONCONSERVATIVE
FORCES
182
4.12 THE GENERAL
LAW OF CONSERVATION OF ENERGY


4.13

173

184

186

POWER

187
CONSERVATION LAWS AND PARTICLE
COLLISIONS
Collisions and Conservation
Collisions,
188; Elastic and Inelastic
Laws,
Collisions in One Dimension, 189; Collisions and Center of MassCoordinates,

4.14

PROBLEMS

5

SOME

MATHEMATICAL

ASPECTS


OF FORCE
AND

ENERGY)

194)

INTRODUCTION
202
PARTIAL DERIVATIVES 202
5.3 HOW TO FIND THE FORCE IF YOU KNOW THE POTENTIAL
206
ENERGY
5.4 THE GRADIENT OPERATOR 207
5.5
THE PHYSICAL MEANING
OF THE GRADIENT 210
Constant
211.
Surfaces and Contour Lines,
Energy
5.6 HOW TO FIND
215
OUT
IF A FORCE IS CONSERVATIVE
5.7 STOKES' THEOREM 225
5.1

5.2


PROBLEMS

6

ANGULAR

6.1

MOMENTUM 6.2
AND

FIXED

AXIS

ROTATION)

228)

INTRODUCTION

232

MOMENTUM

ANGULAR

6.3 TORQUE


OF A PARTICLE

233

238

ANGULAR MOMENTUMAND FIXED AXIS ROTATION 248
OF PURE ROTATION ABOUT AN AXIS
253
DYNAMICS
6.6 THE PHYSICALPENDULUM
255
The Simple Pendulum, 253; The Physical Pendulum, 257.
6.7 MOTION
INVOLVING
ROTATION
BOTH TRANSLATION AND
The Work-energyTheorem,
267.
6.8 THE BOHR ATOM 270
Note 6.1 CHASLES' THEOREM 274
Note 6.2
PENDULUM MOTION 276
6.4

6.5

PROBLEMS

279)))


260

188;
190.


CONTENTS)

7

BODY
MOTION

RIGID

AND

THE

CONSERVATION

OF
ANGULAR

MOMENTUM)

ix)

7.1


INTRODUCTION

7.2

THE VECTOR NATURE

7.4
7.5
7.6

OF

MOMENTUM

ANGULAR
7.3

288

OF GYROSCOPE MOTION 300
ANGULAR MOMENTUM 305
OF A ROTATING RIGID
BODY
ANGULAR
MOMENTUM
308
Axes, 313; Rotational
Angular Momentum and the Tensor of Inertia, 308;Principal
Kinetic

313; Rotation about a Fixed Point, 315.
Energy,
7.7 ADVANCED TOPICS IN THE DYNAM ICS OF RIGID
BODY
ROTATION
316

Note

APPLICATIONS

SOME

OF

CONSERVATION

7.1

AND

FICTITIOUS

FORCES)

Why the

Precession:

ROTATIONS


INFINITESIMAL

GYROSCOPES

ABOUT
Precession,

Earth

Wobbles,

317; Euler's

326

328

2 Torque-free Precession,331;

331; Case

Case

3

334)

PROBLEMS


SYSTEMS

AND

FINITE

Note 7.2 MORE
Case 1 Uniform
Nutation, 331.

NONINERTIAL

AND

295

THE GYROSCOPE

Introduction, 316; Torque-free
Equations, 320.

8

VELOCITY

ANGULAR

288

8.1


INTRODUCTION

8.2

THE

8.3

UNIFORMLY

8.4

THE

8.5

PHYSICS

340
340

TRANSFORMATIONS

GALILEAN

SYSTEMS

ACCELERATING


OF EQUIVALENCE 346
IN A ROTATING
COORDINATE

343

PRINCIPLE

SYSTEM

355

and Rotating Coordinates, 356; Acceleration
Relative to Rotating
Coordinate System, 359.
358; The Apparent Force in a Rotating
Coordinates,
THE EQUIVALENCE PRINCIPLE AND THE
Note
8.1
RED SHIFT 369
GRAVITATIONAL
Note 8.2
ROTATING COORDINATETRANSFORMATION
371
Time

Derivatives

PROBLEMS


9

CENTRAL
FORCE

MOTION)

9.1

INTRODUCTION

CENTRAL

9.4

FINDING

9.5

THE

9.7
Note

FORCE

THE

ENERGY


PLANETARY

MOTION

IN

REAL

AND
EQUATION
MOTION
390

KEPLER'S LAWS 400
9.1
PROPERTIES
OF THE

PROBLEMS

OSCILLATOR)

378

MOTION
AS A ONE BODY
378
PROBLEM
9.3 GENERAL

PROPERTIES
OF CENTRAL FORCE
MOTION
380
The Motion Is Confined
to a Plane, 380; The Energy
and Angular Momentum Are
Constants
of the Motion, 380; The Law of Equal Areas, 382.

