Theoretical Mechanics for Sixth Forms. In Two Volumes

408 Pages·1971·10.544 MB·English

by  C. Plumpton and W. A. Tomkys (Auth.)

Other Titles of Interest
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(two volumes)
THEORETICAL MECHANICS
FOR SIXTH FORMS
IN TWO VOLUMES
VOLUME 1
C. PLUMPTON
Queen Mary College, London
W. A. TOMKYS
Belle Vue Boys* Grammar School,
Bradford
SECOND (SI) EDITION
PERGAMON PRESS
OXFORD · NEW YORK · TORONTO
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U.K. Pergamon Press Ltd., Headington Hill Hall,
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CANADA Pergamon of Canada Ltd., 75 The East Mall,
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FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg/Taunus,
OF GERMANY Pferdstrasse 1, Federal Republic of Germany
Copyright © 1971 C. Plumpton and W. A. Tomkys
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, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publishers.
First edition 1964
Reprinted (with corrections) 1968
Second (SI) edition 1971
Reprinted 1974, 1977
Library of Congress Catalog Card No. 77-131995
Printed in Great Britain by William Clowes & Sons Limited, London, Beccles and Colchester
ISBN 0 08 016268 1 (net)
ISBN 0 08 016269 X (non net)
PREFACE
THIS volume is a rewritten version of the authors' earlier volume of the
same title together with the first chapter, Simple Harmonic Motion, of
Volume 2. This chapter has been added to Volume 1, in response to many
suggestions, so as to make the volume a more complete course for pupils
taking mathematics as a single subject.
We have revised the book so as to present the whole of the subject
matter (text, examples and exercises) in SI units. Our aim has been to
encourage pupils to think metrically and nowhere in the book are F.P.S.
units even mentioned. For the most part we have used only the base
units metre, kilogram and second together with the units derived from
these. Multiple and submultiple units, except for the kilometre and
kilowatt, are used only rarely. We have tried, as far as possible, to use
symbols to denote physical quantities rather than numbers. Thus we refer
to a "mass M", "a length /" and "a time i" and not to a "mass M kilo­
grams", "a length /metres" and a "tim.e / seconds". Included in the book
there is a short discussion of "non-dimensional" equations when this prac­
tice is not rigorously observed. Letter symbols, signs and abbreviations
conform to British Standard 1991, Part 1, 1967.
The number of exercises in the book has been very slightly reduced
from the earlier volume.
The volume aims at providing a first-year course in theoretical mecha­
nics for sixth-form pupils taking mathematics as a double subject and a
two-year course for pupils taking mathematics as a single subject. No
previous knowledge of theoretical mechanics is assumed.
Throughout the book, calculus has been used whenever it was appro­
priate to do so. Elementary ideas of vectors are introduced as early as
possible in the course and the pupil is encouraged to use vector methods
and vector notation. The development of the book is integrated as closely
X PREFACE
as possible with the development of the authors' Sixth Form Pure Mathe­
matics, the worked examples and exercises are graded and increase in
difficulty as the book progresses, and the miscellaneous exercises at the
end of each chapter are chosen to give continual reminders of the work of
earlier chapters.
After examining the nature of theoretical mechanics we first discuss the
statics of a particle in illustration of the techniques of handling vector
quantities, techniques which are developed in successive chapters concern­
ing the kinematics of a particle moving along a straight line, projectiles
and relative velocity. We then discuss the principle of moments, parallel
forces and centres of gravity. The next group of chapters is concerned
with the application of Newton's second law to the dynamics of a particle
and the ideas of work and energy, impulse and momentum, and power.
The chapter on friction, which interrupts this sequence, was delayed in
order to allow a more rapid development of fundamental principles;
when friction is introduced the opportunity is taken to coordinate the
work of the preceding chapters by means of worked examples and exer­
cises involving friction. This volume concludes with chapters concern­
ing motion in a circle and simple harmonic motion. Some sections of the
book, and some exercises, are marked with an asterisk. It is recommended
that these should be omitted in the first reading.
We wish to thank the authorities of the University of London, the Cam­
bridge Syndicate, the Oxford and Cambridge Joint Board and the Northern
Universities Joint Board for permission to include questions (marked L.,
C, O.C., and N. respectively) from papers set by them. We also thank
Mr. J. A. Croft, Mr. G. Hawkes and Mr. J. S. Barnshaw who read the
proofs and made valuable suggestions. Finally, we are deeply indebted to
Prof. E. J. Le Fevre who placed his vast knowledge and experience of units
at our disposal.
C. PLUMPTON
W. A. TOMKYS
CHAPTER I
INTRODUCTION. THE FUNDAMENTAL
CONCEPTS OF THEORETICAL MECHANICS
1.1. The Nature of Theoretical Mechanics
The work of Newton in the seventeenth century forms the basis of the
hypotheses upon which we develop this work on elementary theoretical
mechanics. The results of Newton's theories in so far as they concern
quantities which are not very small, and in so far as they concern distances
and speeds on a terrestrial rather than an astronomical scale, have been
successfully tested by experience. In such a case experience and experi­
ment, while they may disprove a hypothesis, cannot finally confirm its
truth. If experimental results consistently agree with a hypothesis, how­
ever, it is reasonable to accept the truth of the hypothesis within the physical
range implied by the experiments. Modern research, which involves very
small quantities such as those relating to the interior of an atom and
which makes investigations involving astronomical distances and speeds
comparable with the speed of light, cannot make assumptions of abso­
lute time and space which are necessary to Newtonian mechanics.
Nevertheless, results which are derived from the Newtonian hypotheses
discussed here are essential to the development of many technologies
and a proper understanding of Newtonian mechanics is necessary before
the wider problems involving the relative nature of time and space can
be considered.
Our object in this subject of theoretical mechanics is to create a mathe­
matical model of aspects of the physical world (limited as described above).
In the first instance, we simplify the problems involved in this model by
reducing the number of variable factors to as few as possible. In conse­
quence in the early stages we find solutions to problems which refer to the
1
2 THEORETICAL MECHANICS
ideal mathematical system postulated, but these problems have no precise
counterparts in any known physical situation. The application of the
results of theoretical mechanics to real physical problems is a dependent
development.
We begin by stating the hypotheses from which we develop the subject.
Time
The unit of time is the second (1 s). In the Syst£me International
d'Unites (SI) "the second" is that defined by the General Conference of
Weights and Measures. It is defined by reference to the oscillations of
the caesium 133 atom and is in fact the duration of 9 192 631 770
periods of the radiation corresponding to the transition between the
two hyperfine levels of the ground state ofthat atom.
Position
In order to describe the position of a point, it is necessary to have a
frame of reference. Thus the position of a point on the Earth's surface can
be described with reference to the equator and the Greenwich meridian by
means of its latitude and longitude, but this description would be inade­
quate to describe the position of the point in relation, say, to the Sun's
centre. Position, and therefore motion (change of position) are relative
terms depending on the chosen frame of reference. The unit of length, or
difference in position, is arbitrary and refers to a fixed physical standard.
The SI base unit of length is the metre (1 m) as defined by the Gener­
al Conference of Weights and Measures. [1 metre = 102 centimetres
(102 cm).] Any other unit of length is subsidiary and is to be defined in
terms of the metre.
Velocity is defined as rate of change with respect to time of displacement
from a fixed position.
Acceleration is defined as rate of change with respect to time of velocity.
Multiple and Sub-multiple Units
To a limited extent, prefixes to the names of fundamental units are used
to denote multiples or sub-multiples of those units. These prefixes are:
THE FUNDAMENTAL CONCEPTS OF THEORETICAL MECHANICS 3
G M k h da
GIGA MEGA KILO HECTO DECA
109 10« 103 10« 101
d c m μ n P
DECI CENTI MILLI MICRO NANO PICO
10-1 10-« lO"3 io-e 10-9 10"1*
1.2. Newton's Axioms
The hypotheses which Newton formulated and which are the founda­
tions of the subject of theoretical mechanics are known as Newton's
Axioms or Laws of Motion and they may be stated as follows.
