Classical Mechanics E-Book

Martin W. McCall

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 389

Veröffentlichungsjahr: 2011

Beschreibung

This new edition of Classical Mechanics, aimed at undergraduate physics and engineering students, presents ina user-friendly style an authoritative approach to the complementary subjects of classical mechanics and relativity.

The text starts with a careful look at Newton's Laws, before applying them in one dimension to oscillations and collisions. More advanced applications - including gravitational orbits and rigid body dynamics - are discussed after the limitations of Newton's inertial frames have been highlighted through an exposition of Einstein's Special Relativity. Examples given throughout are often unusual for an elementary text, but are made accessible to the reader through discussion and diagrams.

Updates and additions for this new edition include:

• New vector notation in Chapter 1
• An enhanced discussion of equilibria in Chapter 2
• A new section on a body falling a large distance towards a gravitational source in Chapter 2
• New sections in Chapter 8 on general rotation about a fixed principal axes, simple examples of principal axes and principal moments of inertia and kinetic energy of a body rotating about a fixed axis
• New sections in chapter 9: Foucault pendulum and free rotation of a rigid body; the latter including the famous tennis racquet theorem
• Enhanced chapter summaries at the end of each chapter
• Novel problems with numerical answers

A solutions manual is available at: www.wiley.com/go/mccall

Leseprobe

Contents

Preface to Second Edition

Preface to First Edition

Chapter 1: Newton’s Laws

1.1 What is Mechanics?

1.2 Mechanics as a Scientific Theory

1.3 Newtonian vs. Einsteinian Mechanics

1.4 Newton’s Laws

1.5 A Deeper Look at Newton’s Laws

1.6 Inertial Frames

1.7 Newton’s Laws in Noninertial Frames

1.8 Switching Off Gravity

1.9 Finale – Laws, Postulates or Definitions?

1.10 Summary

1.11 Problems

Chapter 2: One-dimensional Motion

2.1 Rationale for One-dimensional Analysis

2.2 The Concept of a Particle

2.3 Motion with a Constant Force

2.4 Work and Energy

2.5 Impulse and Power

2.6 Motion with a Position-dependent Force

2.7 The Nature of Energy

2.8 Potential Functions

2.9 Equilibria

2.10 Motion Close to a Stable Equilibrium

2.11 The Stability of the Universe

2.12 Trajectory of a Body Falling a Large Distance Under Gravity

2.13 Motion with a Velocity-dependent Force

2.14 Summary

2.15 Problems

Chapter 3: Oscillatory Motion

3.1 Introduction

3.2 Prototype Harmonic Oscillator

3.3 Differential Equations

3.4 General Solution for Simple Harmonic Motion

3.5 Energy in Simple Harmonic Motion

3.6 Damped Oscillations

3.7 Light Damping – the Q Factor

3.8 Heavy Damping and Critical Damping

3.9 Forced Oscillations

3.10 Complex Number Method

3.11 Electrical Analogue

3.12 Power in Forced Oscillations

3.13 Coupled Oscillations

3.14 Summary

3.15 Problems

Chapter 4: Two-body Dynamics

4.1 Rationale

4.2 Centre of Mass

4.3 Internal Motion: Reduced Mass

4.4 Collisions

4.5 Elastic Collisions

4.6 Inelastic Collisions

4.7 Centre-of-mass Frame

4.8 Rocket Motion

4.9 Launch Vehicles

4.10 Summary

4.11 Problems

Chapter 5: Relativity 1: Space and Time

5.1 Why Relativity?

5.2 Galilean Relativity

5.3 The Fundamental Postulates of Relativity

5.4 Inertial Observers in Relativity

5.5 Comparing Transverse Distances Between Frames

5.6 Lessons from a Light Clock: Time Dilation

5.7 Proper Time

5.8 Interval Invariance

5.9 The Relativity of Simultaneity

5.10 The Relativity of Length: Length Contraction

5.11 The Lorentz Transformations

5.12 Velocity Addition

5.13 Particles Moving Faster than Light: Tachyons

5.14 Summary

5.15 Problems

Chapter 6: Relativity 2: Energy and Momentum

6.1 Energy and Momentum

6.2 The Meaning of Rest Energy

6.3 Relativistic Collisions and Decays

6.4 Photons

6.5 Units in High-energy Physics

6.6 Energy/Momentum Transformations Between Frames

6.7 Relativistic Doppler Effect

6.8 Summary

6.9 Problems

Chapter 7: Gravitational Orbits

7.1 Introduction

7.2 Work in Three Dimensions

7.3 Torque and Angular Momentum

7.4 Central Forces

7.5 Gravitational Orbits

7.6 Kepler’s Laws

7.7 Comments

7.8 Summary

7.9 Problems

Chapter 8: Rigid Body Dynamics

8.1 Introduction

8.2 Torque and Angular Momentum for Systems of Particles

8.3 Centre of Mass of Systems of Particles and Rigid Bodies

8.4 Angular Momentum of Rigid Bodies

8.5 Kinetic Energy of Rigid Bodies

