Solved Problems in Classical Mechanics potx

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

Great Clarendon Street, Oxford

3ox2 6dp

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1 3 5 7 9 10 8 6 4 2

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 physical equation (dimensional analysis) is stressed. Reference

is also made to the task of measuring these units – metrology, which has 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 –

such as the conservation laws for momentum, angular momentum and energy, and the

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.

All this is familiar to lecturers who teach physics at universities; hence the emphasis

on undergraduate and graduate courses in classical mechanics, and the variety of

excellent textbooks on the subject. It has, furthermore, been recognized that training

in this and related branches of physics is useful also to students whose careers will

take them outside physics. It seems that here the problem-solving abilities that physics

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

formal lectures are kept to a minimum. In our experience, students respond well to this

approach. After all, the rudiments of the subject can be presented quite 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

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.

Some of the numerical questions are suitable for project work in computational physics

(see the Appendix). Most solutions are followed by a set of comments that are intended

to stimulate inductive reasoning (additional analysis of the problem, its possible ex￾tensions and further significance), and sometimes to mention literature we have found

helpful and interesting. We have included questions on bits of ‘theory’ for topics where

students initially encounter difficulty – such as the harmonic oscillator and the 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 (including partial differentiation); and elementary vector analysis. 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
R , version 7.0. In each instance

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

motion on a computer screen, and to study the effects of changing relevant parameters.

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.

In our analytical solutions we have tried to strike a balance between burdening the

reader with too much detail and not heeding Littlewood’s dictum that “ two trivialities

omitted can add up to an impasse”. In this regard it is probably not possible to satisfy

all readers, but we hope that even tentative ones will soon be able to discern footprints

in the mist. After all, it is well worth the effort to learn that (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. Pierru

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 following outline of the rudiments of classical mechanics provides the background

that is necessary in order to use this book. For the reader who finds our presentation

too brief, there are several excellent books that expound on these basics, such as those

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.

That is, space is spanned by the three coordinates of a Cartesian system; 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

that the differential and integral calculus can be applied. A helpful discussion of these

topics is given in Griffiths’s book.[2]

With this background, one selects a coordinate system. Often, this is a rectangular

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.). The position of a particle relative to this coordinate system is specified by

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,

5th edn, 2004.

[4] R. Baierlein, Newtonian dynamics. New York: McGraw-Hill, 1983.

￾ Solved Problems in Classical Mechanics

v(t) = dr(t)

dt , (1)

and the acceleration a(t), which is the time rate of change of the velocity,

a(t) = dv(t)

dt . (2)

It follows from (1) and (2) that the acceleration is also the second derivative

a = d2r

dt2 . (3)

Sometimes use is made of Newton’s notation, where a dot denotes differentiation with

respect to time, so that (1)–(3) can be abbreviated

v = r˙ , a = v˙ = ¨r . (4)

The stage for mechanics – the frame of reference – consists of a coordinate system

together with clocks for measuring time. Initially, we restrict ourselves to an inertial

frame. This is a frame in which an isolated particle (one that is free of any applied

forces) moves with constant velocity v – meaning that v is constant in both magnitude

and direction (uniform rectilinear motion). This statement is the essence of Newton’s

first law of motion. In Newton’s mechanics (and also in relativity) an inertial frame is

not a unique construct: any frame moving with constant velocity with respect to it is

also inertial (see Chapters 14 and 15). Consequently, if one inertial frame exists, then

infinitely many exist. Sometimes mention is made of a primary inertial frame, which

is at rest with respect to the ‘fixed’ stars.

Now comes a central postulate of the entire theory: in an inertial frame, if a particle

of mass m is acted on by a force F, then

F = dp

dt , (5)

where

p = mv (6)

is the momentum of the particle relative to the given inertial frame. Equation (5) is

the content of Newton’s second law of motion: it provides the means for determining

the trajectory r(t), and is known as the equation of motion. If the mass of the particle

is constant then (5) can also be written as

m

dv

dt = F , (7)

or, equivalently,

m

d2r

dt2 = F . (8)

The theory is completed by postulating a restriction on the interaction between

any two particles (Newton’s third law of motion): if F12 is the force that particle 1

exerts on particle 2, and if F21 is the force that particle 2 exerts on particle 1, the

Introduction ￾

F21 = −F12 . (9)

That is, the mutual actions between particles are always equal in magnitude and

opposite in direction. (See also Question 10.5.)

The realization that the dynamics of the physical world can be studied by solving

differential equations is one of Newton’s great achievements, and many of the problems

discussed in this book deal with this topic. His theory shows that (on some level) it is

possible to predict the future and to unravel the past.

The reader may be concerned that, from a logical point of view, two new quantities

(mass and force) are introduced in the single statement (5). However, by using both

the second and third laws, (5) and (9), one can obtain an operational definition of

relative mass (see Question 2.6). Then (5) can be regarded as defining force.

Three ways in which the equation of motion can be applied are:

☞ Use a trajectory to determine the force. For example, elliptical planetary orbits –

with the Sun at a focus – imply an attractive inverse-square force (see Question

8.13).

☞ Use a force to determine the trajectory. For example, parabolic motion in a

uniform field (see Question 7.1).

☞ Use a force and a trajectory to determine particle properties. For example,

the electric charge from rectilinear motion in a combined gravitational and

electrostatic field, and the electric charge-to-mass ratio from motion in uniform

electrostatic and magnetostatic fields (see Questions 3.11, 7.19 and 7.20).

1.2 Multi-particle systems

The above formulation is readily extended to multi-particle systems. We follow stan￾dard notation and let mi and ri denote the mass and position vector of the ith particle,

where i = 1, 2, ··· , N for a system of N particles. The velocity and acceleration of the

ith particle are denoted vi and ai, respectively. The equations of motion are

Fi = dpi

dt (i = 1, 2, ··· , N), (10)

where pi = mivi is the momentum of the ith particle relative to a given inertial frame,

and Fi is the total force on this particle.

In writing down the Fi it is useful to distinguish between interparticle forces, due

to interactions among the particles of the system, and external forces associated with

sources outside the system. The total force on particle i is the vector sum of all

interparticle and external forces. Thus, one writes

Fi =

j=i

Fji + F(e)

i (i = 1, 2, ··· , N), (11)

where Fji is the force that particle j exerts on particle i, and F(e)

i is the external

force on particle i. In (11) the sum over j runs from 1 to N but excludes j = i. The

interparticle forces are all assumed to obey the third la

￾ Solved Problems in Classical Mechanics

Fji = −Fij (i, j = 1, 2, ··· , N). (12)

From (10) and (11) we have the equations of motion of a system of particles in terms

of interparticle forces and external forces:

dpi

dt =

j=i

Fji + F(e)

i (i = 1, 2, ··· , N). (13)

If the masses mi are all constant then (13) can be written as

mi

d2ri

dt2 =

j=i

Fji + F(e)

i (i = 1, 2, ··· , N). (14)

These are the equations of motion for the classical N-particle problem. In general, they

are a set of N coupled differential equations, and they are usually intractable.

Two of the four presently known fundamental interactions are applicable in

classical mechanics, namely the gravitational and electromagnetic forces. For the

former, Newton’s law of gravitation is usually a satisfactory approximation. For

electromagnetic forces there are Coulomb’s law of electrostatics, the Lorentz force,

and multipole interactions. Often, it is impractical to deduce macroscopic forces (such

as friction and viscous drag) from the electromagnetic interactions of particles, and

instead one uses phenomenological expressions.

Another method of approximating forces is through the simple expedient of a

spatial Taylor-series expansion, which opens the way to large areas of physics. Here, the

first (constant) term represents a uniform field; the second (linear) term

encompasses a ‘Hooke’s-law’-type force associated with linear (harmonic) oscillations;

the higher-order (quadratic, cubic, . . . ) terms are non-linear (anharmonic) forces that

produce a host of non-linear effects (see Chapter 13).

Also, there are many approximate representations of forces in terms of various

potentials (Lennard-Jones, Morse, Yukawa, Pöschl–Teller, Hulthén, etc.), which are

useful in molecular, solid-state and nuclear physics. The Newtonian concepts of force

and potential have turned out to be widely applicable – even to the statics and

dynamics of such esoteric yet important systems as flux quanta (Abrikosov vortices)

in superconductors and line defects (dislocations) in crystals.

Some of the most impressive successes of classical mechanics have been in the field

of astronomy. And so it seems ironic that one of the major unanswered questions in

physics concerns observed dynamics – ranging from galactic motion to accelerating

expansion of the universe – for which the source and nature of the force are uncertain

(dark matter and dark energy, see Question 11.20).