1. A body remains in a state of rest or of uniform motion in a straight line,
unless it is acted upon by an external force.
2. The rate of change of the momentum of a body is proportional to the
force acting and takes place in the direction of that force.
3. To every action there is an equal and opposite reaction.
Newton's first law provides us with the concept of force as that which
tends to change the state of motion of a body.
Mass
Momentum, the quantity referred to in the second law quoted above, is
the product of a quantitative property of the body which is called its mass
and its velocity. It is this property which determines the response of a
body to the action of a force, and which is sometimes called the inertia of
that body. The mass of a body is here invariable since a given body under
the action of a given force acquires the same acceleration wherever and
whenever the action takes place.
A particle of matter is defined as a body whose geometrical dimensions
are sufficiently small to be neglected in comparison with the finite distance
with which we shall be concerned.
4 THEORETICAL MECHANICS
1.3. The Unit of Mass and the Unit of Force
We now assign arbitrarily to a particular mass the designation unit mass.
The SI base unit of mass is called one kilogram (1 kg). [1 kilogram
= 10 grams (10 g).] We assume that the quantitative property mass is
3 3
additive and therefore that a single body, which is composed of n bodies
each of mass equal to the arbitrarily chosen unit of mass, has a mass of n
units.
Units of Force
Newton's second law may be stated mathematically as
d^wO
at '
where P is the force acting on a body of mass m and of velocity v at
time t. (Note that the change in v is in the direction of P.)
dv
Then, P oc m -^-
ut
or P oc mf
where/is the acceleration of the body at time /. This statement depends
on the postulate which we have made above that the mass of a body is in­
variable. In problems in which the "body" itself changes and in which,
therefore, there is a variation of mass with time, this variation must be
taken into consideration. Such problems include the case of rocket flight.
Up to this point a force has been qualitatively but not quantitatively
defined. We now define the force P as the product of the physical quantity
called "mass" with the physical quantity called "acceleration", i.e. by the
equation
P = mf (1.1)
where P acts in a straight line and/is in the direction of P. This is the fun­
damental equation of kinetics, that part of theoretical mechanics which
deals with motion and the forces which produce or modify it.
THE FUNDAMENTAL CONCEPTS OF THEORETICAL MECHANICS 5
With this fundamental equation:
1 newton (1 N) is the force which gives to a body of mass 1 kg an acceler­
ation of 1 metre per second, i.e. 1 m/s .
2 2
Notation. The units m/s, (metres per second), m/s are sometimes
2
written m s" , m s~ and there are corresponding alternative unit sym­
1 2
bols.
Dimensions. The dimensions of a physical quantity can be related to the
fundamental quantities mass (M), length (L) and time (T). For example,
the dimensions of force, as defined above, are MLT"". An equation in­
2
volving physical quantities is a dimensional equation.
In subsequent work, when we refer to a body of mass m> a distance s,
a time t, a force P we imply by the symbol a quantity and not a number.
When we are considering numerical problems we shall state the units of
the quantities to which we refer, as, for example: 2 kg, 5 m, 6 s, 10 N.
1.4. Gravitational Units of Force
Newtons Law of Universal Gravitation states that every particle of mat­
ter attracts every other particle of matter with a force which varies directly
as the product of the masses of the particles and inversely as the square of
the distance between them.
The force of attraction P between two particles, of masses m± and m^
which are at a distance d apart is given by
where G is a constant quantity.
If now we assume
(i) that the force exerted by the earth on a body outside it is equal to
the force which would be exerted by a particle of mass equal to that
of the earth situated at its centre, and,
(ii) that for a body whose size is small compared with the earth and
which is near the earth's surface, a sufficient degree of accuracy