8.6 Bats, Cats, Pendula and Gyroscopes

8.7 General Rotation About a Fixed Axis

8.8 Principal Axes

8.9 Examples of Principal Axes and Principal Moments of Inertia

8.10 Kinetic Energy of a Body Rotating About a Fixed Axis

8.11 Summary

8.12 Problems

Chapter 9: Rotating Frames

9.1 Introduction

9.2 Experiments on Roundabouts

9.3 General Prescription for Rotating Frames

9.4 The Centrifugal Term

9.5 The Coriolis Term

9.6 The Foucault Pendulum

9.7 Free Rotation of a Rigid Body – Tennis Rackets and Matchboxes

9.8 Final Thoughts

9.9 Summary

9.10 Problems

Appendix 1: Vectors, Matrices and Eigenvalues

A.1 The Scalar (Dot) Product

A.2 The Vector (Cross) Product

A.3 The Vector Triple Product

A.4 Multiplying a Vector by a Matrix

A.5 Calculating the Determinant of a 3 × 3 Matrix

A.6 Eigenvectors and Eigenvalues

A.7 Diagonalising Symmetric Matrices

Appendix 2: Answers to Problems

Appendix 3: Bibliography

Index

This edition first published 2011

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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 and simple vectors are presumed. The text starts with a careful look at Newton’s Laws, before applying them in one dimension to oscillations and collisions. More advanced applications - including gravitational orbits, rigid body dynamics and mechanics in rotating frames - are deferred until after the limitations of Newton’s laws have been highlighted through an exposition of Einstein’s Special Relativity. Big problems that are tackled using elementary techniques include the stability of the Universe, a body falling from a great height under gravity and Foucault’s pendulum. 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 with an 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–dc222010022396

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

ISBN 9780470715741 (Hbk) 9780470715727 (Pbk)

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 He endures and is present. He endures forever, and is everywhere present; and by existing always and everywhere he constitutes duration and space...And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to natural philosophy.

Isaac Newton, 1687.

Preface to Second Edition

One of the major problems of producing this second edition of Classical Mechanics was deciding what new material to include. The most natural way to extend the work was to cover Lagrangian and Hamiltonian mechanics. Although these techniques are certainly very important, they are rather advanced, and I was keen to maintain the elementary flavour of the first edition. Moreover, as I discovered when I taught these techniques to third year undergraduates, the number of problems that become accessible is rather small; the achievements of these methods is principally conceptual in, for example, paving a path towards quantum mechanics. In fact the only problem I could find that really illustrated the 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, I didn’t feel that it was quite in keeping with the spirit of the first edition, which was to teach mechanics and special relativity ab inito to undergraduate students with minimal mathematics. 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 I haven’t seen in textbooks before. New material on collisions is included to show that snooker balls always scatter at 90°. When I reconsidered the contortions to make the discussion of rigid bodies rotating about a fixed axis ‘simple’, I decided that the labour of introducing the inertia tensor was not so great, and consequently the rotation of arbitrary bodies is now discussed 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 rotating frames rather nicely. Some mathematical extension has been necessary to accommodate these topics, and the brief Appendix of the first edition has now been significantly extended. The chapter 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 been redrawn. New problems have been included, and a comprehensive online solutions manual has been prepared (www.wiley.com/go/mccall). I also took the opportunity of correcting a few typographical errors.

Preface to First Edition

The tale, as Tolkien wrote in the preface to Lord of the Rings, grew in the telling. Having copped the highest profile undergraduate course in the Physics programme at Imperial College, 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 duly contracted with John Wiley & Sons Ltd to produce a camera ready manuscript in a few months. Thus I became pregnant with my first literary child. Little did I understand the pangs of labour.