1.3 Newton and Maxwell

The above outline of Newtonian dynamics relies on the notion of a particle. The theory

can also be formulated in terms of an extended object (a ‘body’). This is the for

Introduction ￾

used originally by Newton, and subsequently by Maxwell and others. In his fascinating

study of the Principia Mathematica, Chandrasekhar remarks that Maxwell’s “is a

rarely sensitive presentation of the basic concepts of Newtonian dynamics” and “is so

completely in the spirit of the Principia and illuminating by itself . . . .”[5]

Maxwell emphasized “ that by the velocity of a body is meant the velocity of its

centre of mass. The body may be rotating, or it may consist of parts, and be capable

of changes of configuration, so that the motions of different parts may be different,

but we can still assert the laws of motion in the following form:

Law I. – The centre of mass of the system perseveres in its state of rest, or of

uniform motion in a straight line, except in so far as it is made to change that state

by forces acting on the system from without.

Law II. – The change of momentum during any interval of time is measured by the

sum of the impulses of the external forces during that interval.”[5]

In Newtonian dynamics, the position of the centre of mass of any object is a unique

point in space whose motion is governed by the two laws stated above. The concept

of the centre of mass occurs in a straightforward manner[5] (see also Chapter 11) and

it plays an important role in the theory and its applications.

Often, the trajectory of the centre of mass

relative to an inertial frame is a simple curve, even

though other parts of the body may move in a more

complicated manner. This is nicely illustrated by the

motion of a uniform rod thrown through the air: to a

good approximation, the centre of mass describes a

simple parabolic curve such as P in the figure, while

other points in the rod may follow a more complicated

three-dimensional trajectory, like Q. If the rod is

P

Q

thrown in free space then its centre of mass will move with constant velocity (that is,

in a straight line and with constant speed) while other parts of the rod may have more

intricate trajectories. In general, the motion of a free rigid body in an inertial frame

is more complicated than that of a free particle (see Question 12.22).

1.4 Newton and Lagrange

The first edition of the Principia Mathematica was published in July 1687, when

Newton was 44 years old. Much of it was worked out and written between about August

1684 and May 1686, although he first obtained some of the results about twenty years

earlier, especially during the plague years 1665 and 1666 “ for in those days I was in

the prime of my age for invention and minded Mathematicks and Philosophy more

than at any time since.”[5]

After Newton had laid the foundations of classical mechanics, the scene for many

subsequent developments shifted to the Continent, and especially France, where

[5] S. Chandrasekhar, Newton’s Principia for the common reader, Chaps. 1 and 2. Oxford: Claren￾don Press, 1995

￾ Solved Problems in Classical Mechanics

important works were published by d’Alembert (1717–1783), Lagrange (1736–1813),

de Laplace (1749–1827), Legendre (1725–1833), Fourier (1768–1830), Poisson (1781–

1840), and others. In particular, an alternative formulation of classical particle

dynamics was presented by Lagrange in his Mécanique Analytique (1788).

To describe this theory it is helpful to consider first a single particle of constant mass

m moving in an inertial frame. We suppose that all the forces acting are conservative:

then the particle possesses potential energy V (r) in addition to its kinetic energy

K = 1

2mr˙2, and the force is related to V (r) by F = −∇V (see Chapter 5). So,

Newton’s equation of motion in Cartesian coordinates x1, x2, x3 has components

mx¨i = Fi = −∂V

∂xi (i = 1, 2, 3). (15)

Also, ∂K

∂xi = 0, ∂K

∂x˙ i = mx˙ i, and ∂V

∂x˙ i = 0. Therefore (15) can be recast in

the form

d

dt

∂L

∂x˙ i

− ∂L

∂xi

=0 (i = 1, 2, 3), (16)

where L = K − V . The quantity L(r, r˙) is known as the Lagrangian of the particle.

The Lagrange equations (16) imply that the action integral

I =

t2

t1

L dt (17)

is stationary (has an extremum – usually a minimum) for any small variation of the

coordinates xi:

δI = 0 . (18)

Equations (16) hold even if V is a function of t, as long as F = −∇V .

This account can be generalized:

☞ It applies to systems containing an arbitrary number of particles N.

☞ The coordinates used need not be Cartesian; they are customarily denoted q1, q2,

··· , qf (f = 3N) and are known as generalized coordinates. (In practice, the

choice of these coordinates is largely a matter of convenience.) The corresponding

time derivatives are the generalized velocities, and the Lagrangian is a function

of these 6N coordinates and velocities:

L = L(q1, q2, ··· , qf ; ˙q1, q˙2, ··· , q˙f ). (19)

Often, we will abbreviate this to L = L(qi, q˙i).

☞ The Lagrangian is required to satisfy the action principle (18), and this implies

the Lagrange equations

d

dt

∂L

∂q˙i

− ∂L

∂qi

=0 (i = 1, 2, ··· , 3N), (20)

where L = K − V , and K and V are the total kinetic and potential energies of

the system.[2