Kindle
edition
Loney’s
Dynamics
of
Rigid Bodies
with
Solution Manual
S. L. LONEY
M-A-T-H
VALLEY
, LONEY’S
DYNAMICS OF RIGID BODIES
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-A-T-H
VALLEY
v
,vi
M-A-T-H
VALLEY
Loney’s Dynamics of Rigid Bodies with Solution Manual
c
°Copyright received by publisher for this Kindle Edition .
The moral rights of the Publisher have been asserted.
First published by Math Valley in September 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 text book is
retyped and carefully checked by the subject experts to make an er-
ror 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.
Loney’s Dynamics of Rigid Bodies with Solution Manual is the
Rigid Dynamics part of the book ’Dynamics of a Particle and of
Rigid Bodies’ by S.L. Loney. At the end the Rigid Dynamics part
of the solution manual is added. This solution manual helps to the
reader to solve the exercises given in the text book.
September 2018 PUBLISHER
, CONTENTS
DYNAMICS OF RIGID BODIES
11 MOMENTS AND PRODUCTS OF INERTIA:
PRINCIPAL AXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
12 D’ ALEMBERT’S PRINCIPLE
THE GENERAL EQUATIONS OF MOTION . . . . . . . . . . 31
13 MOTION ABOUT A FIXED AXIS . . . . . . . . . . . . . . . . . . 43
14 MOTION IN TWO DIMENSIONS. FINITE FORCES . 83
15 MOTION IN TWO DIMENSIONS. IMPULSIVE
FORCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
16 INSTANTANEOUS CENTRE. ANGULAR
VELOCITIES. MOTION IN THREE DIMENSIONS . . . 161
17 ON THE PRINCIPLES OF THE CONSERVATION
OF MOMENTUM AND CONSERVATION OF
ENERGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
18 LAGRANGE’S EQUATIONS IN
GENERALISED COORDINATES . . . . . . . . . . . . . . . . . . . 241
ix
,x CONTENTS
19 SMALL OSCILLATIONS: INITIAL MOTIONS.
TENDENCY TO BREAK . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
20 MOTION OF A TOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
22 MISCELLANEOUS EXAMPLES II . . . . . . . . . . . . . . . . . 307
ON THE SOLUTION OF SOME OF
THE MORE COMMON FORMS OF
DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . 327
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
SOLUTION MANUAL
11 MOMENTS AND PRODUCTS OF INERTIA:
PRINCIPAL AXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
12 D’ ALEMBERT’S PRINCIPLE
THE GENERAL EQUATIONS OF MOTION . . . . . . . . . . 31
13 MOTION ABOUT A FIXED AXIS . . . . . . . . . . . . . . . . . . 43
14 MOTION IN TWO DIMENSIONS. FINITE FORCES . 83
15 MOTION IN TWO DIMENSIONS. IMPULSIVE
FORCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
, CONTENTS xi
16 INSTANTANEOUS CENTRE. ANGULAR
VELOCITIES. MOTION IN THREE DIMENSIONS . . . 161
17 ON THE PRINCIPLES OF THE CONSERVATION
OF MOMENTUM AND CONSERVATION OF
ENERGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
18 LAGRANGE’S EQUATIONS IN
GENERALISED COORDINATES . . . . . . . . . . . . . . . . . . . 241
19 SMALL OSCILLATIONS: INITIAL MOTIONS.
TENDENCY TO BREAK . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
20 MOTION OF A TOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
22 MISCELLANEOUS EXAMPLES II . . . . . . . . . . . . . . . . . 307
,
, Chapter 11
MOMENTS AND PRODUCTS OF
INERTIA: PRINCIPAL AXES
144. If r be the perpendicular distance from any given line of any
element m of the mass of a body, then the quantity ∑ mr2 is called
the moment of inertia of the body about the given line.
In other words, the moment of inertia is thus obtained; take each
element of the body, multiply it by the square of its perpendicular
distance from the given line; and add together all the quantities thus
obtained.
If this sum be equal to Mk2 , where M is the total mass of the body,
then k is called the Radius of Gyration about the given line. It has
sometimes been called the Swing-Radius.
If three mutually perpendicular axes Ox, Oy, Oz be taken, and if the
coordinates of any element m of the system referred to these axes be
x, y and z, then the quantities ∑ myz, ∑ mzx, and ∑ mxy are called the
products of inertia with respect to the axes y and z, z and x, and x
and y respectively.
p
Since the distance of the element from the axis of x is y2 + z2 ,
the moment of inertia about the axis of x
= ∑ m(y2 + z2 ).
