MACMILLAN COLLEGE WORK OUT SERIES Mechanics Titles in this Series Dynamics Mathematics for Economists Electric Circuits Mechanics Electromagnetic Fields Molecular Genetics Electronics Numerical Analysis Elements of Banking Operational Research Engineering Materials Organic Chemistry Engineering Thermodynamics Physical Chemistry FluidMechanics Structural Mechanics Heat and Thermodynamics Waves and Optics MACMILLAN COLLEGE WORK OUT SERIES Mechanics Phil Dyke M MACMILLAN © Phil Dyke 1995 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London WI P 9HE. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published 1995 by MACMILLAN PRESS LTD Houndmills, Basingstoke, Hampshire RG21 2XS and London Companies and representatives throughout the world ISBN 978-0-333-58522-1 ISBN 978-1-349-13074-0 (eBook) DOI 10.1007/978-1-349-13074-0 A catalogue record for this book is available from the British Library. 10 9 8 7 6 5 4 3 2 I 04 03 02 01 00 99 98 97 96 95 To my son Adrian Contents Preface ix Acknowledgements xi 1 Revision of Preliminary Ideas 1.1 Fact Sheet 1 1.2 Worked Examples 2 1.3 Exercises 8 1.4 Outline Solutions to Exercises 8 2 Statics 2.1 Fact Sheet 10 2.2 Worked Examples 11 2.3 Exercises 21 2.4 Outline Solutions to Exercises 22 3 Motion Under Gravity 3.1 Fact Sheet 25 3.2 Worked Examples 25 3.3 Exercises 39 3.4 Outline Solutions to Exercises 39 4 Linear Momentum 4.1 Fact Sheet 42 4.2 Worked Examples 43 4.3 Exercises 54 4.4 Outline Solutions to Exercises 55 5 Variable Mass 5.1 Fact Sheet 58 5.2 Worked Examples 58 5.3 Exercises 66 5.4 Outline Solutions to Exercises 67 6 Vibrations 6.1 Fact Sheet 70 6.2 Worked Examples 71 6.3 Exercises 89 6.4 Outline Solutions to Exercises 90 7 Circular Motion and Rotating Axes 7.1 Fact Sheet 93 7.2 Worked Examples 94 vii 7.3 Exercises 106 7.4 Outline Solutions to Exercises 107 8 Orbits 8.1 Fact Sheet 110 8.2 Worked Examples III 8.3 Exercises 123 8.4 Outline Solutions to Exercises 124 9 Rigid Bodies 9.1 Fact Sheet 127 9.2 Worked Examples 128 9.3 Exercises 145 9.4 Outline Solutions to Exercises 146 10 Energy, Impulse and Stability 10.1 Fact Sheet 150 10.2 Worked Examples 151 10.3 Exercises 163 10.4 Outline Solutions to Exercises 164 11 Lagrangian Dynamics ILl Fact Sheet 168 11.2 Worked Examples 169 11.3 Exercises 186 11.4 Outline Solutions to Exercises 187 12 Non-linear Dynamics 12.1 Fact Sheet 192 12.2 Worked Examples 193 12.3 Exercises 202 12.4 Outline Solutions to Exercises 203 Appendix A: Vector Calculus 206 Appendix B: Differential Equations 209 Bibliography 212 Index 213 viii Preface Since the time of Isaac Newton (1642-1727) and probably before, mechanics has played a central role in mathematics and physics. It is also essential for other subjects such as physical chemistry and civil, mechanical and aeronautical engineering. Mathematics, and those subjects that use mathematics extensively, such as physics and engineering, are based on skill. In much the same way as sportsmen and sportswomen learn by doing, so do mathematicians and allied professionals learn by solving problems. Initially of course the golfer needs to know how to grip the club and how to stand, the pianist needs to know the rudiments of the keyboard and musical notation. Similarly, the student of mechanics needs to know the tools of the trade, in this case some algebra and calculus, a little geometry and trigonometry and some familiarity with vectors. After this, however, there is no substitute for doing problems. Many students have great difficulty getting started on a problem. One of the merits of this Work Out series is that the student learns precisely how to start by watching someone else (in this case me) doing problems. As I do the problems, I talk the reader through my thinking processes, so there should be no doubt why particular steps are being taken and why we are doing what we are doing at any particular stage. It has been said that of all skills-based subjects, mechanics stands out as being unique in benefitting students through exposing them to solved problems. I agree wholeheartedly with this. I learnt most of my mechanics by seeing others solve problems. The ones I have selected for this book include all my favourites. (I used to get so annoyed when the books I consulted never seemed to solve the hard problems. I