9.2

9.6

10 THE
HARMONIC

372)

PROBLEMS

382

ENERGY DIAGRAMS 383

ELLIPSE

403

406)


10.1
INTRODUCTION
AND REVIEW
410
Standard
Form of the Solution,
410; Nomenclature, 411; Energy
412; Time A verage
Values, 413; A verage Energy, 413.
10.2 THE DAMPED HARMONIC OSCILLATOR 414
418.)))
416; The Q of an Oscillator,
Energy,

Considerations,


x)

CONTENTS)

FORCED HARMONIC OSCILLATOR 421
Forced Oscillator, 421; Resonance, 423; The Forced Damped
Harmonic Oscillator,
in a Lightly Damped System: The Quality
424; Resonance
THE

10.3

The

Undamped

Factor

Q, 426.
RESPONSE

10.4

Note 10.1

432
VERSUS RESPONSE IN FREQUENCY
FOR THE
THE EQUATION OF MOTION
433
OSCILLATOR

TIME

IN

OF

SOLUTION

DAMPED


UNDRIVEN

The Useof Complex
Variables,
Note 10.2 SOLUTION OF THE
FORCED OSCILLATOR 437

THE

11

OF

RELA TIVITY)

12

RELATIVISTIC

THE
11.2 THE
11.3 THE
11.1

The

Damped Oscillator,

EQUATION


435.

OF MOTION

THE

FOR

438)

PROBLEMS

SPECIAL
THEORY

433; The

MODE

442
445
450
OF SPECIAL
RELATIVITY
POSTULATES
of Relativity, 451; The
Universal Velocity, 451; The Principle
FOR

NEED


A NEW

452.
Special Relativity,
11.4 THE GALILEAN
11.5 THE LORENTZ
PROBLEMS 459)

TRANSFORMATIONS
TRANSFORMATIONS

INTRODUCTION

12.1

OF

THOUGHT

EXPERIMENT

MICHELSON-MORLEY

Postulates

of

453
455


462
THE ORDER OF EVENTS 463
AND TIME DILATION
CONTRACTION

KINEMATICS) 12.2 SIMULTANEITYAND

12.3

LORENTZ

THE

466

The Lorentz Contraction, 466; Time Dilation, 468.
THE RELATIVISTIC TRANSFORMATIONOF VELOCITY
12.4
12.5 TH E DOPPLER
EFFECT
475
The Doppler Shift in Sound,
475; Relativistic Doppler Effect,
Effect for an Observer off the Line of Motion, 478.

12.6

RELATIVISTIC


MOMENTUM

AND
ENERGY)

14

FOU R-

VECTORS
AND

RELATIVISTIC
I NV ARIANCIE)

13.1
13.2

MOM

477;

The Doppler

480

484)

PROBLEMS


13

PARADOX

TWIN

THE

472

ENTUM

490

493

ENERGY

MASSLESS PARTICLES 500
DOES LIGHT TRAVEL
AT THE VELOCITY
PROBLEMS 512)

13.3
13.4

14.1

INTRODUCTION


OF

LIGHT?

508

516

VECTORS AND TRANSFORMATIONS
516
Rotation about the z Axis, 517; Invariants of a Transformation,
formation Properties of Physical
Laws, 520; Scalar Invariants,
14.3 MINIKOWSKI SPACE AND
521
FOUR-VECTORS
14.4 THE MOMENTUM-ENERGY
527
FOUR-VECTOR
14.5 CONCLUDING REMARKS 534
14.2

536)

PROBLEMS

INDEX

539)))


520;
521.

The Trans-


EXAMPLES,CHAPTER

OF

LIST

EXAMPLES)

1

Wave on

VECTORS

AND

KINEMATICS

-A

FEW

MATHEMATICAL


PRELl

2

M

INARI

1

1.1 Law of Cosines,5; 1.2 Work
and the Dot Product, 5; 1.3 Examples
of
the Vector Product in Physics,
7.
7; 1.4 Area as a Vector,
1.5 Vector Algebra, 9; 1.6 Construction
of a Perpendicular
Vector, 10.
1.7 Finding v from r, 16; 1.8 Uniform Circular Motion, 17.
1.9 Finding Velocity
from Acceleration,
Motion in a Uniform Gravi20; 1.10
tational
Effect
of a Radio
Field, 21; 1.11 Nonuniform Acceleration-The

1.12


an

Motion

1.13 Circular
1.14

NEWTON'S

LAWS-THE

Vectors, 25.

of a

Velocity

38.)

tion,

CHAPTER 2

EXAMPLES,

2.1

and Rotating
and Straight


Line
in Polar Coordinates, 34;
Motion
Bead on a Spoke, 35; 1.15 Off-center
Circle, 35; 1.16 Acof a Bead on a Spoke, 37; 1.17
Radial Motion without Accelera-

Motion

celeration
ES)

Electron, 22.

Ionospheric

Circular

in Space-Inertial
60.
Force,
Systems and Fictitious
The Astronauts' Tug-of-v.ar,
70; 2.3
Freight Train, 72; 2.4 Constraints,
Block on String 1, 75; \037.6 Block on String 2, 76; 2.7 The Whirling
74; 2.5
Block, 76; 2.8 The Conical Pendulum, 77.
2.9 Turtle in an Elevator, 84; 2.10 Block and String 3, 87; 2.11 Dangling
Block and Wedge

88; 2.12 Whirling
89; 2.13 Pulleys, 90; 2.14
Rope,
Rope,
with
in a Viscous
93; 2.15 The Spinning
Friction,
Terror,
94; 2.16 Free Motion
for Simple
Harmonic
Medium, 96; 2.17 Spring and Block-The Equation
Initial Conditions,
Motion, 98; 2.18 The Spring Gun-An Example
Illustrating
Astronauts

2.2

FOUNDATIONS

OF

NEWTONIAN

MECHANICS)

99.)


