Dynamics and Relativity E-Book

Jeffrey Forshaw

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: The Manchester Physics Series
• Sprache: Englisch

Seitenzahl: 552

Veröffentlichungsjahr: 2014

Beschreibung

A new title in the Manchester Physics Series, this introductory text emphasises physical principles behind classical mechanics and relativity. It assumes little in the way of prior knowledge, introducing relevant mathematics and carefully developing it within a physics context. Designed to provide a logical development of the subject, the book is divided into four sections, introductory material on dynamics, and special relativity, which is then followed by more advanced coverage of dynamics and special relativity. Each chapter includes problems ranging in difficulty from simple to challenging with solutions for solving problems.

• Includes solutions for solving problems
• Numerous worked examples included throughout the book
• Mathematics is carefully explained and developed within a physics environment
• Sensitive to topics that can appear daunting or confusing

Leseprobe

Contents

Editors’ Preface to the Manchester Physics Series

Authors’ Preface

Part I Introductory Dynamics

1 Space, Time and Motion

1.1 DEFINING SPACE AND TIME

1.2 VECTORS AND CO-ORDINATE SYSTEMS

1.3 VELOCITY AND ACCELERATION

1.4 STANDARDS AND UNITS

2 Force, Momentum and Newton’s Laws

2.1 FORCE AND STATIC EQUILIBRIUM

2.2 FORCE AND MOTION

2.3 APPLICATIONS OF NEWTON’S LAWS

3 Energy

3.1 WORK, POWER AND KINETIC ENERGY

3.2 POTENTIAL ENERGY

3.3 COLLISIONS

3.4 ENERGY CONSERVATION IN COMPLEX SYSTEMS

4 Angular Momentum

4.1 ANGULAR MOMENTUM OF A PARTICLE

4.2 CONSERVATION OF ANGULAR MOMENTUM IN SYSTEMS OF PARTICLES

4.3 ANGULAR MOMENTUM AND ROTATION ABOUT A FIXED AXIS

4.4 SLIDING AND ROLLING

4.5 ANGULAR IMPULSE AND THE CENTRE OF PERCUSSION

4.6 KINETIC ENERGY OF ROTATION

Part II Introductory Special Relativity

5 The Need for a New Theory of Space and Time

5.1 SPACE AND TIME REVISITED

5.2 EXPERIMENTAL EVIDENCE

5.3 EINSTEIN’S POSTULATES

6 Relativistic Kinematics

6.1 TIME DILATION, LENGTH CONTRACTION AND SIMULTANEITY

6.2 LORENTZ TRANSFORMATIONS

6.3 VELOCITY TRANSFORMATIONS

7 Relativistic Energy and Momentum

7.1 MOMENTUM AND ENERGY

7.2 APPLICATIONS IN PARTICLE PHYSICS

Part III Advanced Dynamics

8 Non-inertial Frames

8.1 LINEARLY ACCELERATING FRAMES

8.2 ROTATING FRAMES

9 Gravitation

9.1 NEWTON’S LAW OF GRAVITY

9.2 THE GRAVITATIONAL POTENTIAL

9.3 REDUCED MASS

9.4 MOTION IN A CENTRAL FORCE

9.5 ORBITS

10 Rigid Body Motion

10.1 THE ANGULAR MOMENTUM OF A RIGID BODY

10.2 THE MOMENT OF INERTIA TENSOR

10.3 PRINCIPAL AXES

10.4 FIXED-AXIS ROTATION IN THE LAB FRAME

10.5 EULER’S EQUATIONS

10.6 THE FREE ROTATION OF A SYMMETRIC TOP

10.7 THE STABILITY OF FREE ROTATION

10.8 GYROSCOPES

Part IV Advanced Special Relativity

11 The Symmetries of Space and Time

11.1 SYMMETRY IN PHYSICS

11.2 LORENTZ SYMMETRY

12 Four-vectors and Lorentz Invariants

12.1 THE VELOCITY FOUR-VECTOR

12.2 THE WAVE FOUR-VECTOR

12.3 THE ENERGY-MOMENTUM FOUR-VECTOR

12.4 ELECTRIC AND MAGNETIC FIELDS

13 Space-time Diagrams and Causality

13.1 RELATIVITY PRESERVES CAUSALITY

13.2 AN ALTERNATIVE APPROACH

14 Acceleration and General Relativity

14.1 ACCELERATION IN SPECIAL RELATIVITY

14.2 A GLIMPSE OF GENERAL RELATIVITY

Appendix A Deriving the Geodesic Equation

Appendix B Solutions to Problems

Index

The Manchester Physics Series

General EditorsF.K. LOEBINGER: F. MANDL: D.J. SANDIFORD:

School of Physics and Astronomy,The University of Manchester

Properties of Matter:

B. H. Flowers and E. Mendoza

Statistical Physics:

Second Edition

F. Mandl

Electromagnetism:

Second Edition

I. S. Grant and W. R. Phillips

Statistics:

R. J. Barlow

Solid State Physics:

Second Edition

J. R. Hook and H. E. Hall

Quantum Mechanics:

F. Mandl

Computing for Scientists:

R. J. Barlow and A. R. Barnett

The Physics of Stars:

Second Edition

A. C. Phillips

Nuclear Physics:

J. S. Lilley

Introduction to Quantum Mechanics:

A. C. Phillips

Particle Physics:

Third Edition

B. R. Martin and G. Shaw

Dynamics and Relativity:

J. R. Forshaw and A. G. Smith

Vibrations and Waves:

G.C. King

This edition first published 2009© 2009 John Wiley & Sons Ltd

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. 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, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

The Publisher and the Author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the Publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the Author or the Publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the Publisher nor the Author shall be liable for any damages arising herefrom.

Library of Congress Cataloging-in-Publication Data

Forshaw, J. R. (Jeffrey Robert), 1968 –Dynamics and relativity/Jeffrey R. Forshaw and A. Gavin Smith.p. cm.

Includes bibliographical references and index.ISBN 978-0-470-01459-2 (cloth : alk. paper) – ISBN 978-0-470-01460-8 (pbk. : alk. paper)1. Special relativity (Physics) 2. Dynamics. I. Smith, A. Gavin. II. Title.QC173.65.F67 2009530.11 – dc22

2008053366

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

ISBN 978-0-470-01459-2 (HB)

ISBN 978-0-470-01460-8 (PB)

Typeset in 10/12 Times by Laserwords Private Limited, Chennai, India

Dedicated to the memory of Howard North and Edward Swallow.

Editors’ Preface to the Manchester Physics Series

The Manchester Physics Series is a series of textbooks at first degree level. It grew out of our experience at the University of Manchester, widely shared elsewhere, that many textbooks contain much more material than can be accommodated in a typical undergraduate course; and that this material is only rarely so arranged as to allow the definition of a short self-contained course. In planning these books we have had two objectives. One was to produce short books so that lecturers would find them attractive for undergraduate courses, and so that students would not be frightened off by their encyclopaedic size or price. To achieve this, we have been very selective in the choice of topics, with the emphasis on the basic physics together with some instructive, stimulating and useful applications. Our second objective was to produce books which allow courses of different lengths and difficulty to be selected with emphasis on different applications. To achieve such flexibility we have encouraged authors to use flow diagrams showing the logical connections between different chapters and to put some topics in starred sections. These cover more advanced and alternative material which is not required for the understanding of latter parts of each volume.

Although these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them are suitable for wider use in other sciences. Each Author’s Preface gives details about the level, prerequisites, etc., of that volume.

The Manchester Physics Series has been very successful since its inception 40 years ago, with total sales of more than a quarter of a million copies. We are extremely grateful to the many students and colleagues, at Manchester and elsewhere, for helpful criticisms and stimulating comments. Our particular thanks go to the authors for all the work they have done, for the many new ideas they have contributed, and for discussing patiently, and often accepting, the suggestions of the editors.

Finally we would like to thank our publishers, John Wiley & Sons, Ltd., for their enthusiastic and continued commitment to the Manchester Physics Series.

F. K. LoebingerF. MandlD. J. SandifordAugust 2008

Authors’ Preface

In writing this book, our goal is to help the student develop a good understanding of classical dynamics and special relativity. We have tried to start out gently: the first part of the book aims to provide the solid foundations upon which the second half builds. In the end, we are able, in the final chapter, to cover some quite advanced material for a book at this level (when we venture into the terrain of Einstein’s General Theory of Relativity) and it is our hope that our pedagogical style will lead the keen student all the way to the denouement. That said, we do not assume too much prior knowledge. A little calculus, trigonometry and some exposure to vectors would help but not much more than that is needed in order to get going. We have in mind that the first half of the book covers material core to a typical first year of undergraduate studies in physics, whilst the second half covers material that might appear in more advanced first or second year courses (e.g. material such as the general rotation of rigid bodies and the role of four-vectors in special relativity).

The classical mechanics of Newton and the theory of relativity, developed by Einstein, both make assumptions as to the structure of space and time. For Newton time is an absolute, something to be agreed upon by everyone, whilst for Einstein time is more subjective and clocks tick at different rates depending upon where they are and how they are moving. Such different views lead to different physics and by presenting Newtonian mechanics alongside relativity, as we do in this book, it becomes possible to compare and contrast the two. Of course, we shall see how Newtonian physics provides a very good approximation to that of Einstein for most everday phenomena, but that it fails totally when things whizz around at speeds approaching the speed of light.

In this era of electronic communications and online resources that can be researched at the push of a button, it might seem that the need for textbooks is diminished. Perhaps not surprisingly we don’t think that is the case. Quiet time spent with a textbook, some paper and a pen, reading and solving problems, is probably still the best way to do physics. Just as one cannot claim to be a pianist without playing a piano, one cannot claim to be a physicist without solving physics problems. It is a point much laboured, but it is true nonetheless. The problems that really help develop understanding are usually those that take time to crack. The painful process of failing to solve a problem is familiar to every successful physicist, as is the excitement of figuring out the way forward. Our advice when solving the problems in this book is to persevere for as long as possible before peeking at the solution, to try and enjoy the process and not to panic if you cannot see how to start a problem.