sincerely hope that this criticism cannot be levelled here.) There are, at the end of each chapter, some exercises you should try for yourself. Outline solutions to all exercises are available at the very end of each set of exercises, but please resist the temptation to cheat! Before running through the contents, some comments about student background and computer algebra are necessary. In the last ten years or so, the amount of algebraic ma nipulation required of schoolchildren in pre-16 mathematics has dramatically decreased. This has occurred in an environment when more assessment is in the form of course work rather than examination. It is much more difficult to assess skills such as algebraic manipulation (or spelling for that matter) outside the formal examination. In my view, the decrease in algebra skills has happened by default rather than by design. Parallel with this has been the loss of geometry at school level. Both of these have changed the kind of mechanics problems that can be tackled by the average student. I have taken this into account in selecting the problems. This brings me to computer algebra. I have, a little reluctantly, decided not to emphasise computer algebra. There are many packages avail able (Macsyrna", Derive", Maple" and Mathernatica" to name but a few). There are also many other excellent software packages, which will increasingly affect the education of mathematicians. (I have used Ornnigraph" extensively in Chapter 12 for example). As yet there is no standard, and to base a problem-solving text such as this on just one compu ter algebra package would not be wise. Also, my colleagues and I can still find faults in most of these packages, in much the same way as no spell-checker is perfect in a word processor. The upshot of this is that I have included much of the algebraic manipulation in the solved problems. In mechanics, there is often an obvious point when all the mech- ix anics has been done and the rest is calculus or algebra. Nevertheless, in order to get an answer, the solving of the equations is necessary. The reader thus has the choice of checking (following) the manipulations by hand or inputting the equations into the chosen package and executing the appropriate menu-driven instructions. This Work Out assumes that the student is familiar with mechanics as presented to 17 to 19 year-olds. If you have zero knowledge or have had a very bad educational experi ence with regard to mechanics, my Guide to Mechanics (with Roger Whitworth) should cover the preliminary material. There is also some overlap (no bad thing, in my view). The organisation of material in a book such as this is never easy, and for mechanics this seems particularly so. Chapter 1is a reprise of preliminary notions in mechanics. It is this chapter that should tell you if you need to see the Guide first. Chapters 2-10 cover mate rial usually found in the first year of an undergraduate course in mechanics for mathema ticians or physicists. It should also cover all the mechanics commonly found in engineering and technology courses. I have decided to include Lagrange's equations and Euler's equa tion (tops and gyroscopes) for completeness. This material usually occupies the second year of mathematics degrees and is absent from engineering degrees; however, it is a crucial precursor for theoretical physicists who wish to study quantum mechanics. It is also, incidently, a wonderfully systematic treatment of mechanics that often appears easy to those who complain at length of the ad hoc nature of earlier mechanics. Lagrange's equations almost seem like pure mathematics. Finally, Chapter 12 is a brief introduction to non-linear mechanics, which is a very fashionable area to study these days. Everyone seems to have heard of 'chaos', although few, I think, understand it. In summary, therefore, you can expect to find the solved problems here representative of those in most undergraduate mechanics courses. In line with other Work Out texts, the material in this book can certainly be used in any order. There is very little cross-refer encing, except perhaps for information. The 'fact sheet' at the beginning of each chapter is a brief summary of relevant formulae and results which is best used as a reference. Phil Dyke December 1993 x