3

MOMENTUM

EXAMPLES, CHAPTER3
Drum Major's
115; 3.2
Nonuniform
Rod,
119; 3.4 Center of

3.1 The Bola,

Center of Mass Motion,
3.6

Spring

Gun

System, 125;3.8
3.9 Rubber
Ball

3.3

117;

Baton,


Mass of a

Center of

Mass of a

Sheet,

120; 3.5

Triangular

122.

Recoil,
123; 3.7
Push
Me-Pull

Earth,

Moon,

and

Sun-A

Three

Body


You, 128.

The

Rebound,
131; 3.10 How to Avoid Broken Ankles, 132.
Car and Hopper, 135;
Momentum,
134; 3.12 Freight
3.13 Leaky Freight Car, 136; 3.14 Rocket in Free Space, 138; 3.15 Rocket
in a Gravitational
Field, 139.
3.16 Momentum
to a Surface, 141;3.17 A Dike at the Bend of a
Transport
Pressure of a Gas, 144.)
River,
143; 3.18

3.11 Mass Flow

4

WORK

AND
ENERGY)

and


EXAM PLES,

CHAPTER 4

the

in a Uniform Gravitational
Upward
of Simple
Harmonic Motion,
in an Inverse
Motion
156.
Field,

4.1 MassThrown

4.3
4.4

Equation
Vertical

154; 4.2

Solving

Square


The Conical Pend ulum,

161;

162.
4.6

The

Inverted

4.8

Work

Done

167; 4.10

Field,

154.

Parametric

Pendulum,
by a Central

4.5


Escape Velocity-The

164;4.7 Work
Force, 167;4.9

Evaluation

of a Line

Done
A

by a Uniform Force, 165;
Line Integral,

Path-dependent

Integral,

General Case,

168.)))


OF EXAMPLES)

LIST

xii)


4.11 Potential

of an

Inverse

Potential Energy
of a Uniform Force Field,
170; 4.12
172.
and Spring,
Force, 171;4.13 Bead,Hoop,

Energy

Square

4.14 Energy and Stability-The Teeter Toy, 175.
4.15 Molecular Vibrations, 179;4.16 Small
181.
Oscillations,
4.17 Block Sliding
down
Inclined Plane, 183.
4.18 Elastic
Collision
of Two Balls, 190; 4.19 Limitations
Scattering Angle, 193.)

SOME


5

MATHEMATICAL

ASPECTS
OF FORCE
AND

ENERGY)

CHAPTER

EXAMPLES,

5.1

Partial

5.3

Gravitational

Attraction

Field, 209; 5.5

5.6
5.7
220;


Gravitational

The

Curl of the

5.9

A Most

Using

Stokes'

of

the

a

Gravitational

How

Theorem,

Partial

Uniform


Derivative,

205.

Gravitational

Masses, 209.
Star System, 212.

Force,

Unusual Force

Function, 222;5.11

Energy

5.12

Applications

5.4
Particle,
208;
Attraction
by Two Point
by

for a Binary


Contours

Energy

Laboratory

5
203; 5.2

Derivatives,

on

A Nonconservative
Force,
Construction of the Potential

219; 5.8

Field,

221; 5.10

the

Curl Got

Its Name, 224.


227.)

ANGULAR
EXAM PLES, CHAPTER 6
MOMENTUM 6.1 Angular
Momentum
Momentum
of a Sliding Block, 236; 6.2 Angular
AXIS
FIXED
of the Conical Pendulum,
237.
ROTATION)
and
6.3 Central Force Motion
the Law of Equal
240; 6.4 Capture
Areas,
on a Sliding
Cross Section of a Planet,
Block,
244; 6.6
241; 6.5 Torque
247.
Pendulum,
245; 6.7
Torque on the Conical
Torque due to Gravity,
The Parallel Axis
6.8 Moments of Inertia

of Some Simple Objects, 250; 6.9

6
AND

Theorem, 252.

6.10 Atwood's

Machine

6.11

Grandfather's

step,

259.

254.
with a Massive Pulley,
6.12
Kater's
Pendulum,

Clock, 256;

258;6.13

The Door-


6.14 Angular Momentum of a Rolling
262; 6.15 Disk on Ice, 264;
Wheel,
down a Plane:
a Plane, 265; 6.17 Drum
6.16 Drum Rolling
down
Rolling
Energy Method, 268; 6.18 The Falling Stick, 269.)