The course entitled ‘Mechanics and Relativity’ is given to incoming undergraduates. I would meet them in the first week of their arrival and finish my forty first and final lecture in about the middle of the second term. The varied level of mathematical preparation of the class of 200 students set special problems for designing the course. Some would be familiar with solving differential equations, whilst others had done very little. I decided to take a ‘lowest common 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 devised geometrical arguments through which some interesting results, such as the instability of the Universe, could be derived. This then was the brief, to develop an interesting, rigorous course covering Newtonian mechanics and relativity, with minimal mathematical prerequisites. It all sounds 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 herein has been produced many times over since the times of Newton and Einstein, and all I could hope 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 suitable supplementary texts. I should acknowledge, however, that the chapters on relativity were undoubtedly influenced by the book from which I learned the subject: the first edition of Taylor and Wheeler’s ‘Spacetime Physics’. But I wanted to tell the story my own way, and so I made a conscious effort to think everything through for myself. I hope my understanding was good enough!

The problems provided at the end of each chapter are taken from those given to the students as classworks and problem sheets. They invariably start with some easy, confidence building exercises, before developing towards harder problems and examination questions. The brief summaries at the end of each chapter are intended to give the most concise exposition for revision; personally I’ve never found such things to be particularly helpful, but they are there for those who do.

There are many who have helped me with this book. I would particularly like to mention Gilbert Satterthwaite who cast his critical eye over the manuscript, and Michael Niblock, a student who endured my first rendering of the course, who provided a valuable student’s perspective on some chapters. Neal Powell and Meilin Sancho colluded to produce Figures 2.3, 3.17, 3.23, 7.12, 8.3, 8.4 and 8.5 – thank you. Thank you also to Keith Butt for giving veterinary supervision of the cat experiment of Figure 8.10, and of course for permitting Charlie to perform in the first place. It is also a pleasure to acknowledge Professor David Websdale from whom I inherited the course. He and his predecessors have undoubtedly influenced the book, not least by allowing me to use the problems and exercises that were passed on to me at the beginning. I have adapted many of these, and if any mistakes have 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 her and 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 have helped immeasurably in this project through your patience, kindness and love.

The West Indian anthropologist and cricket writer C.L.R. James wrote: ‘Anyone who has participated in an electoral campaign will have noticed how a speaker, eyes red from sleeplessness, and sagging with fatigue, will rapidly recover all his power at an uproarious welcome from an expectant crowd.’1 Well, not surprisingly, the class of some 200 students never roared expectantly whenever I entered the lecture theatre (!), but I can vouch for the ephemeral recovery of concentration amidst sleep deprivation – it was shortly after the birth of our son. Life is marginally less stressful now, and I’ve used the space–time to write this book. I hope you like it.

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

1

Newton’s Laws

1.1 What is Mechanics?

Even those uninterested in physics seem to have an intuitive notion about why and how things move. If a ball flies through the air, it does so because we have projected it—a force has been applied to a body which impels it to move. Glossing deftly over any difficulties there may be in defining what the italicised words actually mean, a description of the sequence force–body–motion is, conventionally, what is meant by mechanics. Mechanics is about forces and motion as applied to bodies.

Some people further resolve mechanics into kinematics and dynamics. Kinematics is the description of motion in terms of its trajectory through space as time progresses, or technically as r(t) where r is the position vector of the (dimensionless) body at time t. Typically this trajectory is calculated by solving an equation of motion. Dynamics, on the other hand, relates changes in a body’s motion to their causes, i.e. forces. Dynamics is therefore the ‘why’ of motion, a typical dynamical problem being to find the resultant force acting on a body. Personally I have never found the delineation of kinematics and dynamics as branches of mechanics particularly useful, and so these terms will be avoided in this book.

1.2 Mechanics as a Scientific Theory

So how can we quantify the relation between the forces that act on bodies and their resultant motions? Given the obvious immediacy of this problem to our everyday lives, it is not surprising that progress in this area, firstly by Galileo and then by Newton, heralded the first truly scientific theory. A fair question to ask at this stage is: ‘What are the characteristics of a scientific theory?’ Without becoming too philosophical about the issue, it can be said that a scientific1 theory is a concise summary of scientific ideas describing the results of experiments. Furthermore, a good theory will have predictive power beyond the domain of experimental experience to date. In physics, the theory is usually expressed as a mathematical relationship between experimentally determinable quantities. This means that the acid test which all physical theories must pass is: