S L Loney s Dynamics of a Particle with Solution Manual An Elementary Treatise on the Dynamics of a Particle Math Valley

436 Pages·2019·7.09 MB·English

by  S L Loney Math Vally

Kindle
edition
An Elementary Treatise on the
Dynamics
of
A PARTICLE
with
Solution Manual
S. L. LONEY
M H
-A-T-
VALLEY
LONEY’S
DYNAMICS OF A PARTICLE
WITH
SOLUTION MANUAL
BY
S. L. LONEY
Professor of Mathematics
Royal Holloway College
University of London, Englifield Green, Surrey, UK
Fellow, Sidney Sussex College, Cambridge, UK
Kindle Edition
M H
-A-T-
VALLEY
v
vi
M H
-A-T-
VALLEY
[email protected]
Loney’s Dynamics of a Particle with Solution Manual
(cid:176)c Copyright received by publisher for this Kindle Edition .
The moral rights of the Publisher have been asserted.
First published by Math Valley in August 2018
All rights revised. No part of this publication may be reproduced.
Typeset in Times New Roman
PREFACE
In the following work I have tried to write an elementary class-book
on those parts of Dynamics of a Particle and Rigid Dynamics which
are usually read by Students attending a course of lectures in Applied
Mathematics for a Science or Engineering Degree, and by Junior
Students for Mathematical Honours. Within the limits with which it
professes to deal, I hope it will be found to be fairly complete.
I assume that the Student has previously read some such course as
is included in my Elementary Dynamics. I also assume that he pos-
sesses a fair working knowledge of Differential and Integral Calcu-
lus; the Differential Equations, with which he will meet, are solved
in the Text, and in an Appendix he will find a summary of the meth-
ods of solution of such equations.
In Rigid Dynamics I have chiefly confined myself to two-dimensional
motion, and I have omitted all reference to moving axes.
I have included in the book a large number of Examples, mostly
collected from University and College Examination Papers; I have
verified every question, and hope that there will not be found a large
number of serious errors.
Solutions of the Examples have now been published.
December, 1926 S.L. LONEY
vii
viii PREFACE
NOTE FOR KINDLE EDITION
The book on Dynamics of a Particle and of Rigid Bodies by S.L.
Loney is a world wide acceptable book in Mathematics and Physics.
The latest edition is also passed over a century. This book is retyped
and carefully checked by the subject experts to make an error free.
The overall structure of the book remains unchanged. But the font
size is changed for the Kindle edition suitable for various electronic
devices. Some minor modifications are made for the cross references
and the word ’shew’ is used in the original edition of the book, it has
been changed to ’show’ in this kindle edition. Also added answers
of the given exercises as well as an index at the end of this book.
August 2018 PUBLISHER
CONTENTS
DYNAMICS OF A PARTICLE
1 FUNDAMENTAL DEFINITIONS
AND PRINCIPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 MOTION IN A STRAIGHT LINE . . . . . . . . . . . . . . . . 13
3 UNIPLANAR MOTION WHERE
THE ACCELERATIONS PARALLEL
TO FIXED AXES ARE GIVEN . . . . . . . . . . . . . . . . . . . . 45
4 UNIPLANAR MOTION REFERRED TO POLAR
COORDINATES CENTRAL FORCES . . . . . . . . . . . . . . 63
5 UNIPLANAR MOTION WHEN THE
ACCELERATION IS CENTRAL AND VARYING AS
THE INVERSE SQUARE OF THE DISTANCE . . . . . . . 107
6 TANGENTIAL AND NORMAL ACCELERATIONS:
UNIPLANAR CONSTRAINED MOTION. . . . . . . . . . . . . 137
7 MOTION IN A RESISTING MEDIUM: MOTION
OF PARTICLES OF VARYING MASS . . . . . . . . . . . . . . . 171
8 OSCILLATORY MOTION AND SMALL
OSCILLATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
ix
CONTENTS
x
9 MOTION IN THREE DIMENSIONS . . . . . . . . . . . . . . . . 223
10 MISCELLANEOUS
THE HODOGRAPH: MOTION ON REVOLVING
CURVES. IMPULSIVE TENSIONS OF STRINGS. . . . . 245
11 MISCELLANEOUS EXAMPLES I. . . . . . . . . . . . . . . . . . 267
ON THE SOLUTION OF SOME OF
THE MORE COMMON FORMS OF
DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . 291
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
SOLUTION MANUAL
2 MOTION IN A STRAIGHT LINE . . . . . . . . . . . . . . . . 3
3 UNIPLANAR MOTION WHERE
THE ACCELERATIONS PARALLEL
TO FIXED AXES ARE GIVEN . . . . . . . . . . . . . . . . . . . . 16
4 UNIPLANAR MOTION REFERRED TO POLAR
COORDINATES CENTRAL FORCES . . . . . . . . . . . . . . 20
5 UNIPLANAR MOTION WHEN THE
ACCELERATION IS CENTRAL AND VARYING AS
THE INVERSE SQUARE OF THE DISTANCE . . . . . . . 38
CONTENTS
xi
6 TANGENTIAL AND NORMAL ACCELERATIONS:
UNIPLANAR CONSTRAINED MOTION. . . . . . . . . . . . . 45
7 MOTION IN A RESISTING MEDIUM: MOTION
OF PARTICLES OF VARYING MASS . . . . . . . . . . . . . . . 59
8 OSCILLATORY MOTION AND SMALL
OSCILLATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9 MOTION IN THREE DIMENSIONS . . . . . . . . . . . . . . . . 80
10 MISCELLANEOUS
THE HODOGRAPH: MOTION ON REVOLVING
CURVES. IMPULSIVE TENSIONS OF STRINGS. . . . . 90
11 MISCELLANEOUS EXAMPLES I. . . . . . . . . . . . . . . . . . 97

Chapter 1
FUNDAMENTAL DEFINITIONS
AND PRINCIPLES
1. The velocity of a point is the rate of its displacement, so that, if P
be its position at time t and Q that at time t +(cid:52)t, the limiting value
PQ
of the quantity , as (cid:52)t is made very small, is its velocity.
(cid:52)t
Since a displacement has both magnitude and direction, the ve-
locity possesses both also; the latter can therefore be represented in
magnitude and direction by a straight line, and is hence called a vec-
tor quantity.
2. A point may have two velocities in different directions at the same
instant; they may be compounded into one velocity by the following
theorem known as the Parallelogram of Velocities;
If a moving point possess simultaneously velocities which are rep-
resented in magnitude and direction by the two sides of a parallelo-
gram drawn from a point, they are equivalent to a velocity which is
represented in magnitude and direction by the diagonal of the par-
allelogram passing through the point.
Thus two component velocities AB, AC are equivalent to the resul-
tant velocity AD, where AD is the diagonal of the parallelogram of
which AB, AC are adjacent sides.
1