7

RIGID

BODY

MOTION
AN D TH E

CONSERVATION
OF

ANGULAR

MOMENTUM)

CHAPTER
EXAMPLES,
7.1

Rotations
through

7

289; 7.2 Rotation in the xy Plane, 291;
Angles,
of a
Momentum
of Angular Velocity, 291; 7.4 Angular
Skew Rod, 293; 7.6
Rotating Skew Rod, 292; 7.5 Torque on the Rotating
294.
Torque on the Rotating Skew Rod (GeometricMethod),
7.7 Gyroscope Precession, 298; 7.8 Why a Gyroscope Precesses, 299.
7.9 Precessionof the Equinoxes, 300; 7.10 The Gyrocompass Effect, 301;
304.
7.11 Gyrocompass
Motion, 302; 7.12 The Stability of Rotating
Objects,
for a Rotating Skew
7.13 Rotating Dumbbell,
310; 7.14 The Tensor of Inertia
Do Flying
Saucers
Make Better Spacecraft than
312; 7.15 Why Flying
Rod,

7.3


Vector

Ciga rs,

7.16

Euler's

Finite

Nature

314.

Motion, 322; 7.17 The
Equations and Torque-free Precession,324.)))

Stability

of

Rotational

Rotating

Rod,

323;


\037

.18


OF EXAMPLES)

LIST

8

NONINERTIAL

CHAPTER 8

EXAM PLES,

SYSTEMS
AND

FICTITIOUS

FORCES)

8.1 The Apparent
Plank, 347; 8.3

Force

Pendulum


8.4
The Driving Force of
352.
8.6 Surface of a Rotating

flection of a

Falling

Weather Systems,

9

CENTRAL
FORCE
MOTION)

xiii)

9.2

384;

The

Force, 363;
Earth,

8.8

De366; 8.10

of Comets,

Capture

Satellite Orbit,

393; 9.5

Orbits,

Hyperbolic

The Coriolis

9

Noninteracting Particles,
Perturbed
Circular Orbit, 388.

9.1

9.4

362; 8.7

Liq uid,


364; 8.9 Motion on the Rotating
Mass,
The Foucault Pendulum, 369.)
8.11
366;

CHAPTER

EXAMPLES,

of Gravity, 346; 8.2 Cylinder
on an Accelerating
in an Accelerating Car, 347.
the
Tides,
350; 8.5 Eq ui!ibrium Height of the Tide,

Satellite

9.6

396;

9.3

387;

Maneuver,

398.


9.7 The Law

10

TH

E

EXAM

PLES,

of Period

s, 403.)

CHAPTER

10
and the

HARMONIC

10.1

Initial

OSCILLATOR)


10.2

The Q of

Conditions
Two

Frictionless

Oscillators,

Simple

Harmonic

Oscillator,

419; 10.3

Graphical

411.
of

Analysis

a

Dam ped Oscillator, 420.


10.4 Forced Harmonic
nator,

Oscillator

Demonstration,

424;

10.5

Vibration

Elimi-

428.)

11 THE
SPECIAL

11.1 The Galilean

THEORY

the

CHAPTER

EXAMPLES,


Galilean

11
453; 11.2

Transformations,

A

Light

Pulse

as Described

t;>y

455.)

Transformations,

OF

RELATIVITY)

12

RELATIVISTIC

12


CHAPTER

EXAMPLES,

KINEMATICS) 12.1 Simultaneity,

463;

of

12.2

An Application
and
Timelike

of the

Lorentz

Transformations,

Intervals, 465.
12.4 The Orientation
of a Moving Rod, 467; 12.5
Time Dilation and Meson
Decay,468; 12.6 The Role of Time Dilation in a n Atomic Clock, 470.
12.7 TheSpeedof Light in a Moving Medium, 474.
464;


12.3

The Order

12.8 DopplerNavigation,

13

RELATIVISTIC

EXAMPLES,

MOMENTUM

AND
ENERGY)

13.1

Velocity

Events:

Spacelike

479.)

CHAPTER 13
Dependence


of the

Electron's Mass, 492.

13.2 Relativistic
and Momentum in an Inelastic
Collision, 496; 13.3
Energy
The Equivalence
of Mass and Energy, 498.
13.4 The Photoelectric
502; 13.5 Radiation Pressure of Light,
502;)))
Effect,


LIST

xiv)

OF EXAMPLES)

The Compton Effect,
503; 13.7
Picture of the Doppler Effect, 507.
13.9
The Rest Mass of the
Photon,


13.6

14

FOUR.

EXAMPLES,

CHAPTER

Pair Production,
510; 13.10

Light

505;

13.8

from

The

Photon

a Pulsar,

510.)

14


VECTORS 14.1 Transformation Properties of the Vector Product, 518; 14.2 A NonAND
vector, 519.
14.3 Time Dilation, 524; 14.4 Construction of a Four-vector: The FourRELATIVISTIC
526.
INVARIANCE)
525; 14.5 The Relativistic Addition of Velocities,
velocity,
14.6 The Doppler Effect,
Once
More, 530; 14.7 Relativistic Center of Mass
in Electron-electron
Collisions, 533.)))
Systems, 531;14.8 Pair Production


There is good reasonfor
PREFACE)

the

tradition

that students
with the study

engineeringstart college
physics
mechanics is the cornerstoneof


pure

concept

is essential for

of

energy,

of the

evolution

for

example,

and

the properties

universe,

applied
of

of science and
of mechanics:
science.

