A. E. H. (Augustus Edward Hough) Love.

Theoretical mechanics: an introductory treatise on the principles of dynamics with applications and numerous examples online

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A. E. H. LOVE, M.A., F.R.S.,




\_All Rights reserved.]

(Catnbrilfge :



The foundations of Mechanical Science were laid by Newton,
and his achievements in this department constitute perhaps his
most enduring title to fame. Later writers have developed his
principles analytically, and have extended the region of their
application, but, in regard to the principles themselves, they have
acted the part of commentators. Nevertheless we may trace a
tendency in modern investigations, which is of the nature of a
gradual change in the point of view : there is less search for
causes, more inclination to regard the object to be attained as a
precise formulation of observed facts. On another side there is
an important respect in which modern writers have departed
from the form of the Newtonian theory. The philosophical
dictum that all motion is relative stands in pronounced contra-
diction with Newton's dynamical apparatus of absolute time,
absolute space, and absolute motion. It has been necessary to
reconsider in detail the principles, and the results deduced from
them, in order to ascertain what modification would be needed
to bring the theory of Rational Mechanics founded by Newton
into harmony with the doctrine of the relativity of motion.

The purpose of this book is didactic ; it is meant to set before
students an account of the principles of Mechanics, which shall
be as precise as possible, and which shall be in accordance with
modem ideas.

The book is divided into three parts. The first part is pre-
liminary in character, and is intended to accustom the student to
the idea of acceleration, and to the fact that a precise description
of any motion can be given by a statement of the accelerations
involved. In the first Chapter attention has been paid to the
determination of position, the essential relativity of position being
the key to much that follows. In the second Chapter is intro-
duced the idea of a Vector, and it proves useful to recognise classes


of Vectors separated by degrees of localisation. In the third
Chapter care has been taken to give precise definitions of velocity
and acceleration. The fourth Chapter treats of the simpler
motions — uniformly accelerated rectilinear motion, parabolic
motion, simple harmonic motion, elliptic motion, and central

The second part is devoted to an exposition of the Principles
of Dynamics. Chapter V. contains a statement of the principles.
The standpoint adopted is that for which the notion of Mass is
the central idea of the subject. In Chapter VI. will be found an
analytical formulation of the general theory, so far as to include the
equations of motion and the theory of the motion of the centre of
inertia. It does not include the equations of Lagrange. Chapter
VII. treats of systems of forces, the main point dealt with being
the resultant of forces applied to a rigid body. An indication
is given here of the application of theoretical mechanics to elastic
bodies. Chapter VIII. deals with the theory of work and energy,
the equation of energy being regarded primarily as an integral
of the equations of motion. A note at the end of Chapter V.
indicates very briefly the history of the fundamental principles of
Dynamics, and a note at the end of Chapter VIII. describes
critically the transformation of the subject from a theory of force
to a theory of energy. To avoid interruptions of the argument,
the discussion of Units and Dimensions is postponed to an
Appendix, and some matters which offer special difficulties, when
not treated from the old "absolute" standpoint, are reserved for
the last Chapter (XIIL).

The third part of the book is devoted to exemplifying the
applications of the general theory. Chapters IX. and X. deal
with Dynamics of a Particle ; the former referring to free motions
of particles, and the latter to constrained and resisted motions.
Chapter XI. contains the elements of Rigid Dynamics. Experience
shows that students appreciate the theory of momentum most
easily in its application to rigid bodies. The subject is treated
only in its elementary stages, the geometrical difficulties inherent
in the consideration of three dimensional motions making it
advisable to postpone their discussion. Chapter XII. contains a


number of miscellaneous methods and subsidiary theories, relating
to impulses, initial motions, small oscillations, and the motion
of chains, and it includes also some further applications of the
principles of energy and momentum. The order is very different
from that adopted in most English text-books; in the ordinary
course Mechanics is subordinated to Geometry, the order being
that of geometrical difficulty. The order here adopted is meant
to be that of the difficulty of the physical notions involved.
There can be no doubt that the mechanical meaning of equations
of motion in general is easier to grasp than that of equations
of initial motion, and the theory of collision involves special
hypotheses subsidiary to the general principles of dynamics.