1
edition
Loney’s
Dynamics
of
Rigid Bodies
with
Solution Manual
S. L. LONEY
M-A-T-H
VALLEY
, LONEY’S
DYNAMICS OF RIGID BODIES
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-A-T-H
VALLEY
v
,vi
M-A-T-H
VALLEY
Loney’s Dynamics of Rigid Bodies with Solution Manual
c
°Copyright received by publisher for this Kindle Edition .
The moral rights of the Publisher have been asserted.
First published by Math Valley in September 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 text book is
retyped and carefully checked by the subject experts to make an er-
ror 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.
Loney’s Dynamics of Rigid Bodies with Solution Manual is the
Rigid Dynamics part of the book ’Dynamics of a Particle and of
Rigid Bodies’ by S.L. Loney. At the end the Rigid Dynamics part
of the solution manual is added. This solution manual helps to the
reader to solve the exercises given in the text book.
September 2018 PUBLISHER
, CONTENTS
DYNAMICS OF RIGID BODIES
11 MOMENTS AND PRODUCTS OF INERTIA:
PRINCIPAL AXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
12 D’ ALEMBERT’S PRINCIPLE
THE GENERAL EQUATIONS OF MOTION . . . . . . . . . . 31
13 MOTION ABOUT A FIXED AXIS . . . . . . . . . . . . . . . . . . 43
14 MOTION IN TWO DIMENSIONS. FINITE FORCES . 83
15 MOTION IN TWO DIMENSIONS. IMPULSIVE
FORCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
16 INSTANTANEOUS CENTRE. ANGULAR
VELOCITIES. MOTION IN THREE DIMENSIONS . . . 161
17 ON THE PRINCIPLES OF THE CONSERVATION
OF MOMENTUM AND CONSERVATION OF
ENERGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
18 LAGRANGE’S EQUATIONS IN
GENERALISED COORDINATES . . . . . . . . . . . . . . . . . . . 241
ix
,x CONTENTS
19 SMALL OSCILLATIONS: INITIAL MOTIONS.
TENDENCY TO BREAK . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
20 MOTION OF A TOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
22 MISCELLANEOUS EXAMPLES II . . . . . . . . . . . . . . . . . 307
ON THE SOLUTION OF SOME OF
THE MORE COMMON FORMS OF
DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . 327
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
SOLUTION MANUAL
11 MOMENTS AND PRODUCTS OF INERTIA:
PRINCIPAL AXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
12 D’ ALEMBERT’S PRINCIPLE
THE GENERAL EQUATIONS OF MOTION . . . . . . . . . . 31
13 MOTION ABOUT A FIXED AXIS . . . . . . . . . . . . . . . . . . 43
14 MOTION IN TWO DIMENSIONS. FINITE FORCES . 83
15 MOTION IN TWO DIMENSIONS. IMPULSIVE
FORCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
, CONTENTS xi
16 INSTANTANEOUS CENTRE. ANGULAR
VELOCITIES. MOTION IN THREE DIMENSIONS . . . 161
17 ON THE PRINCIPLES OF THE CONSERVATION
OF MOMENTUM AND CONSERVATION OF
ENERGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
18 LAGRANGE’S EQUATIONS IN
GENERALISED COORDINATES . . . . . . . . . . . . . . . . . . . 241
19 SMALL OSCILLATIONS: INITIAL MOTIONS.
TENDENCY TO BREAK . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
20 MOTION OF A TOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
22 MISCELLANEOUS EXAMPLES II . . . . . . . . . . . . . . . . . 307
,
, Chapter 11
MOMENTS AND PRODUCTS OF
INERTIA: PRINCIPAL AXES
144. If r be the perpendicular distance from any given line of any
element m of the mass of a body, then the quantity ∑ mr2 is called
the moment of inertia of the body about the given line.
In other words, the moment of inertia is thus obtained; take each
element of the body, multiply it by the square of its perpendicular
distance from the given line; and add together all the quantities thus
obtained.
If this sum be equal to Mk2 , where M is the total mass of the body,
then k is called the Radius of Gyration about the given line. It has
sometimes been called the Swing-Radius.
If three mutually perpendicular axes Ox, Oy, Oz be taken, and if the
coordinates of any element m of the system referred to these axes be
x, y and z, then the quantities ∑ myz, ∑ mzx, and ∑ mxy are called the
products of inertia with respect to the axes y and z, z and x, and x
and y respectively.
p
Since the distance of the element from the axis of x is y2 + z2 ,
the moment of inertia about the axis of x
= ∑ m(y2 + z2 ).
1