The

the study

of

the

particles,

elementary

the mechanisms of biochemical reactions. The concept
of
also
to
of
is
essential
the
a
cardiac
and
energy
design
pacemaker
to
of the limits of growth
the
of industrial

Howanalysis
society.
there
are
difficulties
in
an
introd
course
in
ever,
presenting
uctory
which
mechanics
is both exciting and intellectually
rewarding.
Mechanics
is a mature science and a satisfying
of its
discussion
is
lost
a
in
At
the
treatment.
other
superficial

principles
easily
and

extreme,
advanced

\"enrich\"

to

attempts

can produce

topics

the

subject

by

emphasizing

a false sophistication which

empha-

techniq ue rather than

understanding.
This text was developed from
a first-year
course which we taught
for a number of years at the Massachusetts
Institute of Technology
We have tried to present
and, earlier, at Harvard
University.
form which offers a strong
for
mechanicsin an engaging
work in pure and applied science. Our
future
departs
approach
from tradition more in depth
of topics;
and style than in the choice
a view of mechanics held by twentiethit
reflects
nevertheless,
sizes

base

century

physicists.


who come to the course
to differentiate and integrate simple functions.! It has also been used successfully in courses
requiring
registration in calculus.
(For a course
only concurrent
of this nature, Chapter 1 should
be treated
as a resource chapter,
for a time.
deferring the detaileddiscussionof vector kinematics
Our
Other suggestions are listed
in To The
experiTeacher.)
stuence has beenthat the principal source of difficulty
for
most
dents is in learning
how to apply mathematics to physical problems,
not
with
mathematical
as such. The elements of caltechniques
of
culus
can be mastered relatively easily, but the development
careful
We have proproblem-solving ability
guidance.

requires
vided
numerous
worked
the text to help
examples
throughout
this guidance.
Some of the examples,particularly
in the
supply
howexamples,
early chapters, are essentially pedagogical. Many
to probever, illustrate principles and techniq ues by application
lems of real physical interest.
on vecThe
is a mathematical introduction, chiefly
first
chapter
of a vector,)
tors and kinematics. The concept of rate of change
Our book

1

The

is written

some


knowing

primarily

calculus,

background
provided
John Wiley & Sons,

Ramsey,

for students

enough

in

\"Quick Calculus\" by Daniel Kleppner
New York, 1965, is adequate.)))

and

Norman


xvi)

PREFACE)


plays

most

the

probably
an

role

important

this topic is developed
The

in the text,
concept
mechanics.
Consequently,
throughout
care, both analytically and geometrically.
in particular, later proves to be invalumathematical

difficult

with

approach,


geometrical

momentum.
visualizing the dynamics of angular
2 discusses inertial
Newton's
laws, and some
Chapter
systems,

able for

common forces.

of the

Much

ton's laws, since analyzing
general principles can be a
a complex system in terms
inertial

and

coordinates,

erations are all


acquired

ples in

have

text

the

according to
problems
at first.
task
Visualizing

simple

challenging

its essentials,

of

between
numerous

distinguishing
skills.
The


been carefully

on applying New-

centers

discussion
even

chosen

to

selecting suitable
forces and accelillustrative examthese

develop

help

skills.

Momentum and energy are developed
two chapin the following
ters. Chapter 3, on momentum,
applies Newton's laws to extended
become
when they try to
confused

systems. Studentsfrequently
momentum
to rockets and other systems
considerations
apply
flow
of mass.
Our approach is to apply
a differential
involving
method to a system defined so that no mass crosses its boundary
the chosen
time interval.
This ensures that no contribution
during
to the total momentum is overlooked. The chapter
with
concludes
a discussion of

the

work-energy

theorem

and

forces.


The

nonconservative

and

illustrated

are

energy

Chapter 5 deals with

forces and
the

in

potential

but

text,

by

be of interest

will


usually

lies so far from
their
As a result, introductory
their

made understandable
mathematical

than

ments, and

by

often

texts

by

found

emphasizing

formalism,
providing


numerous

the propertiesof

they cannot
slight these
that

rotational

physical

angular

motion

rotational

because

partly

that

experience

have

subject.


to grasp

body motion,

We

importance.

of the

it difficult

find

and rigid

momentum

material

this

matically complete treatment
Students

of conservative
aspects
is not needed elsewhere
to students who want a math e-


mathematical

some
energy;

it

4, on energy, develops
Chapter
its application to conservative and
conservation
laws for momentu m
a discussion
of collision
problems.

flux.

momentum

rely

on

intuition.

topics, despite
motion

be

rather

can

reasoning

to geometric arguworked examples. In Chapter

by appealing

6 angular
axis rotation

momentum is introduced, and
is treated. Chapter 7 develops

of

by applying vector arguments to systems
angular momentum. An elementary
treatment
rigid body motion is pr\037sented
in the
last sections of
to show
how Euler's equations can be developed
from)))

rigid


body

of general

Chapter

motion

by spin

dominated

7

the dynamics
the

important

of

fixed

features


PREFACE)

xvii)


is
This more advanced material
course.
own
our
do
it
in
we
not
treat
however;
usually
coordinate systems, completes the
8, on noninertial
mechanics.
of the
of newtonian
Up to
principles

arguments.

physical

simple

optional
Chapter
development


in the
inertial systems have been used exclusively
text,
to avoid confusion between forces and
accelerations.
value
of noninertial systems emphasizes their
discussion

this point
order

in

Our

tools and their

computational

implications

for

the

foundations

as


of

mechanics.
the harmonic
and
Chapters 9 and 10treat central force motion
new physical
no
concepts are
respectively. Although
of the principles
the
involved, these chapters illustrate
application
of mechanics to topics of general
in
interest
and importance
phyoscillator
sics.
Much of the algebraic complexity
harmonic
of the
is avoided by focusing
the discussion
on energy, and by using sim-

oscillator,


ple approximations.