We have deliberately tried to keep the figures as simple as possible. A good drawing can often be an important step to solving a physics problem, and we encourage you to make them at every opportunity. For that reason, we have illustrated the book with the sorts of drawings that we would normally use in lectures or tutorials and have deliberately avoided the sort of embellishments that would undoubtedly make the book look prettier. Our aim is to present diagrams that are easy to reproduce.

A comment is in order on our usage of the word “classical”. For us “classical” refers to physics pre-Einstein but not everyone uses that terminology. Sometimes, classical is used to refer to the laws of physics in the absence of quantum mechanics and in that sense, special relativity could be said to be a classical theory. We have nothing to say about the quantum theory in this book, except that quantum theories that are also consistent with relativity lie at the very heart of modern physics. Hopefully this book will help whet the appetite for further studies in that direction.

We should like to express our gratitude to all those who have read the manuscript and provided helpful suggestions. In particular we thank Rob Appleby, Richard Battye, Mike Birse, Brian Cox, Joe Dare, Fred Loebinger, Nicola Lumley, Franz Mandl, Edward Reeves, David Sandiford and Martin Yates.

Finally, we would like to express particular gratitude to our parents, Thomas & Sylvia Forshaw and Roy & Marion Smith, for their constant support. For their love and understanding, our heartfelt thanks go to Naomi, Isabel, Jo, Ellie, Matt and Josh.

Jeffrey R. ForshawA. Gavin SmithOctober 2008

Part I

Introductory Dynamics

1

Space, Time and Motion

1.1 DEFINING SPACE AND TIME

If there is one part of physics that underpins all others, it is the study of motion. The accurate description of the paths of celestial objects, of planets and moons, is historically the most celebrated success of a classical mechanics underpinned by Newton’s laws1. The range of applicability of these laws is vast, encompassing a scale that extends from the astronomical to the microscopic. We have come to understand that many phenomena not previously associated with motion are in fact linked to the movement of microscopic objects. The absorption and emission spectra of atoms and molecules arise as a result of transitions made by their constituent electrons, and the random motion of ensembles of atoms and molecules forms the basis for the modern statistical description of thermodynamics. Although atomic and subatomic objects are properly described using quantum mechanics, an understanding of the principles of classical mechanics is essential in making the conceptual leap from continuous classical systems with which we are most familiar, to the discretised quantum mechanical systems, which often behave in a manner at odds with our intuition. Indeed, the calculational techniques that are routinely used in quantum mechanics have their roots in the classical mechanics of particles and waves; a close familiarity with their use in classical systems is an asset when facing problems of an inherently quantum mechanical nature.

As we shall see in the second part of this book, when objects move at speeds approaching the speed of light classical notions about the nature of space and time fail us. As a result, the classical mechanics of Newton should be viewed as a low-velocity approximation to the more accurate relativistic theory of Einstein2. To look carefully at the differences between relativistic and non-relativistic theories forces us to recognise that our intuitive ideas about how things move are often incorrect. At the most fundamental level, mechanics of either the classical or the quantum kind, in either the relativistic or non-relativistic limit, is a study of motion and to study motion is to ask some fundamental questions about the nature of space and time. In this book we will draw out explicitly the different underlying structures of space and time used in the approaches of Newton and Einstein.

1.1.1 Space and the classical particle

We all have strong intuitive ideas about space, time and motion and it is precisely because of this familiarity that we must take special care in our attempts to define these fundamental concepts, so as not to carry too many unrecognised assumptions along with us as we develop the physics. So let us start by picking apart what we mean by position. We can usually agree what it means for London to be further away than Inverness and we all know that in order to go to London from Inverness we must also know the direction in which to travel. It may also seem to be fairly uncontentious that an object, such as London, has a position that can be specified, i.e. it is assumed that given enough information there will be no ambiguity about where it is. Although this seems reasonable, there is immediately a problem: day-to-day objects such as tennis balls and cities have finite size; there are a number of ‘positions’ for a given object that describe different parts of the object. Having directions to London may not be enough to find Kings Cross station, and having directions to Kings Cross station may not be enough to find platform number nine. To unambiguously give the position of an object is therefore only possible if the object is very small – vanishingly small, in fact. This hypothetical, vanishingly small object is called a particle. It might be suggested that with the discovery of the substructure of the atom, true particles, with mass but no spatial extent, have been identified. However, at this level, the situation becomes complicated by quantum uncertainty which makes the simultaneous specification of position and momentum impossible. The classical particle is therefore an idealisation, a limit in which the size of an object tends to zero but in which we ignore quantum phenomena. Later we shall see that it is possible to define a point called the centre of mass of an extended object and that this point behaves much like a classical particle. The collection of all possible positions for a particle forms what we call space.