Classical Mechanics

Second Edition

Classical Mechanics: From Newton to Einstein: A Modern Introduction, Second Edition Martin W. McCall© 2011 John Wiley & Sons, Ltd. ISBN: 978-0-470-71574-1

Classical MechanicsSecond Edition

From Newton to Einstein:A Modern Introduction

Martin W. McCallImperial College London, UK

A John Wiley and Sons, Ltd., Publication

This edition first published 2011

c© 2011 John Wiley & Sons, Ltd.

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Library of Congress Cataloging-in-Publication Data

McCall, Martin W.Classical mechanics : from Newton to Einstein : a modern introduction / Martin W. McCall. – 2nd ed.

p. cm.Summary: “Classical Mechanics provides a clear introduction to the subject, combining a user-friendly style

with an authoritative approach, whilst requiring minimal prerequisite mathematics - only elementary calculus andsimple vectors are presumed. The text starts with a careful look at Newton’s Laws, before applying them in onedimension to oscillations and collisions. More advanced applications - including gravitational orbits, rigid bodydynamics and mechanics in rotating frames - are deferred until after the limitations of Newton’s laws have beenhighlighted through an exposition of Einstein’s Special Relativity. Big problems that are tackled using elementarytechniques include the stability of the Universe, a body falling from a great height under gravity and Foucault’spendulum. Many new problems are included together with a supplementary web link to the solutions manual.”–Provided by publisher.

Summary: “Classical Mechanics will be a clear introduction to the subject, combining a user-friendly style withan authoritative approach, whilst requiring minimal prerequisite mathematics”– Provided by publisher.

Includes bibliographical references and index.ISBN 978-0-470-71574-1 (hardback)

1. Mechanics. I. Title.QC125.2.M385 2011531–dc22

2010022396

A catalogue record for this book is available from the British Library.

ISBN 9780470715741 (Hbk) 9780470715727 (Pbk)

Typeset by the author.

Printed in Singapore by Markono Print Media Pte Ltd.

The author examining a rare second edition of Newton’s Principia at the Specola Vaticana,Castel Gandolfo, May 1999.

He is not eternity, or infinity, but eternal and infinite. He is not duration and space, but Heendures and is present. He endures forever, and is everywhere present; and by existing alwaysand everywhere he constitutes duration and space. . .And thus much concerning God; to dis-course of whom from the appearances of things, does certainly belong to natural philosophy.

Isaac Newton, 1687.

Contents

Preface to Second Edition xi

Preface to First Edition xiii

1 Newton’s Laws 11.1 What is Mechanics? 11.2 Mechanics as a Scientific Theory 11.3 Newtonian vs. Einsteinian Mechanics 21.4 Newton’s Laws 31.5 A Deeper Look at Newton’s Laws 51.6 Inertial Frames 71.7 Newton’s Laws in Noninertial Frames 101.8 Switching Off Gravity 111.9 Finale – Laws, Postulates or Definitions? 121.10 Summary 121.11 Problems 13

2 One-dimensional Motion 152.1 Rationale for One-dimensional Analysis 152.2 The Concept of a Particle 162.3 Motion with a Constant Force 172.4 Work and Energy 172.5 Impulse and Power 192.6 Motion with a Position-dependent Force 192.7 The Nature of Energy 212.8 Potential Functions 222.9 Equilibria 252.10 Motion Close to a Stable Equilibrium 252.11 The Stability of the Universe 262.12 Trajectory of a Body Falling a Large Distance Under Gravity 302.13 Motion with a Velocity-dependent Force 322.14 Summary 342.15 Problems 35

viii Contents

3 Oscillatory Motion 393.1 Introduction 393.2 Prototype Harmonic Oscillator 393.3 Differential Equations 403.4 General Solution for Simple Harmonic Motion 413.5 Energy in Simple Harmonic Motion 433.6 Damped Oscillations 443.7 Light Damping – the Q Factor 473.8 Heavy Damping and Critical Damping 493.9 Forced Oscillations 513.10 Complex Number Method 573.11 Electrical Analogue 603.12 Power in Forced Oscillations 613.13 Coupled Oscillations 623.14 Summary 673.15 Problems 69

4 Two-body Dynamics 754.1 Rationale 754.2 Centre of Mass 754.3 Internal Motion: Reduced Mass 764.4 Collisions 774.5 Elastic Collisions 784.6 Inelastic Collisions 814.7 Centre-of-mass Frame 834.8 Rocket Motion 884.9 Launch Vehicles 904.10 Summary 924.11 Problems 93