Chapters 11 through
emphasize

believing,
classical
at

length

14

and

relativity

special

a discussion of the
of its applications. We

present

some

principles of
attempt

to


classical
the harmony between relativistic
and
thought,
for example, that it is more valuable to show how the
than to dwell
laws are unified
conservation
in relativity
is contreatment
on the so-called \"paradoxes.\" Our

cise and minimizes
ideas of symmetry

how
14 shows
algebraic complexities. Chapter
of
formulation
role
the
in
fundamental
playa
in
students
we
the

have
mind,
relativity.
Although
kept
beginning
the concepts
here are more subtle than
in the
previous chapters.
but by illustrating how symif desired;
Chapter 14can be omitted
an exciting
bears on the principles of mechanics,
it offers
metry
mode of thought
and a powerful new tool.
no subis absolutely
there
Physics cannot be learned passively;
stitute for tackling challenging problems. Hereis where
students
the sense of satisfaction and involvement
by a
gain
produced
The collecgenuine understanding of the principlesof physics.

tion of problems

classroom
use.
for

few

book
problems

was developed over
are straightforward

many

and

thought

this

Problems worthy
attack

of

prove
by
1

From


years

of

and intended

serious

basic principles and require
emphasize
effort.
We have tried to choose problems which
effort worthwhile in the spirit of Piet Hein's aphorism

most

drill;

make

this

in
A

their
hitting

Grooks /,


worth

back
by

Piet

1)

Hein, copyrighted

T. Press.)))
1966,The M.1.


xviii)

PREFACE)

to
contributions
us pleasure to acknowledge the many
I n parbook from our colleaguesand
from
our students.
David
E.
we thank Professors George B. Benedekand
ticular,

should
We
Pritchard
for a number of examples
and
problems.
for their
also like to thank Lynne Rieck and Mary Pat Fitzgerald
It gives

this

cheerful fortitude
Daniel Kleppner
Robert

J.

Kolenkow)))

in

typing

the

manuscript.)


to

form a comprehensive introduction
chapters
and constitute the heart of a one-semester
mechanics
classical
the
covered
In a 12-week semester, we have
course.
generally
9 or 10. However, Chapter
first 8 chapters and parts of Chapters
7 and 8 are usually
of the advanced topics in Chapters
5 and
some
first

The

TO

THE

eight

TEACHER)

some


although

omitted,

Chapters11,12,and
14,

Chapter

relativity.

insight

deeper

provides

We have

used the

pursue them independently.
introduction to special
on transformation
theory and four-vectors,
students.
into the subject for interested
students

a complete


13 present

chapters on

course and alsoas part

of the

relativity

second-term

in

a three-week

course

in

electricity

short
and

magnetism.

The problems at


the end of

each

chapter

are generally

graded

are also cumulative; concepts and techniq ues
in
They
difficulty.
sections
are repeatedly called upon in later
from
earlier
chapters
The hope is that by the end of the course the student
of the book.
new problems,
for
a good intuition
will
have
tackling
developed

that he


will

be

about whether
approach,

energy

tack

able
to

to make an intelligent estimate, for instance,
the
from the momentum approach or from
and that he will know how to set off on a new

start

students
is unsuccessful.
report
first approach
Many
skills.
these
sense of satisfaction from acquiring

than a numerical
rather
Many of the problems requirea symbolic
the importance of numerito minimize
solution. This is not meant
cal work but to reinforce the habit of analyzing problems symbolia numerica!
Answers
are given to some problems; in others,
cally.
to allow
the student to check his sym\"answer clue\" is provided
bolic
Some of the problems are challengingand
req uire
result.
such
serious
problems
thought and discussion. Sincetoo many
should have a
each
at once can result in frustration,
assignment
of easier
mix
and harder problems.
we would
prefer to start a course in mechanChapter1 Although
are real
there

ics by discussing
physics rather than mathematics,
mathematics
the
to
to
lectures
the
few
first
advantages
devoting
for
are straightforward
of motion.
of kinematics
The
concepts
the most part, and it is helpful to have them clearly in hand
before
the much subtler problems presentedby newtackling
in this
from tradition
in Chapter
tonian
2. A departure
dynamics
coordinates.
is
the

of
discussion
kinematics
polar
using
chapter
students
find this topic troublesome at first, requiring serious
Many
rewarded.
effort.
we feel that the effort will be amply
However,
In the first place, by being
able to use polar coordinates freely\"
to understand;)))
easier
the
of rotational
kinematics
motion are much
if

a deep

his


xx)