The class of students for whom the book is intended may
be described as beginners in Mathematical Analysis. The reader
is supposed to have a slight acquaintance with the elements of
the Differential and Integral Calculus, and some knowledge of
Plane Coordinate Geometry. He is not assumed to have read
Solid Geometry or Differential Equations. The apparatus of
Cartesian Coordinates in three dimensions is described, and the
solutions of the differential equations that occur are explained.
It not infrequently happens that analytical methods are preferred
to geometrical ones, as likely to be more helpful to the students
whose wants are in view. Chapter XI., and isolated Articles in
other chapters, are marked with an asterisk to indicate that in a
first reading they may with advantage be omitted. These Articles
usually contain matter of such a degree of difficulty that they
are likely to be more easily understood after the rest of the book
has been read, and further progress in Pure Mathematics has
been made. A student attempting to read the book without the
guidance of a teacher is recommended to pay the greatest attention
in the first place to the unmarked Articles in Chapters IV., IX., X.,
and XII., and to work out the Examples inserted in the text and
attached to such Articles, passing lightly over the more theoretical
Chapters, and reverting to them wherever they are referred to.
He caimot expect to grasp the whole subject at once in its logical
order, and he will find it advisable to read some parts two or three
times, connecting them with different special applications.


In addition to the Examples in the text, some of Avhich are
well-known theorems and are referred to in subsequent demon-
strations, large collections have been appended to some of the
Chapters. It is hoped that these may prove useful to teachers,
and to students occupied in revising their work. These Examples
are for the most part taken from University and College Examina-
tion papers ; others, in very small number, which I have not
found in such papers, are taken from the well-known collections
of Besant, Routh, and Wolstenholme.

The works which have been most useful to me in connexion
with matters of principle are Kirchhoff's Vorlesungen ilher Mathe-
matische Physik {Mechanik), Pearson's Grammar of Science, and
Mach's Science of Mechanics. The last should be in the hands of
all students who desii'e to follow the history of dynamical ideas.
In regard to methods for the treatment of particular questions, I am
conscious of a deep obligation to the teaching of Mr R. R. Webb.

I am much indebted to the kindness of friends who have
assisted me in the production of this book. Mr J. Larmor has
read a large part of the manuscript, and his criticisms and sug-
gestions have been of the greatest value. He has also pointed
out a number of errors in the proofs, and has shown a deep and
encouraging interest in the progress of the work. Mr J. Greaves
has read all the proofs with untiring punctuality and care, and
the painstaking industry he has expended upon them leads me to
hope that the book may be found free from serious misprints. He
has also helped with many valuable suggestions. Prof. Greenhill
also has assisted me with a number of corrections and criticisms.
Mr R. Hargreaves has most kindly performed the distasteful task
of verifying a large number of the Examples. His work shows
that a sensible proportion of those submitted to him were originally
either ambiguous or incorrect, and I fear that many inaccuracies
may remain among the others. I shall be grateful for a notice of
any correction that ought to be made.

A. E. H. LOVE.


August, 1897.





1. Nature of the subject 1 — 2

2 — 6. Position, Time, and Motion. Frame of Eeferenee . . 2 — 7


7 — 13. Definition of a Vector. Composition and Eesolution of

Vectors 8 — 15

14 — 27. Localised Vectors, Moments, and Couples. Reduction of a

system of Vectors 16 — 29


28 — 33. Definitions of Displacement, Velocity, and Acceleration . 30 — 34

34 — 38. Notation and Formulaa for Acceleration. Circular Motion 34 — 38

39, 40. Relative Motion 39—41


41 — 44. Uniformly Accelerated Motion. Gravity .... 42 — 44

45, 46. Parabolic Motion 44—47

47 — 49. Simple Harmonic Motion 47—51

50—52. Central Orbits 51—54

53 — 57. Elliptic Motion. Law of Inverse Square .... 54 — 59
58 — 62. Eadial and Transversal Accelerations. Differential Equa-
tion of Central Orbits 60 — 64

63 — 66. Apses. Stability of Circular Orbits 65 — 68

Examples 68 — 84




67 — 74. Conception of a Body. Mutual Actions. Mass. Density 85 — 88

75 — 77. Force. Motion of a Body under given Forces . . . 88 — 90

78 — 80. Momentum. Kinetic Reaction. Equations of Motion . 90

81—83. Impulses 90—92

84 — 88. Rational Mechanics and its applications. Postulates of

Mechanics 92, 93

89—94. Field of Force, Weight, Validity of Concept of Mass,

Gravitation 93 — 98

95—97. Quantity of Matter. Inertia 98, 99

Historical Note 100, 101


98—104. Theory of Parallel Axes and Motion of Centre of Inertia 102—105
105 — 110. Equations of Motion. D'Alembert's Principle. Independ-
ence of Translation and Rotation ..... 106 — 108
111 — 113. Conservation of Momentum. Impulses .... 108 — 110


114 — 117. Forces acting on Rigid Systems Ill — 114

118 — 120. Constraints and Resistances. Friction .... 114, 115

121 — 124. Deformable Bodies. Stress and Strain. Surface Tractions 116—118

125—130. Strings and Springs 119—122


131 — 138. Work done by Forces. Conservative System. Potential

Energy 123—127

139—150. Examples of Calculation of Work 127—133

151. Equation of Energy ........ 133 — 135

152 — 154. Positional and Motional Forces. Conservation of Energy.

Power 135—138

155. Kinetic Energy produced by Impulses .... 138, 139

156 — 159. Virtual Work. Variational Equation of Motion . . 139 — 141

Critical Note 141—144







Dynamics of a Particle. Given Fields of Force
Problem of Two Bodies. Planetary Motions

Disturbed Elliptic Motion





182 — 187. Constrained Motion under Gravity. Atwood's Machine . 172 — 177

188—190. Simple Cii-cular Pendulum. Small Oscillation . . . 178—180

191—194. Finite Motion of Pendulum 180—182

195—202. Motion on Smooth Curve. Kotating Tube. Rough Curve 182—189

203—207. Motion on a Surface. Conical Pendulum .... 189—195

208—213. Resisting Medium 195—200

Examples 201—226


214. Two-dimensional Motion of Rigid Body. Angular Velocity 227, 228

215—217. Moments of Inertia 228—231

218 — 220. Momentum, Kinetic Energy, and Kinetic Reaction of Rigid

Body 232—235

221—224. Equations of Motion. Pendulum 235—238

225. Illustrative Problems. Inertia, Friction, Stress . . 238 — 247

Examples 247 — 255























Introductory Remarks . . . . . . . 256

Impulses. Restitution of Form in collision. Impulsive

Changes of Motion 256—269

Impulsive motion of rigid bodies 269 — 278

Initial motions of systems of particles .... 278 — 282

Initial motions of sj'stems of rigid bodies .... 282 — 286

Small oscillations of systems of particles .... 286 — 289

Small oscillations of systems of rigid bodies . . . 289 — 292

Applications of the Principles of Energy and Momentum 293 — 295
Stability of steady motions. Principles of Energy and

Momentum applied to systems of rigid bodies . . 295 — 301

Motion of a chain, restricted freedom .... 301 — 304
General motion of a chain in two dimensions. Initial

Motion and Impulsive Motion 304 — 310

Examples 310—350




275 — 284. Rotation of the Earth, Weight, Acceleration due to Gravity,

Pendulum 351—360

285—289. Relativity of Force. Universal Gravitation . . . 360—366
290, 291. Measurement of time. Effect of change of time-measuring

process 366 — 370

292—299. Measurement, Units, and Dimensions .... 371—375

Index 377—379


p. 3, line 25 from the top, insert positive before number.

p. 68, Example 2, the path is a rectangular hyperbola.

p. 69, ExamjDle 14 /or cos~i { read [cos^^

p. 70, Example 17, the path is a conic.

p. 70, Example 18, for sine read hyiDerbolic sine.

p. 71, Example 24, the third component should be ih - {mcx + vicy)jr-.

p. 72, Example 36, the range is 2i?, not R.

p. 75, Example 60, for 2hv^a read 2hv~g.

p. 76, Example 66, for \ir + e read hir - 6.

p. 79, Example 94, result should be— ( 1 + - j */(■'■"") J^ars.

p. 80, Example 105, result should be e^h-XPI(lSM'^).

p. 82, Exami^le 121, the acceleration contains an additional term varying in-
versely as the cube of the distance,
p. 84, Example 138 for 4k'- (2a-»- ...) read 4k' (2;-a- ...).
p. 164, Example 17, I'esult should read

h^-( sin d^j^-u COB dj - i/x j(sin3^- sin ^) "-2cos0[ =C.

p. 164, Example 18, result sliould read

dhi („ ^ , , du] I .,,„,,,.,

13. 165, line 3 from the bottom, for uja, read ^au'-^.
p. 224, Example 155,

for ^= r P'p dp read ^l=j"p~'^dp.

p. 263, line 9 from the top,

for {{U-uy^-(U'-ur-}
read {{U- V') - {u-u')'}.
p. 289, line 11 from the bottom, for kinectic read kinetic.



1. Mechanics is a Natural Science; its data are facts of
experience, its principles are generalisations from experience.
The possibility of Natural Science depends on a principle which
is itself derived from multitudes of particular expei-iences — the
" Principle of the Uniformity of Nature." This principle may
be stated as follows — Natural events take place in invariable
sequences. The object of Natural Science is the description of
the facts of nature in terms of the rules of invariable sequence
which natural events are observed to obey. These rules of sequence,
discovered by observation, suggest to our minds certain general
notions in terms of which it is possible to state the rules in
abstract forms. Such abstract formulas for the rules of sequence
which natural events obey we call the " Laws of Nature." When
any rule has been established by observation, and the corresponding
Law formulated, it becomes possible to predict a certain kind of
future events.

The Science of Mechanics is occupied with a particular kind
of natural events, viz. with the motions of material bodies. Its
object is the description of these motions in terms of the rules
of invariable sequence which they obey. For this purpose
it is necessary to introduce and define a number of abstract
notions suggested by observations of the motions of actual
bodies. It is then possible to formulate laws according to which
such motions take place, and these laws are such that the
L. 1


future motions and positions of bodies can be deduced from them,
and predictions so made are verified in experience. But in the
process of formulation the Science assumes a highly abstract
character. The fundamental notions involved have, for the most
part, names in common use, such as " force " for example, but the
common use of such names is never precise, and for scientific
purposes the meaning to be attached to them must be made
definite. This is done, as in Geometry, by means of definitions
and postulates. Except in the statement of the postulates nothing
ought to be taken from experience, all the results ought to be
logically deduced. There is thus an abstract logical theory of
Mechanics, of the same nature as Geometry, in which all that is
assumed is suggested by experience, all that is found is proved by
reasoning. The test of the validity of a theory of this kind is
its consistency with itself, the test of its value is its ability to
furnish rules under which natural events actually fall. In what
follows we shall be mainly occupied with the exposition of the
theory, we shall not detail the observations and experiments by
which the fundamental notions were suggested, nor shall we do
more than indicate in particular cases the kinds of natural events
to which applications of our theory can be made.

2. Motion of a point. We have said that our object is
the description of the motions of bodies. The necessity for a
simplification arises from the fact that, in general, all parts of a
body have not the same motion, and the simplification we make
is to consider the motion of so small a portion of a body that
the differences between the motions of its parts are unimportant.
How small the portion must be in order that this may be the
case we cannot say beforehand, but we avoid the difficulty thus
arising by regarding it as a geometrical point. We think then in
the first place of the motion of a point.

Motion may be defined as change of position taking place in

In regard to this definition it is necessary to make clear two
things: one is the measurement of time, and the other is the
meaning of the phrase " change of position."

3. Measurement of time. Time may be measured by
any process which goes on continually. The amount of the

1-4] TIME. 3

process that is effected in any interval of time is supposed to
be measurable, and the measure of this amount can be taken to
measure the interval, so that equal intervals of time are those in
which equal amounts of the process selected as time-measurer
take place, and different intervals are in the ratio of the measures
of the amounts of the process that take place in them. In any
interval of time many processes may be going on. Of these one
is selected as a time-measurer; we shall call it the standard
process. " Uniform processes " are such that equal amounts of
them are effected in equal intervals of time, that is, in intervals
in which equal amounts of the standard process are effected.
Processes which are not uniform are said to be " variable." It is
clear that processes which are uniform when measured by one
standard may be variable when measured by another standard.
The choice of a standard being in our power, it is clearly desirable
that it should be so made that a number of processes uncontroll-
able by us should be uniform or approximately uniform ; it is also
clearly desirable that it should have some relation to our daily
life. The process actually adopted for measuring time is the
average rotation of the Earth relative to the Sun*, and the unit
in terms of which this process is measured is called the " mean
solar second." In the course of this book we shall assume that
time is measured in this way, and we shall denote the measure of
the time which elapses between two particular instants by the
letter t, then Hs a real number (in the most general sense of the
word " number ") and the interval it denotes is t seconds.

4. Determination of Position. Position of a point relative
to a set of points f is not definite until the set includes four points
which do not all lie in one plane. Suppose 0, A, B, G to be four
such points ; one of them, 0, is chosen and called the origin, and
the three planes OBC, OCA, OAB are the faces of a trihedral
angle having its vertex at 0. The position of a point P with
reference to this trihedral angle is determined as follows : — we
draw PN parallel to OG to meet the plane AOB in N, and we
draw NM parallel to OB to meet OA in M; then the lengths
OM, MN, NP determine the position of P. Any particular length,
e.g. one centimetre, being taken as the unit of length, each of these

* See Chapter XIII.

t The phrase " position of a point " means its position relative to other points.



[chap. 1.

lengths is represented by a number (in the general sense) viz. by
the number of centimetres contained in it. It is clear that OP

Fig. 1.

is a diagonal of a parallelepiped and that OM, MN, NP are three
edges no two of which are parallel. The position of a point is
therefore determined by means of a parallelepiped whose edges
are parallel to the lines of reference, and one of whose diagonals
is the line joining the origin to the point.

It is generally preferable to take the set of lines of reference
to be three lines mutually at right angles, then the faces of the
trihedral angle are also at right angles; sets of lines so chosen
are called systems of rectangular axes, and the planes that contain
two of them are coordinate planes*. It is clear from the figure

Fig. 2.
We shall, in the course of this book, make use of rectangular coordinates




that a set of rectangular coordinate planes divide the space about
a point into eight compartments, the particular trihedral angle
OABC being one compartment. The lengths OM, MN, NP of
Fig. 1, taken with certain signs, are called the coordinates of the
point P, and are denoted by the letters x, y, z. The rule of signs
is that X is equal to the number of units of length in the length
OM when P and A are on the same side of the plane BOC, and
is equal to this number with a minus sign when P and A are on
opposite sides of the plane BOC, and similarly for y and z. We
can express the conventions as regards sign by means of the
following table, in which A', B', C denote points va. AO, BO, CO
produced : —

sign of X

sign of y

sign of z

iu trihedral angle OABC




„ „ „ OA'BC




„ „ „ OAB'C




„ „ „ OA'B'C




„ „ „ OABC




„ „ „ OA'BC




„ „ „ OAB'C




„ „ „ OA'B'C




It is clear that the coordinates x, y, z determine the position
of the point P with reference to the lines OA, OB, OC.

A set of lines of reference such as OA, OB, OC with respect