5 Relativity 1: Space and Time 975.1 Why Relativity? 975.2 Galilean Relativity 985.3 The Fundamental Postulates of Relativity 995.4 Inertial Observers in Relativity 1025.5 Comparing Transverse Distances Between Frames 1035.6 Lessons from a Light Clock: Time Dilation 1055.7 Proper Time 1075.8 Interval Invariance 1085.9 The Relativity of Simultaneity 1095.10 The Relativity of Length: Length Contraction 1105.11 The Lorentz Transformations 1115.12 Velocity Addition 1155.13 Particles Moving Faster than Light: Tachyons 116

Contents ix

5.14 Summary 1185.15 Problems 119

6 Relativity 2: Energy and Momentum 1236.1 Energy and Momentum 1236.2 The Meaning of Rest Energy 1296.3 Relativistic Collisions and Decays 1306.4 Photons 1316.5 Units in High-energy Physics 1336.6 Energy/Momentum Transformations Between Frames 1346.7 Relativistic Doppler Effect 1366.8 Summary 1376.9 Problems 139

7 Gravitational Orbits 1437.1 Introduction 1437.2 Work in Three Dimensions 1437.3 Torque and Angular Momentum 1447.4 Central Forces 1477.5 Gravitational Orbits 1517.6 Kepler’s Laws 1577.7 Comments 1597.8 Summary 1607.9 Problems 160

8 Rigid Body Dynamics 1658.1 Introduction 1658.2 Torque and Angular Momentum for Systems of Particles 1668.3 Centre of Mass of Systems of Particles and Rigid Bodies 1678.4 Angular Momentum of Rigid Bodies 1698.5 Kinetic Energy of Rigid Bodies 1748.6 Bats, Cats, Pendula and Gyroscopes 1758.7 General Rotation About a Fixed Axis 1818.8 Principal Axes 1868.9 Examples of Principal Axes and Principal Moments of Inertia 1878.10 Kinetic Energy of a Body Rotating About a Fixed Axis 1918.11 Summary 1928.12 Problems 193

9 Rotating Frames 1999.1 Introduction 1999.2 Experiments on Roundabouts 2009.3 General Prescription for Rotating Frames 2029.4 The Centrifugal Term 204

x Contents

9.5 The Coriolis Term 2059.6 The Foucault Pendulum 2079.7 Free Rotation of a Rigid Body – Tennis Rackets and Matchboxes 2119.8 Final Thoughts 2139.9 Summary 2149.10 Problems 214

Appendix 1: Vectors, Matrices and Eigenvalues 217A.1 The Scalar (Dot) Product 217A.2 The Vector (Cross) Product 218A.3 The Vector Triple Product 219A.4 Multiplying a Vector by a Matrix 220A.5 Calculating the Determinant of a 3 × 3 Matrix 220A.6 Eigenvectors and Eigenvalues 221A.7 Diagonalising Symmetric Matrices 223

Appendix 2: Answers to Problems 225

Appendix 3: Bibliography 229

Index 230

Preface to Second Edition

One of the major problems of producing this second edition of Classical Mechanics wasdeciding what new material to include. The most natural way to extend the work was tocover Lagrangian and Hamiltonian mechanics. Although these techniques are certainly veryimportant, they are rather advanced, and I was keen to maintain the elementary flavourof the first edition. Moreover, as I discovered when I taught these techniques to thirdyear undergraduates, the number of problems that become accessible is rather small; theachievements of these methods is principally conceptual in, for example, paving a pathtowards quantum mechanics. In fact the only problem I could find that really illustratedthe progression from Lagrangian to Hamiltonian mechanics is the nutation of a gyroscope.Another option was to include a chapter on four-vectors. Again, whilst interesting, Ididn’t feel that it was quite in keeping with the spirit of the first edition, which was toteach mechanics and special relativity ab inito to undergraduate students with minimalmathematics. In the end I decided to embellish what I already had in the first edition. So here,I have included a new section on a body free-falling a large distance under gravity, which Ihaven’t seen in textbooks before. New material on collisions is included to show that snookerballs always scatter at 90◦. When I reconsidered the contortions to make the discussion ofrigid bodies rotating about a fixed axis ‘simple’, I decided that the labour of introducingthe inertia tensor was not so great, and consequently the rotation of arbitrary bodies is nowdiscussed in Chapter 8. The Foucault pendulum is now discussed in detail, together with the‘tennis racket theorem’ which pulls together material on rigid bodies, stability and rotatingframes rather nicely. Some mathematical extension has been necessary to accommodate thesetopics, and the brief Appendix of the first edition has now been significantly extended. Thechapter summaries have been extended where necessary to include the new material.

The complete manuscript has been re-typeset in LaTex, and a number of figures have beenredrawn. New problems have been included, and a comprehensive online solutions manualhas been prepared (www.wiley.com/go/mccall). I also took the opportunity of correcting afew typographical errors.

Preface to First Edition

The tale, as Tolkien wrote in the preface to Lord of the Rings, grew in the telling. Havingcopped the highest profile undergraduate course in the Physics programme at ImperialCollege, I set about writing the lectures on my laptop. With everything available in software,I felt it would be a relatively simple task to cut and paste the material into a book, and dulycontracted with John Wiley & Sons Ltd to produce a camera ready manuscript in a fewmonths. Thus I became pregnant with my first literary child. Little did I understand the pangsof labour.

The course entitled ‘Mechanics and Relativity’ is given to incoming undergraduates. Iwould meet them in the first week of their arrival and finish my forty first and final lecture inabout the middle of the second term. The varied level of mathematical preparation of the classof 200 students set special problems for designing the course. Some would be familiar withsolving differential equations, whilst others had done very little. I decided to take a ‘lowestcommon denominator’ approach in which more or less everything was derived in the lectures.I didn’t want it to degenerate into dry mathematical machination, though, so I devisedgeometrical arguments through which some interesting results, such as the instability of theUniverse, could be derived. This then was the brief, to develop an interesting, rigorous coursecovering Newtonian mechanics and relativity, with minimal mathematical prerequisites. It allsounds like a contradiction in terms, but I gave it my best shot.

I decided not to consult any books, so this one has very few references. Everything hereinhas been produced many times over since the times of Newton and Einstein, and all I couldhope for was that my approach could be individual and fresh to the reader. The bibliography,however, lists the books I recommended at the beginning of the course as being suitablesupplementary texts. I should acknowledge, however, that the chapters on relativity wereundoubtedly influenced by the book from which I learned the subject: the first edition ofTaylor and Wheeler’s ‘Spacetime Physics’. But I wanted to tell the story my own way, andso I made a conscious effort to think everything through for myself. I hope my understandingwas good enough!

The problems provided at the end of each chapter are taken from those given to the studentsas classworks and problem sheets. They invariably start with some easy, confidence buildingexercises, before developing towards harder problems and examination questions. The briefsummaries at the end of each chapter are intended to give the most concise exposition forrevision; personally I’ve never found such things to be particularly helpful, but they are therefor those who do.

There are many who have helped me with this book. I would particularly like tomention Gilbert Satterthwaite who cast his critical eye over the manuscript, and Michael

xiv Preface to First Edition

Niblock, a student who endured my first rendering of the course, who provided a valuablestudent’s perspective on some chapters. Neal Powell and Meilin Sancho colluded to produceFigures 2.3, 3.17, 3.23, 7.12, 8.3, 8.4 and 8.5 – thank you. Thank you also to Keith Buttfor giving veterinary supervision of the cat experiment of Figure 8.10, and of course forpermitting Charlie to perform in the first place. It is also a pleasure to acknowledgeProfessor David Websdale from whom I inherited the course. He and his predecessors haveundoubtedly influenced the book, not least by allowing me to use the problems and exercisesthat were passed on to me at the beginning. I have adapted many of these, and if any mistakeshave crept in as a consequence, then I am responsible.

I never told my wife that I was writing this book I thought it would be fun to send herand her teaching colleagues a copy to review on publication so I can’t give the customary‘without her tireless assistance, etc., etc.’ acknowledgement. Nevertheless, Lulu, you havehelped immeasurably in this project through your patience, kindness and love.

The West Indian anthropologist and cricket writer C.L.R. James wrote: ‘Anyone whohas participated in an electoral campaign will have noticed how a speaker, eyes red fromsleeplessness, and sagging with fatigue, will rapidly recover all his power at an uproariouswelcome from an expectant crowd.’1 Well, not surprisingly, the class of some 200 studentsnever roared expectantly whenever I entered the lecture theatre (!), but I can vouch for theephemeral recovery of concentration amidst sleep deprivation – it was shortly after the birthof our son. Life is marginally less stressful now, and I’ve used the space–time to write thisbook. I hope you like it.

1James, C.L.R. (1966), Beyond a Boundary, Stanley Paul and Co.