TO THE TEACHER)

the

mystery

this

topic

of

gives

radial acceleration
valuable insights

vector, insights
in

motion

2 but

Chapter

momentum
and 7, and

not


which

flux

in

the

simplify

only

More important,

disappears.
into

which are

of a

time-varying
the dynamics of particle
nature

invaluable to

the


of

discussion

6
momentum in Chapters
3, angular
noninertial
coordinates
in Chapter 8. Thus,

Chapter

the use of

vectors
into understanding
the nature of time-varying
course.
1 pays important dividends throughout
the
If the
is intended for students who are concurrently
course
beginning their study of calculus, we recommend that parts of Chapter 1
be deferred. Chapter 2 can be started after having
covered
only
the first six sections of Chapter
1. Starting

with
2.5, the
Example
kinematics of rotational motion are needed; at this point the ideas
1.9 should
be introduced. Section 1.7,on the
presented in Section
of vectors,
can be postponed until the class has become
integration
familiar with integrals.
Occasional
and problems involvexamples
will have to be omitted until
that
time.
Section 1.8,
ing integration
of vector
on the geometric interpretation
is essendifferentiation,
tial
6
for
7
discussed
but
not
be
and

need
preparation
Chapters

the effort
in

put

Chapter

earlier.
Chapter

2

material

The

in

dent's first serious
attempt
crete situations.
Newton's
most people unconsciously

laws


the sturepresents
abstract principles to conmotion are not self-evident;
2 often

Chapter

to apply
of

We find
thought.
students become accustomed to analyzing
to principles rather than
problems
according
A common source of difficulty
is to conintuition.
at first
vague
fuse force and acceleration. We therefore
the use of
emphasize
inertial
and
recommend
systems
strongly that noninertial coordinate systems be reserved until Chapter
their correct
8, where
use is discussed. In particular,

the
use of centrifugal force in
the
can lead to endless confusion
inertial
between
early chapters
and noninertial systems and, in any case, it is not adeq uate for the
that

after

analysis

an

initial

of motion

follow

in

coordinate

rotating

Chapters 3 and 4 There


rocket equations.
ones

in which

However,

there is

aristotelian

of uncertainty,

period

are

rocket

a mass flow,

systems.
different

many

so

ways


problems
that it

are

to derive

not the

the
only

is important to adopt
that the
also desirable

a method
which
is easily generalized.
It is
method be in harmony
with the laws of conservation of momentum
The
involved.
or, to put it more crudely, that there is no swindle
differential approach used in Section
to meet
3.5 was developed
these req uirements. The approach
it is

may not be elegant, but
straightforward

and quite

general.)))


xxi)

TO THE TEACHER)

the
In Chapter
4, we attempt to emphasize
the work-energy theorem and the difference
and

tive

forces.

nonconservative

introduced and explained,only
evaluated, and general computational
given

und


ue

Although
line

simple

of

nature

general

betweenconservathe line integral is
need to be

integrals

not

should

techniques

be

attention.

and
This chapter completesthe discussionof energy

useful introd uction to potential theory and vector calculus. However,
it is relatively
advanced and will appeal
only to
The results are not
students with an appetite
for mathematics.
this
needed
elsewhere
in the text, and we recommend leaving
for optional use, or as a specialtopic.
chapter
is
momentum
6 and 7
Most students find that angular
Chapters
mechanics. The
the most difficult
in elementary
concept
physical
hurdle
is visualizing the vector properties of
major
conceptual
nature
momentum.
We therefore emphasize the vector

angular
In
of angular
these
momentum repeatedly throughout
chapters.
understood
be
can
particular, many features of rigid body motion
vectors
on the understanding of time-varying
by relying
intuitively
to
It
more
in
earlier
is
emphasize
profitable
developed
chapters.
formal
the qualitative
of rigid body motion than
features
aspects
such as the tensor of inertia.

If desired,
these qualitative arguments can be pressedquite far, as in the analysis of gyroscopic
nutation
7.2. The elementary discussion of Euler's
in Note
equations in Section
7.7 is intended as optional readingonly.
Although
6 and 7 req uire hard work, many students develop a phyChapters
sical insight into angular momentum
and
rigid body motion which
is seldom
at the introd
and which is often
level
gained
uctory

Chapter 5
a
provides

obscuredby
Chapter 8
springboard
formation

practical


The

courses.
systems

in advanced

mathematics

noninertial

of

subject

such speculative
and
and the principle of
theory
to

point of view, the

tant techniq ue
Chapters 9 and

for

use
many


solving

offers a natural
interesting topics as transequivalence. From a more

noninertial

of

physical

systems

is an

impor-

problems.

chapters the principles developed
applied
important
problems, central force motion
and
are generally
the harmonic
both
oscillator.
topics

Although
treated rather formally,
we have
tried to simplify the mathematical
of central force motion relies heavily
development. The discussion
on the conservation
laws and on energy diagrams. The treatment
of the harmonic oscillator sidesteps much
of the
usual algebraic

earlierare

10

to

In these

two

complexity by focusing
tions and examples play

on the
an

lightly


important

oscillator.

damped

role

in

both

chapters.)))

Applica-


xxii)

TO THE TEACHER)

Chapters

11 to

14 Special

pace to a course in

size the connection of


usedthe Michelson-Morley
the

Although

offers
an exciting
approach
attempts
with classical
relativity
thought.
to motivate
the
experiment
of this experiment in Einstein's

mechanics.

prominence

relativity

Our

change of
to empha-

We have

discussion.
thought

been much exaggerated, this approach has the advantage
of
the discussion on a real experiment.
grounding
to focus on the ideas of eventsWe
have
tried
and their transformations without
aids such as diaemphasizing
computational
methods.
This
allows us to deemphasize
grammatic
approach
of the so-called
paradoxes.
many
For many students, the real mystery
of relativity
lies not in the
or transformation
laws but in why transformation
postulates
prinfor genciples should suddenlybecomethe fundamental
concept
on the deepest and most

erating new physical laws. This touches
of Einstein's
provocative
aspects
thought.
Chapter 14, on fouran introduction to transformation theory which
vectors,
provides
unifies
and summarizes
the preceding development. The chapter
is intended
to be optional.)
has

Daniel

Kleppner

Robert

J.

Kolenkow)))


AN

INTRODUCTION


TO

MECHANICS)))



VECTORS

AND

KINEMATICS-

A

FEW

MATHEMATICAL

PRELIMINARIES)))


2)

VECTORS

1.1

I

KINEMATICS-A


AND

the

MATHEMATICAL

help

you

PRELIMINARIES)

uction

ntrod

The goal of this
of

FEW

book is to

of mechanics.

principles

the very heart of physics; its
the everyday physical

standing

acquire a deep understanding
is at
The subject of mechanics

concepts are essentialfor
well

as

world

the
as

role

in

atomic and cosmic scales. The concepts
and
momentum, angular momentum,
energy,
of
area
physics.
practically
every
use


shall

We

mathematics

freq

ideas

and

quickly

Furthermore,
is based on quantitative

physics
For these

reasons, we

necessarymathematical
principlesof mechanics
1.2

and

new


in

experiment

measurement.

and

prediction

to

way

this chapter to developing some
our discussion of the
postpone

devote

shall

complicated

points the

and it often

of


discussion

our

the interplay of theory

insights.

vital

us express

lets

transparently,

playa
in

uently

physical principles, since mathematics

under-

as phenomena
on
of mechanics,
such


tools

and

until

Chap.

2.)

Vectors)

to the role of
of vectors provides a good introd
uction
study
mathematics in physics.
By using vector notation, physical laws
a matter
can often be written in compact and simple form.
(As
was
notation
invented
of fact, modern vector
by a physicist,
The

of Yale


Gibbs

Willard

ance of equations.)
law

F

shall discuss
notation:)
century

in

the

simplify

is how

here

next

the

appear-


Newton's second

chapter)

appears

in

may

= ma z.)

Fz

F

to

primarily

example,

max

y =

In

For


we

(which

nineteenth

Fx =

University,

vector
=

Our

form

notation,

one simply

1na.)

for introducing vectors
motivation
principal
of equations.
However, as we shall see in

book, vectors have

are closely related to the
of the

their
unknown

writes)

use

can
laws.)))

is to
the

simplify

last

the

chapter

Vectors
significance.
fundamental
ideas of symmetry and
forms of
lead to valuable insights into

the
possible
a

much

deeper


SEC. 1.2

VECTORS)

3)

of a Vector)

Definition

Vectors can

be approachedfrom

of view-geometric,
points
th ree points of view
are
use-

three


all
analytic, and axiomatic. Although
the
ful, we shall need only
geometric

From the

label

geometric

In writing,
with
a letter

segment.
it

and

of

point

\037)

the same


have

vectors

two

If

equal.

a

in

arrow
In print,

and

bold-

both its length and
shall assume that
Thus the arrows

specify
we

vector.


the

and

e are

directed line

by an

vector

arrow.

length

Band

vectors

The

is a

a vector

view,

we can represent
capped by a symbolic


faced letters are traditionally
used.
In order to describe a vector
we must
its direction. Unless indicated
otherwise,
a
parallel translation does not change
at left all represent
the same vector.
they are

analytic approaches

of mechanics.

discussion

our

same

direction

equal:)

B = e.)

The


A.

or simply
the

If

length

labeled

unit vector is
to A is A. It
A

=-,

A

is called its

a vector

magnitude.

The

magnitude


will occur,
For example, the magnitude of A is written
IAI,
If the
of A is V2, then IAI = A = V2.
length
A
of a vector is one unit,
vector.
we call it a unit

italics.

using

by

of

length

vector is indicated

of a

bars

vertical

by


confusion

no

if

the vector

a caret;

by

or,

of

unit

length

parallel

that)

follows

A
IAI)


and

conversely)

A =

The

IAIA.)

Algebra

of Vectors)

Muniplication of a
C=bA)

to A,

parallel

lei

/)

=

by

and its


a Scalar

If

is b times

length

we

multiply

= bA.

e

vector

new

greater.

A by a positive
e is
The vector
A
A.
Thus C = A, and


b/A/.

The result of
\037)

Vector

scalar b, the result is a

in direction

change

a vector

(anti parallel) to the
of

Multiplication
both

multiplying

the

a vector

magnitude

by


and

by

original

a

-1 is a

new

vector

opposite

vector.

negative

the direction

scalar
sense.)))

evidently

can