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An Introduction to Mechanics
For 40 years, Kleppner and Kolenkow’s classic text has introduced stu-
dents to the principles of mechanics. Now brought up-to-date, this re-
vised and improved Second Edition is ideal for classical mechanics
courses for first- and second-year undergraduates with foundation skills
in mathematics.
The book retains all the features of the first edition, including numer-
ous worked examples, challenging problems, and extensive illustrations,
and has been restructured to improve the flow of ideas. It now features
- New examples taken from recent developments, such as laser slowing
of atoms, exoplanets, and black holes
- A “Hints, Clues, and Answers” section for the end-of-chapter prob-
lems to support student learning
- A solutions manual for instructors at www.cambridge.org/kandk
daniel kleppner is Lester Wolfe Professor of Physics, Emeritus, at
Massachusetts Institute of Technology. For his contributions to teaching
he has been awarded the Oersted Medal by the American Association
of Physics Teachers and the Lilienfeld Prize of the American Physical
Society. He has also received the Wolf Prize in Physics and the National
Medal of Science.
robert kolenkow was Associate Professor of Physics at Mas-
sachusetts Institute of Technology. Renowned for his skills as a teacher,
Kolenkow was awarded the Everett Moore Baker Award for Outstanding
Teaching.
University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is a part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence.
www.cambridge.org Information on this title: www.cambridge.org/ 9780521198110
©c D. Kleppner and R. Kolenkow 2014
This edition is not for sale in India.
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First edition previously published by McGraw-Hill Education 1973
First published by Cambridge University Press 2010 Reprinted 2012
Second edition published by Cambridge University Press 2014
Printed in the United States by Sheridan Inc.
A catalogue record for this publication is available from the British Library
ISBN 978-0-521-19811-0 Hardback
Additional resources for this publication at www.cambridge.org/kandk
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
CONTENTS PREFACE page xi
TO THE TEACHER xv
PREFACE An Introduction to Mechanics^ grew out of a one-semester course at the
Massachusetts Institute of Technology—Physics 8.012—intended for
students who seek to understand physics more deeply than the usual
freshman level. In the four decades since this text was written physics
has moved forward on many fronts but mechanics continues to be a
bedrock for concepts such as inertia, momentum, and energy; fluency
in the physicist’s approach to problem-solving—an underlying theme of
this book—remains priceless. The positive comments we have received
over the years from students, some of whom are now well advanced in
their careers, as well as from faculty at M.I.T. and elsewhere, reassures
us that the approach of the text is fundamentally sound. We have received
many suggestions from colleagues and we have taken this opportunity to
incorporate their ideas and to update some of the discussions.
We assume that our readers know enough elementary calculus to dif-
ferentiate and integrate simple polynomials and trigonometric functions.
We do not assume any familiarity with differential equations. Our expe-
rience is that the principal challenge for most students is not with un-
derstanding mathematical concepts but in learning how to apply them to
physical problems. This comes with practice and there is no substitute
for solving challenging problems. Consequently problem-solving takes
high priority. We have provided numerous worked examples to help pro-
vide guidance. Where possible we try to tie the examples to interesting
physical phenomena but we are unapologetic about totally pedagogical
problems. A block sliding down a plane is sometimes mocked as the
quintessentially dull physics problem but if one allows the plane to ac-
celerate, the system takes on a new complexion.
PREFACE xiii
momentum. Chapter 5 introduces the concepts of kinetic energy, po-
tential energy, and the conservation of energy, including heat and other
forms. Chapter 6 applies the preceding ideas to phenomena of general in-
terest in mechanics: small oscillations, stability, coupled oscillators and
normal modes, and collisions. In Chapter 7 the ideas are extended to ro-
tational motion. Fixed axis rotation is treated in this chapter, followed by
the more general situation of rigid body motion in Chapter 8. Chapter 9
returns to the subject of inertial systems, in particular how to understand
observations made in non-inertial systems. Chapters 10 and 11 present
two topics that are of general interest in physics: central force motion and
the damped and forced harmonic oscillator, respectively. Chapters 12–
provide an introduction to non-Newtonian physics: the special theory of
relativity.
When we created Physics 8.012 the M.I.T. semester was longer than
it is today and there is usually not enough class time to cover all the ma-
terial. Chapters 1–9 constitute the intellectual core of the course. Some
combination of Chapters 9–14 is generally presented, depending on the
instructor’s interest.
We wish to acknowledge contributions to the book made over
the years by colleagues at M.I.T. These include R. Aggarwal, G. B.
Benedek, A. Burgasser, S. Burles, D. Chakrabarty, L. Dreher, T. J.
Greytak, H. T. Imai, H. J. Kendall (deceased), W. Ketterle, S. Mochrie,
D. E. Pritchard, P. Rebusco, S. W. Stahler, J. W. Whitaker, F. A. Wilczek,
and M. Zwierlein. We particularly thank P. Dourmashkin for his help.
Daniel Kleppner
Robert J. Kolenkow
xvi TO THE TEACHER
the language of vectors and provides a background in kinematics that is
used throughout the text. Students are likely to return to Chapter 1, using
it as a resource for later chapters.
On a few occasions we have been able to illustrate concepts by ex-
amples based on relatively recent advances in physics, for instance exo-
planets, laser-slowing of atoms, the solar powered space kite, and stars
orbiting around the cosmic black hole at the center of our galaxy.
The question of student preparation for Physics 8.012 at M.I.T. comes
up regularly. We have found that the most reliable predictor of per-
formance is a quiz on elementary calculus. At the other extreme, oc-
casionally a student takes Physics 8.012 having already completed an
AP physics course. Taking a third introductory physics course might be
viewed as cruel and unusual, but to our knowledge, these students all felt
that the experience was worthwhile.
LIST OF
EXAMPLES
Chapter 1 VECTORS AND KINEMATICS
1.1 The Law of Cosines 5; 1.2 Work and the Dot Product 5; 1.3 Ex-
amples of the Vector Product in Physics 7; 1.4 Area as a Vector 8;
1.5 Vector Algebra 10; 1.6 Constructing a Vector Perpendicular to a
Given Vector 10; 1.7 Finding Velocity from Position 17; 1.8 Uniform
Circular Motion 18; 1.9 Finding Velocity from Acceleration 19; 1.
Motion in a Uniform Gravitational Field 21; 1.11 The Effect of Radio
Waves on an Ionospheric Electron 21 1.12 Circular Motion and Rotat-
ing Vectors 24; 1.13 Geometric Derivation of dˆr/dt and d θˆ/dt 30; 1.
Circular Motion in Polar Coordinates 31; 1.15 Straight Line Motion in
Polar Coordinates 32; 1.16 Velocity of a Bead on a Spoke 33; 1.
Motion on an Off-center Circle 33; 1.18 Acceleration of a Bead on a
Spoke 34; 1.19 Radial Motion without Acceleration 35
Chapter 2 NEWTON’S LAWS
2.1 Inertial and Non-inertial Systems 55; 2.2 Converting Units 63;
2.3 Astronauts’ Tug-of-War 67; 2.4 Multiple Masses: a Freight Train
69; 2.5 Examples of Constrained Motion 70; 2.6 Masses and Pulley
71; 2.7 Block and String 1 73; 2.8 Block and String 2 73; 2.9 The
Whirling Block 74; 2.10 The Conical Pendulum 75
Chapter 3 FORCES AND EQUATIONS OF MOTION
3.1 Turtle in an Elevator 87; 3.2 Block and String 89; 3.3 Dangling
Rope 90; 3.4 Block and Wedge with Friction 93; 3.5 The Spinning
