Theoretical Mechanics for Sixth Forms. in Two Volumes

424 Pages·1972·4.996 MB·English

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

THEORETICAL MECHANICS
FOR SIXTH FORMS
IN TWO VOLUMES
VOLUME 2
C. PLUMPTON
Queen Mary College, London
W. A. TOMKYS
Belle hue Boys' Grammar School
Bradford
SECOND (SI) EDITION
PERGAMON PRESS
OXFORD • NEW YORK • TORONTO
SYDNEY • BRAUNSCHWEIG
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Copyright © 1972 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
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Pergamon Press Ltd.
First edition 1964
Second (SI) edition 1972
Library of Congress Catalog Card No. 77-131995
Printed in Hungary
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PREFACE
Tns volume completes the revised edition of the course in Theoretical
Mechanics for sixth-form pupils taking mathematics as a double subject
using SI units throughout. We believe that some of the chapters,
notably Chapters XXI and XXII on Virtual Work and Stability, Chap-
ter XVI on the free motion of a rigid lamina in a plane, and Chapter XXV
on the motion of bodies with variable mass are useful material for pupils
in the third year of a sixth-form course and as preparation for their first
year at University or at one of the Polytechnics. The chapters of the
book are arranged in a logical order of development, but exercises,
sections of chapters, and whole chapters, consideration of which might
well be postponed until the third year of the course, are marked with an
asterisk.
Many of the topics discussed in Volume I are reintroduced in this
volume where they are considered in more detail and with more rigour.
In particular the ideas of vectors are developed further in Chapters XVIII
and XIX and, because of the importance of vectors in modern mathemat-
ics, we have included Chapter XXVI on vector algebra and its applica-
tions. We hope that this will serve as a useful introduction to more
advanced work.
The first two chapters of this book concern the motion of a rigid
lamina in a plane. We then discuss the dynamics of a particle under the
action of variable forces and follow with a chapter concerning the uni-
planar motion of a particle with two degrees of freedom. The next group
of chapters develops the ideas of Statics which were introduced in Vol-
ume I. After a new consideration of the theoretical aspects of force ana-
lysis, the sets of conditions of equilibrium for a body under the action
of coplanar forces are reclassified and applied to more difficult problems
of equilibrium. We then consider stability and introduce the special
ix
x PREFACE
technique of Virtual Work. Then graphical statics applied particularly
to light frameworks is considered and finally in this group we discuss
continuously distributed forces in the particular cases of loaded beams
and the catenary.
In this volume, as in Volume I, references are made wherever necessary
to the authors' Sixth Form Pure Mathematics and particularly to Volume
II of that book. In A Course of Mathematics, Volume III by Chirgwin and
Plumpton, the topics of the present volume are developed further and
third year sixth-form pupils in particular would be well advised to read
that book.
We wish to thank the Authorities of the University of London, the
Cambridge Syndicate, the Oxford and Cambridge Joint Board, the
Northern Universities Joint Board, the Oxford Colleges and the Cam-
bridge Colleges for permission to include questions (marked L., C., O.C.,
N., O.S. and C.S., respectively) from papers set by them. We also thank
Mr. J. A. Croft who read the proofs and made valuable suggestions.
C. PLUMPTON
W. A. Toikns
CHAPTER XV
THE MOTION OF A RIGID BODY ABOUT
A FIXED AXIS
15.1. The Definition of a Rigid Body
We define a rigid body here as an aggregate of particles in which the
distance between any two particles is invariable. A rigid body, thus
defined, is a mathematical ideal which is unattainable in practice; most
bodies are distorted under the action of external forces, but the mathe-
matical models which are produced on the basis of the assumption of
rigidity give results of value in applications to practical problems.
We assume further that the particles of a rigid body thus defined,
are held together by internal forces between them and that the action
and reaction between any pair of particles are equal in magnitude and
opposite in direction. We also assume that, because the distance between
any two particles of a rigid body remains constant and because the action
and reaction between them are equal and opposite, the total work done
by these internal forces in a displacement of the body is zero and that
therefore the increase of the kinetic energy of a rigid body in any period
is equal to the work done by the external forces in that period.
15.2. The Kinetic Energy of a Rigid Body Rotating about a Fixed Axis
