Solved Problems in Classical Mechanics: Analytical and Numerical Solutions with Comments

608 Pages·2010·11.8 MB·English

by  Owen de Lange

Solved Problems in Classical Mechanics
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Solved Problems in Classical
Mechanics
Analytical and numerical solutions
with comments
O.L. de Lange and J. Pierrus
School of Physics, University of KwaZulu-Natal,
Pietermaritzburg, South Africa
1
3
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Preface
It is in the study of classical mechanics that we first encounter many of the basic
ingredients that are essential to our understanding of the physical universe. The
concepts include statements concerning space and time, velocity, acceleration, mass,
momentum and force, and then an equation of motion and the indispensable law
of action and reaction – all set (initially) in the background of an inertial frame of
reference. Units for length, time and mass are introduced and the sanctity of the
balance of units in any physicalequation(dimensional analysis)is stressed.Reference
is alsomade to the task of measuringthese units – metrology,whichhas become such
an astonishing science/art.
The rewards of this study are considerable. For example, one comes to
appreciate Newton’s great achievement – that the dynamics of the classical universe
can be understood via the solutions of differential equations – and this leads on
to questions regarding determinism and the effects of even small uncertainties or
disturbances. One learns further that even when Newton’s dynamics fails, many of
the concepts remain indispensable and some of its conclusions retain their validity –
suchastheconservationlawsformomentum,angularmomentumandenergy,andthe
connection between conservation and symmetry – and one discusses the domain of
applicability of the theory. Along the way, a student encounters techniques – such as
the use of vector calculus – that permeate much of physics from electromagnetism to
quantum mechanics.
Allthisisfamiliartolecturerswhoteachphysicsatuniversities;hencetheemphasis
on undergraduate and graduate courses in classical mechanics, and the variety of
excellent textbooks on the subject. It has, furthermore, been recognizedthat training
in this and related branches of physics is useful also to students whose careers will
takethemoutsidephysics.Itseemsthatheretheproblem-solvingabilitiesthatphysics
students develop stand them in good stead and make them desirable employees.
Our book is intended to assist students in acquiring such analytical and
computational skills. It should be useful for self-study and also to lecturers and
students in mechanics courses where the emphasis is on problem solving, and
formallecturesarekepttoaminimum.Inourexperience,studentsrespondwelltothis
approach.After all,the rudimentsofthe subjectcanbe presentedquite succinctly(as
we have endeavoured to do in Chapter 1) and, where necessary, details can be filled
in using a suitable text.
With regard to the format of this book: apart from the introductory chapter, it
consists entirely of questions and solutions on various topics in classical mechanics
that are usually encountered during the first few years of university study. It is
Solved Problems in Classical Mechanics
(cid:0)(cid:1)
suggested that a student first attempt a question with the solution covered, and
only consult the solution for help where necessary. Both analytical and numerical
(computer) techniques are used, as appropriate,in obtaining and analyzing solutions.
Someofthenumericalquestionsaresuitableforprojectworkincomputationalphysics
(seetheAppendix).Mostsolutionsarefollowedbyasetofcommentsthatareintended
to stimulate inductive reasoning (additional analysis of the problem, its possible ex-
tensions andfurther significance),andsometimes to mentionliteraturewe havefound
helpfulandinteresting.Wehaveincludedquestionsonbitsof‘theory’fortopicswhere
students initially encounterdifficulty – suchas the harmonicoscillatorandthe theory
of mechanical energy – because this can be useful, both in revising and cementing
ideas and in building confidence.
The mathematical ability that the reader should have consists mainly of the
following: an elementary knowledge of functions – their roots, turning points, asymp-
totic values and graphs – including the ‘standard’ functions of physics (polynomial,
trigonometric, exponential, logarithmic, and rational); the differential and integral
calculus(includingpartialdifferentiation);andelementaryvectoranalysis.Also,some
knowledge of elementary mechanics and general physics is desirable, although the
extent to which this is necessary will depend on the proclivities of the reader.
For our computer calculations we use Mathematica(cid:1)R, version7.0.In eachinstance
the necessary code (referred to as a notebook) is provided in a shadebox in the text.
Notebooks that include the interactive Manipulate function are given in Chapters
6, 10, 11 and 13 (and are listed in the Appendix). They enable the reader to observe
motiononacomputerscreen,andtostudytheeffectsofchangingrelevantparameters.
A reader without prior knowledge of Mathematica should consult the tutorial
(‘First Five Minutes with Mathematica’) and the on-line Help. Also, various useful
tutorials can be downloaded from the website www.Wolfram.com. All graphs of
numerical results have been drawn to scale using Gnuplot.
Inouranalyticalsolutionswehavetriedtostrikeabalancebetweenburdeningthe
readerwithtoomuchdetailandnotheedingLittlewood’sdictumthat“twotrivialities
omittedcanadduptoanimpasse”.Inthisregarditisprobablynotpossibletosatisfy
allreaders,butwehopethatevententativeoneswillsoonbeabletodiscernfootprints
in the mist. After all, it is well worth the effort to learnthat (on some level) the rules
of the universe are simple, and to begin to enjoy “the unreasonable effectiveness of
mathematics in the natural sciences” (Wigner).
Finally, we thank Robert Lindebaum and Allard Welter for their assistance with
our computer queries and also Roger Raab for helpful discussions.
Pietermaritzburg, South Africa O. L. de Lange
January 2010 J. Pierrus
Contents
1 Introduction 1
2 Miscellanea 11
3 One-dimensional motion 30
4 Linear oscillations 60
5 Energy and potentials 92
6 Momentum and angular momentum 127
7 Motion in two and three dimensions 157
8 Spherically symmetric potentials 216
9 The Coulomb and oscillator problems 263
10 Two-body problems 286
11 Multi-particle systems 325
12 Rigid bodies 399
13 Non-linear oscillations 454
14 Translation and rotation of the reference frame 518
15 The relativity principle and some of its consequences 557
Appendix 588
Index 590
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1
Introduction
The followingoutline ofthe rudimentsofclassicalmechanicsprovidesthe background
that is necessary in order to use this book. For the reader who finds our presentation
toobrief,thereareseveralexcellentbooksthatexpoundonthesebasics,suchasthose
listed below.[1−4]
1.1 Kinematics and dynamics of a single particle
The goal of classical mechanics is to provide a quantitative description of the motion
of physical objects. Like any physical theory, mechanics is a blend of definitions and
postulates. In describing this theory it is convenient to first introduce the concept of
a point object (a particle) and to start by considering the motion of a single particle.
To this end one must make an assumption concerning the geometry of space. In
Newtonian dynamics it is assumed that space is three-dimensional and Euclidean.
Thatis,space is spannedby the threecoordinatesofa Cartesiansystem;the distance
between any two points is given in terms of their coordinates by Pythagoras’s
theorem, and the familiar geometric and algebraic rules of vector analysis apply. It
is also assumed – at least in non-relativistic physics – that time is independent of
space. Furthermore, it is supposed that space and time are ‘sufficiently’ continuous
thatthe differentialandintegralcalculus canbe applied.A helpful discussionofthese
topics is given in Griffiths’s book.[2]
Withthis background,oneselectsacoordinatesystem.Often,this isarectangular
or Cartesian system consisting of an arbitrarily chosen coordinate origin O and three
orthogonal axes, but in practice any convenient system can be used (spherical, cylin-
drical,etc.).Thepositionofaparticlerelativetothiscoordinatesystemisspecifiedby
a vector function of time – the position vector r(t). An equation for r(t) is known as
the trajectory of the particle, and finding the trajectory is the goal mentioned above.
In terms of r(t) we define two indispensable kinematic quantities for the particle:
the velocity v(t), which is the time rate of change of the position vector,
[1] L. D. Landau, A.I. Akhiezer, and E. M.Lifshitz,General physics: mechanics and molecular
physics. Oxford:Pergamon, 1967.
[2] J. B. Griffiths, The theory of classical dynamics. Cambridge: Cambridge University Press,
1985.
[3] T. W. B. Kibble and F. H. Berkshire, Classical mechanics. London: Imperial College Press,
5thedn,2004.
[4] R.Baierlein,Newtonian dynamics. NewYork:McGraw-Hill,1983.