Figure 15.1 represents a section perpendicular to the axis of a rigid
body rotating about a fixed axis at right angles to the section and meet-
ing the section at O. The rigid body is considered as consisting of an
aggregate of particles, each of which is in one such section, and in which a
particle of mass m is distant r from the axis. When the body is rotating
about the axis with angular velocity w, the linear velocity of the particle
of mass m is r w and the kinetic energy of this particle is mrrPw2.
l
i
405
406 THEORETICAL MECHANICS
FIG. 15.1
The kinetic energy of the whole body is therefore
SZm rPw2 = Zw2Sm rR. (15.1)
P R
The summation sign indicates summation over all the particles of the
body.
The quantity Sm rP is defined as the moment of inertia of the rigid
P
body about the given axis. The quantity k, where Sm rP = 1k2 and M is
P
the total mass of the body, is defined as the radius of gyration of the body
about the given axis.
15.3. Calculation of Moments of Inertia in Particular Cases
1. The moment of inertia of a thin uniform rod of length 21 and mass
M about an axis through its centre perpendicular to its length. Let the line
density of the rod be p (Fig. 15.2). Then the M. of!. of a small increment
bx of the rod, distance x from the axis, is approximately pbx. x2. Hence
FIG. 15.2
THE MOTION OF A RIGID BODY ABOUT A FIXED AXIS 407
the total M. of I. of the rod about the axis is
1= Rx2 dx = 233 _ 112 (15.2)
_Jl
2. The moment of inertia of a thin uniform rod of length 21 and mass
I about an axis through its centre at an angle Q with its length. Let the
line density of the rod be p and let its mid-point be O (Fig. 15.3). Then
FiG. 15.3
2a
M
7 C
2b
FIG. 15.4
the M. of I. about the given axis of a small increment dx of the rod,
distance x from 0, is approximately rdc. x2 sin2 Q. Hence the total M. of
I. of the rod about the axis is
+i
I = f rc2 sin2 Q dx = 2r13 Sin2 M12 sin2 Q (15.3)
3 3
-i
408 THEORETICAL MECHANICS
3. The moment of inertia of a lamina in the shape of a rectangle of
length 2a, breadth 2b, and mass M about the axis in its plane, through
its centre and parallel to the side of length 2b. Figure 15.4 shows that the
rectangle can be divided into thin strips each parallel to a side of length
2a, a typical strip being distance x from that side and of width dc. Then
if the surface density of the lamina is r, the M. of I. of the lamina about
the axis is
2b
I — 2ra. 32 dx— 4r3 3b Ma2 (15.4)
0
4. The moment of inertia of a thin uniform ring of mass M and radius r
about the axis through its centre perpendicular to its plane. Since every
particle of the ring is at a distance r from the axis,
I Mr2. (15.5)
5. The moment of inertia of a uniform disc of radius r and mass M
about the axis through its centre and perpendicular to its plane. Let the
surface density of the disc be r. Consider a thin ring of the disc, con-
centric with the disc, of radius x and width bx (Fig. 15.5). Then the
FIG. 15.5
THE MOTION OF A RIGID BODY ABOUT A FIXED AXIS 409
M. of I. of this thin ring about the axis is approximately 2prxdx.x2,
r
4
2
.'. I= 2prc.c2dx = n2 = Z . (15.6)
r
o
6. The moment of inertia of a uniform sphere of radius R and mass M
about a diameter. Let the density of the sphere be p (Fig. 15.6). Con-
sider a thin disc (of the sphere) whose plane faces are perpendicular to
FIG. 15.6
the axis at a distance c from the centre of the sphere. Let the thickness
of the disc be bx. Then the mass of the disc is approximately (R2 — c2)rdx
and hence the M. of I. of the disc about the axis is approximately
p(R2—x 2)rdc. ~(R2—x 2). Hence the M. of I. of the sphere about the
axis is
+R
I = =2 pr(R2— c2)2dx = 2 ttr GLR 4x— 23 2~ + 55, +— RR
-R
_ 8prR5 _ 21R2
(15.7)
15 5 ~
410 THEORETICAL MECHANICS
Two Theorems
(a) The theorem of perpendicular axes. If Ox and Oy are any two
rectangular axes in the plane of a lamina and Oz is an axis at right angles
to the plane, then the moment of inertia of the lamina about Oz is
equal to the sum of the moments of inertia of the lamina about Ox and
Oy.
t
Fio. 15.7
For, with reference to Fig. 15.7, if m,. is the mass of a particle P whose
coordinates referred to Ox, Oy are (x, y), the M. of I. of the particle
r r
about Ox is myr and the M. of I. of the particle about Oy is m x2. But
r
m y+m x; = m(y;+x;) = m,OP2 and this is equal to the M. of I. of
r r r
P about Oz. Hence, for the whole lamina, the sum of the moments of
inertia about Ox and Oy is equal to the moment of inertia about Oz.
(b) The theorem of parallel axes. The moment of inertia of a body about
any axis is equal to the moment of inertia about a parallel axis through
the centre of mass together with the mass of the whole body multiplied
by the square of the distance between the axes.
Figure 15.8 shows a section of the body through the centre of mass G.
This section is at right angles to the axis and meets the axis at T where