THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
THEORETICAL MECHANICS
AN INTRODUCTORY TREATISE
ON THE
PEINCIPLES OF DYNAMICS
CAMBRIDGE UNIVERSITY PRESS
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THEORETICAL MECHANICS
AN INTRODUCTORY TREATISE
ON THE
PKINCIPLES OF DYNAMICS
WITH APPLICATIONS AND NUMEROUS EXAMPLES
BY
A. E. H. LOVE, M.A., D.Sc., F.R.S.
SEDLEIAN PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF OXFORD.
HONORARY FELLOW OF QUEEN'S COLLEGE, OXFORD. FORMERLY FELLOW
AND LECTURER OF ST JOHN'S COLLEGE, CAMBRIDGE
THIRD EDITION
CAMBRIDGE
AT THE UNIVERSITY PRESS
1921
First Edition, 1897
Second Edition, 1906
Third Edition, 1921
Engineering
Library
PEEFACE
book, of which this is the third edition, is intended as a
- text-book of Dynamics, for the use of students who have some
acquaintance with the methods of the Differential and Integral
Calculus. Its scope includes the subjects usually described as
Elementary Dynamics, Dynamics of a Particle, and Dynamics of a
Rigid Body moving in two dimensions. It also includes an attempt
to trace the logical development of the theory.
Within the chosen range of subject-matter there are many
topics, such as the oscillation of a pendulum through wide angles,
which it would be inappropriate to omit although they are difficult
for a beginner. Articles dealing with such topics are marked with
an asterisk. A student reading the subject for the first time, and
without the guidance of a teacher, is advised to confine his atten-
tion to the unmarked Articles and the unmarked collections of
Examples inserted in the text. Many of these Examples are well-
known theorems, and some of them are referred to in subsequent
demonstrations. Collections of miscellaneous Examples are ap-
pended to most of the Chapters. It is hoped that these will be
useful to teachers and to students engaged in revising their work.
A few of them, which I have not found in Examination papers, are
taken from the well-known collections of Besant, Routh, and
Wolstenholme.
The works which were most useful to me in regard to matters
of theory, when I was writing the first edition of this book, were
KirchhofFs Vorlesungen uber mathematische Physik (Mechanik),
Pearson's Grammar of Science, and Mach's Science of Mechanics.
The last ought to be in the hands of all students who wish 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 RJL. Webb.
ENGINEERING LIBRARY
VI PREFACE
For the second edition the book was largely re-written and
entirely re-arranged. In the present edition few changes have been
made in the text. New Articles, which have been added, are
marked with the letter A, thus " 102 A." The number of miscel-
laneous Examples has been reduced considerably, but it is hoped
that the most interesting have been retained, and that these are
still sufficiently numerous.
A. E. H. LOVE.
OXFORD.
April, 1921.
CONTENTS
ART. PAGE
1 Nature of the science 1
2 Motion of a particle 2
3 Measurement of time 2
4 Determination of position ....... 3
5 Frame of reference ......... 5
6 Choice of the time-measuring process and of the frame of re-
ference 6
CHAPTER I
DISPLACEMENT, VELOCITY, ACCELERATION
7 Introductory 7
8 Displacement .7
9 Definition of a vector 8
10 Examples of equivalent vectors 9
11 Components and resultant 10
12 Composition of any number of vectors 13
13 Vectors equivalent to zero 14
14 Components of displacement . . . . . .14
15 Velocity in a straight line . . . . . . .15
16 Velocity in general 16
17 Localized vectors 17
18 Formal definition of velocity . . . . . . .19
19 Measurement of velocity 19
20 Moment of localized vector 19
21 Lemma 20
22 Theorem of moments 20
23 Acceleration 21
24 Measurement of acceleration 23
25 Notation for velocities and accelerations .... 23
26 Angular velocity and acceleration ...... 24
27 Relative coordinates and relative motions .... 24
28 Geometry of relative motion 25
CONTENTS
THE MOTION OF A FEEE PARTICLE IN A FIELD OF FORCE
ART. PAGE
29 Gravity 27
30 Field of force . 27
31 Rectilinear motion in a uniform field 27
32 Examples 28
33 Parabolic motion under gravity 28
34 Examples 30
35 Motion in a curved path 32
36 Acceleration of a point describing a plane curve . . .32
37 Examples 33
38 Simple harmonic motion 34
39 Composition of simple harmonic motions . . . .35
40 Examples 36
41 Kepler's laws of planetary motion 37
42 Equable description of areas 37
43 Radial and transverse components of velocity and acceleration 38
44 Examples . . . 39
45 Acceleration in central orbit ....... 40
46 Examples 40
47 Elliptic motion about a focus 41
48 Examples 42
49 Inverse problem of central orbits 44
50 Determination of central orbits in a given field ... 44
51 Orbits described with a central acceleration varying inversely
as the square of the distance 45
52 Additional examples of the determination of central orbits in
given fields 46
53 Conic described about a focus. Focal chord of curvature . 47
54 Law of inverse square. Rectilinear motion .... 48
55 Examples . . . . . . . . . .49
56 Field of the Earth's gravitation 50
57 Examples 50
Miscellaneous Examples 51
CHAPTER III
FORCES ACTING ON A PARTICLE
58 The force of gravity 57
59 Measure of force 58
60 Units of mass and force . 59
CONTENTS IX
ART. PAGE
61 Vectorial character of force ....... 60
62 Examples 61
63 Definitions of momentum and kinetic reaction ... 62
64 Equations of motion ........ 62
EQUATIONS OF MOTION IN SIMPLE CASES
65 Motion on a smooth guiding curve under gravity ... 63
66 Examples 65
67 Kinetic energy and work ....... 65
68 Units of energy and work 66
69 Power 66
70 Friction 67
71 Motion on a rough plane 67
72 Examples 68
73 Atwood's machine . 68
74 Examples 70
75 Simple circular pendulum executing small oscillations . . 70
76 Examples 71
77 One-sided constraint 72
78 Examples 72
79 Conical pendulum ......... 73
80 Examples . . . . 73
THEORY OF MOMENTUM
81 Impulse 74
82 Sudden changes of motion 74
83 Constancy of momentum ....... 75
84 Moment of force, momentum and kinetic reaction about an
axis ........... 75
85 Constancy of moment of momentum 77
85A Note (on Moments about a moving axis) .... 78
WORK AND ENERGY
86 Work done by a variable force 79
87 Calculation of work 79
88 Work function 80
89 Potential function ......... 81
90 Forces derived from a potential ...... 82
91 Energy equation 83
92 Potential energy of a particle in a field of force ... 84
93 Forces which do no work 84
94 Conservative and non-conservative fields .... 85
Miscellaneous Examples 86
CONTENTS
CHAPTER IV
ART. PAGE
95 Introductory 88
96 Formation of equations of motion ...... 88
97 Acceleration of a point describing a tortuous curve . . 89
98 Polar coordinates in three dimensions 90
99 Integration of the equations of motion 91
100 Example . .91
101 Motion of a body attached to a string or spring ... 91
102 Examples 93
102A Force of simple harmonic type 93
103 The problem of central orbits 95
104 Apses 95
105 Examples 97
106 Apsidal angle in nearly circular orbit . . . . ' . 97
107 Examples 98
108 Examples of equations of motion expressed in terms of polar
coordinates ......... 99
109 Examples of motion under several central forces ... 99
110 Disturbed elliptic motion 101
111 Tangential impulse 101
112 Normal impulse 103
113 Examples 103
Miscellaneous Examples 103
CHAPTER V
MOTION UNDER CONSTRAINTS AND RESISTANCES
114 Introductory 108
115 Motion on a smooth plane curve under any forces . .108
116 Examples 109
117 Motion of two bodies connected by an inextensible string . 109
118 Examples 109
119 Oscillating pendulum 110
120 Complete revolution 112
121 Limiting case 112
122 Examples 113
123 Smooth plane tube rotating in its plane . . . .113
124 Newton's revolving orbit 114
125 Examples 115
126 Motion on a rough plane curve under gravity . . .116
CONTENTS XI
ART. PAGE
127 Examples .117
128 Motion on a curve in general 117
129 Motion on a smooth surface of revolution with a vertical axis 118
130 Examples 119
131 Motion on a surface in general 120
132 Osculating plane of path ....... 121
133 Examples 122
134 Motion in resisting medium . . . . . . . 123
135 Resistance proportional to the velocity 123
136 Damped harmonic motion 124
136A Effect of damping on forced oscillation . . . .124
137 Examples 125
138 Motion in a vertical plane under gravity . . . .126
139 Examples 128
Miscellaneous Examples 129
CHAPTER VI
THE LAW OF REACTION
140 Direct impact of spheres 137
141 Ballistic balance . . . . . . . ... 137
142 Statement of the law of reaction 138
143 Mass-ratio 138
144 Mass 139
145 Density . 139
146 Gravitation . . . 140
147 Theory of Attractions 141
148 Mean density of the Earth 141
149 Attraction within gravitating sphere 142
150 Examples . . 142
THEORY OF A SYSTEM OF PARTICLES
151 Introductory 143
152 Centre of mass 143
153 Resultant momentum . . . . . . . . 143
154 Resultant kinetic reaction . . . . . . .144
155 Relative coordinates 144
156 Moment of momentum . . . . . . . .145
157 Moment of kinetic reaction . . . . . . .146
158 Kinetic energy 146
159 Examples . .147
160 Equations of motion of a system of particles . . .147
161 Law of internal action ........ 148
162 Simplified forms of the equations of motion . . . .148
Xll CONTENTS
ART. PAGE
163 Motion of the centre of mass . . . . . . .149
164 Motion relative to the centre of mass 149
165 Independence of translation and rotation . . . .150
166 Conservation of momentum . . . . . . .150
167 Conservation of moment of momentum .... 150
168 Sudden changes of motion . . . . . . .150
169 Work done by the force between two particles . . . 151
170 Work function 152
171 Potential energy 152
172 Potential energy of gravitating system 153
173 Energy equation 153
174 Kinetic energy produced by impulses ..... 154
THE PROBLEM OF THE SOLAR SYSTEM
175 The problem of n bodies 154
176 The problem of two bodies ....... 155
177 Examples . 156
178 General problem of planetary motion . . . . .157
BODIES OF FINITE SIZE
179 Theory of the motion of a body 158
180 Motion of a rigid body 160
181 Transmissibility of force . . . . . . .160
182 Forces between rigid bodies in contact . . . . .161
183 Friction 161
184 Potential energy of a body 162
185 Energy of a rigid body 163
186 Potential energy of a stretched string 163
187 Localization of potential energy 164
188 Power 165
189 Motion of a string or chain 165
190 String or chain of negligible mass in contact with a smooth
surface .......... 166
Miscellaneous Examples 167
APPENDIX TO CHAPTER VI
REDUCTION OF A SYSTEM OF LOCALIZED VECTORS
Vector couple 171
Equivalence of couples in the same plane .... 171
Parallel vectors 172
Equivalence of couples in parallel planes . . . .173
Composition of couples 174
Systems of localized vectors in a plane 175
Reduction of a system of vectors localized in lines . . 176
CONTENTS Xlll
CHAPTER VII
MISCELLANEOUS METHODS AND APPLICATIONS
ART. PAGE
191 Introductory . . . . . . . . . .177
SUDDEN CHANGES OF MOTION
192 Nature of the action between impinging bodies . . . 177
193 Newton's experimental investigation ..... 178
194 Coefficient of restitution ....... 178
195 Direct impact of elastic spheres 179
196 Generalized Newton's rule 180
197 Oblique impact of smooth elastic spheres .... 180
198 Deduction of Newton's rule from a particular assumption . 181
199 Elastic systems 181
200 General theory of sudden changes of motion . 182
201 Illustrative problems 182
202 Examples 185
INITIAL MOTIONS
203 Nature of the problems . .187
204 Method for initial accelerations . . . . . .187
205 Illustrative problem '. . .188
206 Initial curvature 189
207 Examples 190
APPLICATIONS OF THE ENERGY EQUATION
208 Equilibrium 191
209 Machines 192
210 Examples 193
211 Small oscillations . .193
212 Examples 195
213 Principles of energy and momentum ..... 195
214 Examples 196
Miscellaneous Examples . . . . . . .198
CHAPTER VIII
MOTION OF A RIGID BODY IN TWO DIMENSIONS
215 Introductory ..........
216 Moment of inertia
217 Theorems concerning moments of inertia ....
218 Calculations of moments of inertia .....
219 Examples ..........
220 Velocity and momentum of rigid body .....
XIV CONTENTS
ART. PAGE
221 Kinetic reaction of rigid body 211
222 Examples 213
223 Equations of motion of rigid body 213
224 Continuance of motion in two dimensions .... 214
225 Rigid pendulum 214
226 Examples 215
227 Illustrative problems. (Note on motion of a train) . . 216
228 Examples 220
229 Kinematic condition of rolling 220
230 Examples 222
231 Stress in a rod 226
232 Impulsive motion 227
233 Kinetic energy produced by impulses 228
234 Examples 228
235 Initial motions 229
236 Small oscillations 229
237 Illustrative problem 230
238 Examples 231
Miscellaneous Examples 232
CHAPTER IX
RIGID BODIES AND CONNECTED SYSTEMS
239 Impact of two solid bodies 240
240 Impact of smooth bodies 241
241 Impact of rough bodies 242
242 Case of no sliding 243
243 Examples 244
244 Impulsive motion of connected systems .... 244
245 Examples 247
246 Initial motions and initial curvatures 248
247 Illustrative problem 249
248 Examples 251
249 Small oscillations 251
250 Examples 252
251 Stability of steady motions ....... 253
252 Examples 254
253 Illustrative problem. (Energy and momentum) . . . 255
254 Kinematical Note 256
255 Examples. (Note on moments about a moving axis) . . 257
MOTION OF A STRING OR CHAIN
256 Inexteusible chain 259
257 Tension at a point of discontinuity 259
258 Illustrative problems . . . . . . . 260
259 Constrained motion of a chain under gravity . . . 261
CONTENTS XV
ART. PAGE
260 Examples . . 262
261 Chain moving freely in one plane. Kinematical equations . 263
262 Chain moving freely in one plane. Equations of motion . 265
263 Invariable form 266
264 Examples . . . ^ 266
265 Initial motion . . * 268
266 Impulsive motion 268
267 Examples 269
Miscellaneous Examples ....... 270
CHAPTER X
THE ROTATION OF THE EARTH
268 Introductory .279
269 Sidereal time and mean solar time 279
270 The law of gravitation 280
271 Gravity 281
272 Variation of gravity with latitude 282
273 Mass and weighing ........ 283
274 Lunar deflexion of gravity 283
275 Examples 284
276 Motion of a free body near the Earth's surface . . . 285
277 Initial motion 286
278 Motion of a pendulum .... ... 287
279 Foucault's pendulum 288
280 Examples . . . 288
CHAPTER XI
SUMMARY AND DISCUSSION OF THE PRINCIPLES
OF DYNAMICS
Newton's laws of motion. Field of force. Definition of force.
Mass. Constitution of bodies. Stress. Conservation of energy.
" Energetic " method. Frame of reference and time-measuring
process 290
APPENDIX
Measurement and Units 304
INDEX . 308
INTRODUCTION
1. MECHANICS is a Natural Science ; its data are facts of
experience, its principles are generalizations from experience.
The possibility of Natural Science depends on a principle which
is itself derived from multitudes of particular experiences — 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 future motions and
positions of bodies can be deduced from them, and predictions so
made are verified in experience. In the process of formulation
the Science acquires the character of an abstract logical theory, 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.
The study of such a science ought to be partly "experimental ;
it ought also to be partly historical. Something should be known
2 INTRODUCTION
of the kind of experiments from which were derived the abstract
notions of the theory, and something also of the processes of
inductive reasoning by which these notions were reached. It will
be assumed here that some such preliminary study has been
made*. The purpose of this book is to formulate the principles
and to exemplify their application.
2. Motion of a particle. 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.
A moving point considered as defining the position from time
to time of a very small part of a body will be called a " particle."
Motion may be defined as change of position taking place in
time.
In regard to this definition it is necessary to attend to two
things : the measurement of time, and the determination of
position.
3. Measurement of time. Any instant of time is separated
from any other instant by an interval. The duration of the interval
may be measured by the amount of any process which is effected
continuously during the interval. For the purposes of Mechanics
it is generally more important that time should be conceived as
measurable than that it should be measured by an assigned
process.
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 generally assume that time is
measured in this way, and we shall denote the measure of the time
* Historical accounts are given by E. Mach, The Science of Mechanics (Trans-
lation), Chicago, 1893, and by H. Cox, Mechanics, Cambridge, 1904.
1-4] POSITION 3
which elapses between two particular instants by the letter t, then
t is a real positive number (in the most general sense of the word
" number ") and the interval it denotes is t seconds.
4. Determination of position. The " position of a point "
means its position relative to other points. Position of a point
relative" to a set of points is not definite until the set includes
four points which do not all lie in one plane. Suppose 0, A, B, C
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. (See Fig. 1.) The
position of a point P with reference to this trihedral angle is
determined as follows :— we draw PN parallel to 00 to meet the
plane AOB in N, and we draw NM parallel to OB to meet OA
Fig. 1.
in M ; then the lengths ON, MN, NP determine the position of P.
Any particular length, e.g. one centimetre, being taken as the
unit of length, each of these lengths is represented by a number
(in the general sense), viz. by the number of centimetres contained
in it. It is clear that OP 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 at right angles to each other, then the faces of
the trihedral angle are also at right angles to each other; sets of
1—2
4 INTRODUCTION
lines so chosen are called systems of rectangular axes, and the
planes that contain two of them are coordinate planes*.
It is clear from Fig. 2 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
Fig. 2.
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 BOG, and is equal to this number with a
minus sign when P and A are on opposite sides of the plane BOG,
and similarly for y and z.
Axes drawn and named as in Fig. 2 are said to be "right-handed." If the
letters x and y are interchanged the axes are left-handed. In most applications
of mathematics to physics right-handed axes are preferable to left-handed
axest. To fix ideas we may think of the compartment in which x, y, z are all
positive as being bounded by two adjacent walls of a room and the floor of the
room. If we look towards one wall with the other wall on the left-hand, and
name the intersection of the walls the axis of z, the intersection of the floor
with the wall on our left the axis of x, and the intersection of the floor with
the wall in front of us the axis of y, the axes are right-handed. An ordinary,
or right-handed, screw, turned so as to travel in the positive direction of the
axis of x (or y, or 2), will rotate in the sense of a line turning from the positive
direction of the axis of y (or z, or x) to the positive direction of the axis of
z (or x, or y). The senses of rotation belonging to the three screws are
indicated in Fig. 3.
* We shall, in the course of this book, make use of rectangular coordinates only,
t In the course of this book the axes will be taken to be right-handed unless a
statement to the contrary is made.
4, 5]
FRAME OF REFERENCE
Fig. 3.
5. Frame of reference. A triad of orthogonal lines OA,
OB, OC, with respect to which the position of a point P can be
determined, will be called -A frame of reference.
To determine a frame of reference we require to be able to
mark a point, a line through that point, and a plane through that
line. Suppose 0 to be the point, OA a line through the point,
AOB a plane through the line. We can draw on the plane a line
at right angles to OA meeting it in 0, and we can erect at 0 a
perpendicular to the plane. The three lines so determined can be
a frame of reference.
In practice we cannot mark a point but only a small part of a body, for
example we may take as origin a place on the Earth's surface ; then at the
place we can always determine a particular line, the vertical at the place, and,
at right angles to it, we have a particular plane, the horizontal plane at the
place ; on this plane we may mark the line which points to the North, or in
any other direction determined with reference to the points of the compass,
we have then a frame of reference. Again we might draw from the place lines
in the direction of any three visible stars, these would determine a frame of
reference. Or again we might take as origin the centre of the Sun, and as
lines of reference three lines going out from thence to three stars.
When we are dealing with the motions of bodies near a place
on the Earth's surface, for example, the motion of a train, or a
cannon-ball, or a pendulum, we shall generally take the frame of
reference to be determined by lines which are fixed relatively to
the Earth, and we shall generally take one of these lines to be
the vertical at the place. When we are dealing with the motion
of the Earth, or a Planet, or the Moon, we shall generally take
6 . INTRODUCTION
the frame of reference to be determined by means of the " fixed "
stars.
A point, or line, or plane which occupies a fixed position
relatively to the chosen frame of reference will be described as
" fixed."
6. Choice of the time-measuring process and of the
frame of reference. Time may be measured by any process which
goes on continually. 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-measure; we shall call it the st'.!it<l<i.rd
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 uncontrollable by us should be
uniform or approximately uniform ; it is also clearly desirable that it should
have some relation to our daily life. The choice of the mean solar second as
a unit of time satisfies these conditions. So long as these conditions are not
violated, we are at liberty to choose a different reckoning of time for the
purpose of simplifying the description of the motions of bodies.
The choice of a suitable frame of reference, like the choice of the time-
measuring process, is in our power, and it is manifest that some motions which
we wish to describe will be more simply describable when the choice is made
in one way than when it is made in another. We shall return to this matter
in Chapter XI.
CHAPTER I
DISPLACEMENT, VELOCITY, ACCELERATION
7. THE history of the Science of Mechanics shows how, through
the study of the motions of falling bodies, importance came to be
attached to the notions of variable velocity and acceleration, and
also how, chiefly through the proposition called " the parallelogram
of forces," the vectorial character of such quantities as force and
acceleration came to be recognized. We shall now be occupied
with precise and formal definitions of some vector quantities and
with some of the immediate consequences of the definitions.
8. Displacement. Suppose that a point which, at any
particular instant, had a position P with reference to any frame,
has at some later instant a position Q relative to the same frame.
The point is said to have undergone a " change of position " or a
displacement Let the line PQ be drawn. It is clear that the
displacement is precisely determined by this line ; we say that it
is represented by this line. Let the line PQ drawn through P
be produced indefinitely both ways, and let a parallel line
be drawn through any other point, for
instance through 0. Then this line de-
termines a particular direction ; this is
the direction of "the displacement. Of the
two senses in which this line may be
described one, OR, is the sense from 0
towards that point (R) which is the fourth
corner of a parallelogram having OP, PQ
as adjacent sides ; this is the sense of the
displacement. The measure of the length
of PQ is the number of units of length it
contains ; this number is the magnitude
of the displacement. The subsequent position, Q, is entirely
determined by (1) the previous position, P, (2) the direction of
the displacement. (3) the sense of the displacement, (4) the
magnitude of the displacement.
8 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
Further it is clear that exactly the same change of position
• is effected in moving a point from P to K by
the straight line PK, and from K to Q by the
straight line KQ, as in moving the point from
P to Q directly by the straight line PQ. That
is to say, displacements represented by lines
PK, KQ are equivalent to the displacement
represented by the line PQ.
Displacement is a quantity, for one dis-
placement can be greater than, equal to, or
less than another; but two displacements in
Fig. 5.
different directions, or in different senses, are
clearly not equivalent to each other, even when they are equal in
magnitude ; and thus displacement belongs to the class of mathe-
matical quantities known as vectors or directed quantities.
9. Definition of a vector. A vector may be defined as a
directed quantity which obeys a certain rule of operation*.
By a "directed quantity" we mean an object cf mathematical
reasoning which requires for its determination (1) a number
called the magnitude of the quantity, (2) the direction of a line
called the direction of the quantity, (3) the sense in which the
line is supposed drawn from one of its points, called the sense of
the quantity.
Let any particular length be taken as unit of length. Then
from any point a straight line can be drawn to represent the
vectorf in magnitude, direction, and sense. The sense of the line
is indicated when two of its points are named in the order in which
they are arrived at by a point describing the line.
The rule of mathematical operation to which vectors are subject
is a rule for replacing one vector by other vectors to which it is
(by definition) equivalent.
* The rule of operation is an essential part of the definition. For example,
rotation about an axis is not a vector, although it is a directed quantity.
f The line is not the vector. The line possesses a quality, described as exten-
sion in space, which the vector may not have. From our complete idea of the
line this quality must be abstracted before the vector is arrived at. On the other
hand the vector is subject to a rule of operation to which a line can only be sub-
jected by means of an arbitrary convention.
8-10] EQUIVALENT VECTORS 9
This rule may be divided into two parts and stated as
follows : —
(1) Vectors represented by equal and parallel lines drawn
from different points in like senses are equivalent.
(2) The vector represented by a line AC is equivalent to the
vectors represented by the lines AB, BC, the points A, B, C being
any points whatever.
Among vector quantities, as here defined, we note (i) displacement of a
particle, (ii) couple applied to a rigid body (see Appendix to Chapter VI).
10. Examples of equivalent vectors. If AC, A'C' are
equal and parallel lines, their ends can be joined by two lines AA'y
Fig. 6.
CC' which are equal and parallel ; then the vectors represented by
AC, A'C' are equivalent ; vectors represented by AC, C'A' are not
equivalent.
Again if A, B, C are any three points, and a parallelogram
A, B, C, D is constructed having AB, BC as adjacent sides, AD
Fig. 7.
and BC are equivalent vectors. Also the vector A C is equivalent
to the vectors AB, BC, or AD, DC, or AB, AD.
Further if a polygon (plane or gauche) is constructed, having
AC as one side, and having any points P, Q, ... T as corners, the
10 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
vector represented by AC is equivalent to the vectors represented
by AP, PQ, . . . TC. This is clear because by definition the vectors
AP, PQ can be replaced by AQ, and so
on. The statement is independent of the
number of sides of the polygon, and of
the order in which its corners are taken,
no corner being taken more than once,
provided that the points A, (7 are regarded
as the first and last corners. [The restric-
tion that no corner is to be taken more
than once will be removed presently.]
In particular, if the polygon is a
gauche quadrilateral ABDC, a parallele-
piped can be constructed having its edges
parallel to AB, BD, DC, and having AC
as one diagonal. Then the vector AC is equivalent to the vectors
represented by the edges AB, AP, AQ which meet in A. (See
Fig. 9.)
The case of this which is generally most useful is the case
where the edges of the parallelepiped are the axes of reference
relatively to which the positions of points are determined.
11. Components and resultant. A set of vectors equivalent
to a single vector are called components, and the single vector to
which they are equivalent is called their resultant.
The operation of deriving a resultant vector from given com-
ponent vectors is called composition, we compound the components
to obtain the resultant ; the operation of deriving components in
10, ll] COMPOSITION AND RESOLUTION OF VECTORS 11
particular directions from a given vector is called resolution, we
resolve the vector in the given directions to obtain the components
in those directions.
It is clear from the constructions in the preceding article that
we can resolve a vector in one way into components parallel to
any two given lines which are in a plane to which the vector is
parallel, and again we can resolve the vector in one way into
components parallel to any three given lines not in the same
plane.
When the directions of the component vectors are at right
angles to each other the components are called resolved parts of
the resultant vector in the corresponding directions.
Thus, if we take a system of rectangular coordinate axes, any
vector parallel to a coordinate plane, e.g. the plane of (#, y), can be
resolved into components parallel to the axes of x and y, these are
the resolved parts of the vector in the directions of the axes of x
and y.
Again, if we take a three-dimensional system of rectangular
axes, any vector can be resolved
into components parallel to the
axes of x, y, and z, and these are
the resolved parts of the vector
in the directions of these axes.
In the former case (Fig. 10)
we take OP to represent the
vector, and draw PM at right
angles to Ox, then OM and MP Fi^ 10-
represent the resolved parts of the vector parallel to the axes. If
R is the magnitude of the vector represented by OP, and 6, <f> the
angles* between the lines OP and Ox, Oy, then R cos 6 and
* In Fig. 10 cos 0 is sin 6, but it is easy
to draw a figure, e.g. Fig. 11, which makes it
appear that cos 0 is - sin 6. With the usual
conventions in regard to the signs of trigono-
metrical functions we shall always have
cos 0 = sin 6
provided that 0 is the angle traced out by a
line OP starting from Ox and turning round 0
in the direction from Ox to Oy.
Fig. 11.
12 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
Rcos<j) are the magnitudes of the resolved parts respectively, and
these are the projections of OP on the axes.
More generally, we take OP to represent the vector, and
construct a parallelepiped with 0 and P as opposite corners and
with its faces parallel to the coordinate planes, then the resolved
parts of the vector in the directions of the axes are numerically
equal to the projections of OP on the axes. If R is the magnitude
of the vector represented by OP, and if I, m, n are the cosines of
Fig. 12.
the angles which OP makes with Ox, Oy, Oz respectively, the
resolved parts in these directions are Rl, Rm, Rn respectively.
This rule determines the senses as well as the magnitudes of
the resolved parts ; thus, when cos 6, in the first case, and I, in
the second case, are negative, the resolved part parallel to the x
axis is in the negative direction of that axis, i.e. in the direction
xO produced.
It is clear from this rule that, when the magnitudes and signs
of the resolved parts of a vector in the directions of three mutually
rectangular lines are given, the vector is uniquely determinate,
that is to say there is one and only one vector which has given
resolved parts parallel to three such lines.
11, 12] COMPOSITION AND RESOLUTION OF VECTORS 13
Fig. 13.
The construction in the former of these cases is a construction
for the resolved parts of a
vector parallel and perpen-
dicular to a line. As before,
let OP be a line representing
the vector, and OA a line
parallel and perpendicular
to which the vector is to be
resolved. Draw PM at right
angles to OA. Then the
vector is equivalent to vectors represented by OM, MP, and the
magnitudes of these are respectively R cos 6 and R sin 6, where
R is the magnitude of the vector to be resolved, and 6 is the angle
between its direction and OA,
The vector represented by MP is the resolved part of the
vector represented by OP at right angles to the line OA,
12. Composition of any number of vectors. I. Consider
first the case where all the vectors are
parallel to a plane, and take it to be
the plane of (x, y). Let OP15 OP2,
. . . OPn be lines representing the
vectors, (supposed to be n in num-
ber,) in magnitude, direction, and
sense, and let ff1} 02, ... 6n be the
angles which the lines OPl} OP2,
. . . OPn make with Ox, i.e. the angles
Fig. 14.
traced out by a revolving line turning about 0 from Ox towards
Oy. Let i\, r.2, ... rn denote the magnitudes of the vectors.
Then the vector represented by OPj may be replaced by vectors
TI cos 0l parallel to Ox, and 1\ sin #j parallel to Oy, and similarly
for the others.
All the resolved parts parallel to Ox are equivalent to a single
vector X parallel to Ox given by
X — 1\ cos #j + r., cos #2 + ••.+?'« cos 6n = 2 (r cos 0).
All the resolved parts parallel to Oy are equivalent to a single
vector Y parallel to Oy given by
Y= i\ sin 0L + r., sin 02 + . . . + rn sin 0n = S (r sin 6\
14 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
The vector whose resolved parts parallel to Ox and Oy are X
and F is the resultant of all the vectors. Let the magnitude of
this vector be R, and let its direction and sense be those of a line
going out from 0 and making an angle ^r with Ox.
Then we have R cos -fy = X, and R sin ty = Y.
These two equations determine the magnitude R and the angle
i/r. R is the numerical value of \I(X2 + F2), and i/r is that one
among the angles whose tangents are YfX for which the sine has
the same sign as F and the cosine has the same sign as X.
II. Consider the more general case where the vectors are not
parallel to a plane. Let i\, r2,...rn be the magnitudes of the
vectors, and call any one of these numbers r. Let I, in, n be the
cosines of the angles which the line representing this vector in
direction and sense makes with the axes Ox, Oy, Oz. Then this
vector may be resolved into rl, rm, rn parallel to the lines Ox, Oy,
Oz, and the whole set of vectors is equivalent to a vector whose
resolved parts parallel to the axes are X. F, Z, where X = 2r/,
F=Srm, Z=^m, the summations extending to all the vectors of
the set. The resultant is therefore a vector whose magnitude, R,
is the numerical value of »J(XZ + F2 + Zz}, and such that the line
representing it in direction and sense makes with the axes Ox, Oy,
Oz angles whose cosines are X/R, Y/R, Z/R.
13. Vectors equivalent to zero. When the magnitude of
the resultant of any set of vectors is zero the set of vectors is said
to be equivalent to zero. Thus two equal vectors parallel to the
same line, and in opposite senses, are equivalent to zero.
It is clear that the sum of the resolved parts, in any direction,
of a set of vectors equivalent to zero is equal to zero.
Again vectors parallel and proportional to the sides of a closed
polygon, and with senses determined by the order of the corners
when a point travels round the polygon, are equivalent to zero.
This last statement enables us to do away with the restriction
(Art. 10) that in the resolution of a vector into components
parallel to the sides of a polygon not more than two sides of the
polygon may meet in a point.
14. Components of displacement. Let x, y, z be the
coordinates of a moving point at any particular instant with
12-15] DEFINITION OF VELOCITY 15
reference to any particular frame, x, y', z' the coordinates of the
point at a subsequent instant, with reference to the same frame,
then x — x, y — y, z — z are the components, parallel to the axes,
of a vector 'quantity which is the displacement of the point.
<Cf. Art. 8.)
15. Velocity in a straight line. Consider in the first place
a, point moving in a straight line, e.g, one of the lines of reference,
and let s be the number of units of length it passes over in t units
of time. Then it may happen that the two numbers s and t have
a constant ratio whatever number we take for t. The point is then
o
said to move uniformly in the line, and the fraction - is defined to
t
be the measure of its velocity. A point moving uniformly describes
equal lengths in equal times.
Again consider the case where the point moves in a straight line, but the
number of units of length passed over in any interval of time does not bear a
constant ratio to the number of units of time in the interval. In this case
there will be equal intervals of time in which the point describes unequal
lengths ; in the one of two equal intervals in which it describes the greater
length we should say it was moving faster, in the other, in which it describes
the shorter length, we should say it was moving more sldwly. We have thus
an idea of velocity of a point not moving uniformly, and we seek to make it
precise.
For a point moving in a straight line we may define the
average velocity in any interval of time to be the fraction
number of units of length described in an interval
number of units of time in the interval
When the point is not moving uniformly this fraction is a
variable number, which has a definite value when the measure of
the interval is given and the first instant of the interval is given.
Taking the first instant of the interval always the same, and
taking for the measure of the interval a series of diminishing
numbers, we obtain a series of fractions, which approach a limit-
ing value as the measure of the interval is indefinitely diminished.
This limiting value is defined to be the velocity of the point at
the first instant of the interval. We might in the same way define
the velocity of a point at the last instant of an interval.
We can now define the velocity of a point moving in a straight
line at any instant. It is the limit of the average velocity in an
16 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
interval of time beginning or ending at the instant, the interval
being diminished indefinitely.
The two limits are in general the same; when they are different
we call them the velocity just after the instant and the velocity
just before the instant respectively.
Let t be the measure of the interval of time which has elapsed
since some particular instant, chosen as the origin of time, and
suppose that at the end of this interval the point has described a
length s measured from some particular point in the line of its
motion. We say that the point is at s at time t. In the same way
suppose that it is at s' at time t'. Then in the interval t' — t it
'describes a length s' — s, and its average velocity in the interval is
S — ~ S
-, — 7 . The number s is a function of the number t, and the
t — t
limit of the fraction just written is the number known as the
differential coefficient of s with respect to t. The velocity of the
moving point is accordingly measured by -j- .
The number s' — s is the measure of the displacement of the point during
the interval t' - 1. When the velocity is uniform it is measured by the
displacement in a unit of time. If the unit of time were replaced by a
smaller unit the displacement in it would be replaced by a shorter length,
and this length would measure the velocity in terms of the new unit of time.
However short an interval is taken for the unit of time the length described
in it measures the velocity in terms of it. When we wish to recall this fact,
and to bring it into connexion with the definition of variable velocity we say
that the latter is measured by " the rate of displacement per unit of time,"
but we must not attach to this phrase any other meaning than that which has
just been explained, i.e. the phrase means nothing but the limit of the fraction
number of units of length described in an interval
number of units of time in the interval
when the interval is diminished indefinitely.
16. Velocity in general. When the point is not moving in a
straight line it will have a component of displacement in any interval
t' — t parallel to each of the three axes of reference. Let these
components be x' — x, y — y, z — z. Then each of the fractions
x — x y' — y z — z , ,. .
—7 — - , ~ — - , —, — - has a limit, and these limits are, as above,
V "~~ V V """ ~* It V ~~~ V
the rates of displacement per unit time parallel to the axes. They
are defined to be the component velocities parallel to the axes. As
15-17] DEFINITION OF VELOCITY 17
before x, y, z are functions of t, and the component velocities parallel
to the axes are
dx dy dz
~di' ~di' dt'
The velocity at an instant is the limit of the average velocity in an
interval. This limit has a definite magnitude, and is associated with a
definite straight line. At any instant the point is moving along the tangent
to a curve, called its path or trajectory. The velocity is associated with this
particular line, drawn in a definite sense. Let * be the arc of the curve
measured from some particular point of the curve up to the position of the
moving point at time t, and let s' be the corresponding arc for time t'. Then
the length of the chord joining the two positions is the magnitude of the
vector whose components parallel to the axes are X'-A; y' -y, z' — z. From
the definition of s we have the equation
ds
Thus the magnitude of the velocity of the moving point at time t is -T-, where
Cvt
s is the length of the arc of the path measured, in the sense of description of
the path, from some particular point of it to the position of the moving point
at time t. The magnitude of the velocity of a point is often called its speed,
and, when it is independent of the time, the point is said to move with uniform
speed whether its path is straight or curved.
It is manifest that the velocity of a moving particle can be represented in
many respects by a vector, of which the components parallel to the axes are
df'tit'dt' ^U^ ^e vec^or does u°t exPress the association of the velocity
with a particular line — the tangent to the path of the particle.
17. Localized vectors. The vectors we have so far considered
have no relation to any particular point, they are equally well repre-
sented by lines drawn from any point ; and they have no relation
to any particular line, they are equally well represented by seg-
ments of all lines parallel to their direction. They may be called
unlocalized vectors. But it is often important to consider quantities
which, in other respects, have the properties of vectors, but which
have relations to particular points or particular lines.
A vector localized at a point is defined by its magnitude, direction,
and sense, and also by a point and by a rule of equivalence, viz. : —
two sets of vectors localized at the same point are equivalent if two
sets of unlocalized vectors with the same magnitudes, directions,
and senses are equivalent.
L. M. 2
18 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
There is in general no rule of equivalence for vectors localized
at different points.
A vector localized in a line is a vector localized at any point in
a particular line, which is in the direction of the vector, with the
additional rules of equivalence, (i) Two vectors localized in the same
line are equivalent if they have the same magnitude and the same
sense, (ii) Two vectors localized in lines which meet are equivalent
to a single vector localized in a line.
All the constructions in the previous Articles apply to vectors
localized at points and to vectors localized in lines, provided that
all components and resultants are localized at the proper points or
in the proper lines. In particular a vector localized at a point is
equivalent to components (or resolved parts) of the same magni-
tudes, directions, and senses as if it were unlocalized, provided that
these components and resolved parts are localized at the same point ;
also a vector localized in a line is equivalent to components (or
resolved parts) of the same magnitudes, directions, and senses as if
it were unlocalized, provided that these components and resolved
parts are localized in lines which meet in a point on the line of the
resultant.
Thus a vector localized at 0 may be represented (as in Fig. 12)
by a line OP, and is equivalent to vectors localized at 0 and repre-
sented by lines OH, OK, OM; and a vector localized in the line OP,
having the same magnitude and sense, is equivalent to vectors
localized in any three lines parallel to Ox, Oy, Oz, meeting in a point
on OP, and having the magnitudes and senses of OH, OK, OM.
The differences between the three classes of vectors may be
expressed thus : —
A vector (unlocalized) is equivalent to any parallel vector of
equal magnitude and like sense. Thus the line representing the
vector may be drawn from any point.
A vector localized in a line is equivalent to any vector of equal
magnitude and like sense localized in the same line. The line repre-
senting it may be drawn from any point in a particular line, and is
a segment of that line.
A vector localized at a point is not equivalent to any other single
vector. The line representing it must be drawn from the point.
17-20] LOCALIZED VECTORS 19
A vector localized in a line is clearly determined by its com-
ponents parallel to three given lines and by one point of the line,
in particular the line in which it is localized is thereby determined.
As examples of vectors localized in lines we may cite (i) velocity of a
moving particle, (ii) force applied to a rigid body (Chapter VI). Force
applied to a particle is an example of a vector localized at a point (Chapter III).
18. Formal definition of velocity. We may now define the
velocity of a moving point to be a vector, localized in a line through
the position of the point, whose resolved part in any direction is
the rate of displacement of the point in that direction per unit of
time.
19. Measurement of velocity. The measure of any particular
velocity is a number expressing the ratio of the velocity to the unit
velocity.
The unit velocity is that with which a point describes one unit
of length uniformly in each unit of time.
The number expressing a velocity is the ratio of a number ex-
pressing a length to a number expressing an interval of time. It
therefore varies inversely as the unit of length and directly as the
unit of time.
Velocity is accordingly said to be a quantity of one dimension
in length, and of minus one dimension in time ; or its dimension
symbol is LT~l, where L stands for length, and T for time.
20. Moment of localized vector. The reason for defining
velocity as a localized vector is that special significance is found
to attach to a certain quantity called the "moment of the velocity."
We shall attend at present to the cases of vectors localized in lines
that lie in a plane and vectors localized at points in a plane, and
having their directions parallel to the plane*. We define the mo-
ment of such a vector about a point in the plane as follows : —
Draw a line L' in the direction of the vector, so that if the vector
is localized in a line that line is L', and if the vector is localized at
a point the line L' passes through the point. The moment of the
* A more general discussion will be given in Chapter III.
2 2
20 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
vector about a point 0 is the product, with a certain sign, of the
magnitude of the vector and the perpendicular to L' from 0. The
rule of signs is this : Draw a line L through 0 at right angles to
the plane containing 0 and L', and choose a sense of description of
this line ; then, if the senses of L and the vector are the same as
those of translation and rotation in an ordinary right-handed screw,
the sign is +, otherwise it is — .
The rule of signs may also be stated thus: Let a watch be
placed in the plane of 0 and L', so that a line drawn from the
back to the face is in the sense of L ; when the sense of the vector
is opposite to that of the motion of the hands the sign is +, other-
wise it is — .
21. Lemma. The moment about a point 0 of a vector localized
at a point A is identical with the moment about 0 of the resolved
part of the vector at right angles to OA.
Let 0 be the angle which the direction of the vector makes
with the line A 0, and draw
ON at right angles to the
line of the vector. The
magnitude of the resolved
part of the vector at right
angles to A 0 is R sin 6,
where R is the magnitude
of the vector. The perpen-
dicular from 0 on the line
of the vector is the line
ON, and it is equal to
Fig. 15. 0 A . sin 6.
Now moment of R about 0 = R. ON
= R.OAsin0
= R. sin 0.0 A
= moment about 0 of resolved part at right angles to OA.
22. Theorem of moments. The sum (with proper signs) of
the moments about a point 0 of two vectors localized at a point A is
equal to the moment of their resultant about 0.
20-23]
MOMENTS
21
Let P1 and P2 be the magnitudes of the vectors,
angles which the lines re-
presenting them drawn from °\
A make with AO, R the
magnitude of the resultant,
<j) the angle which the line
representing it makes with
AO. Then the magnitudes
of the resolved parts at right
angles to A 0 are PI sin 6l ,
P2 sin 0.2, and R sin <f>, and
we know (Article 12) that
.Rsin <j>=Pism61 + P.,s'm02.
Now sum of moments of
and 0.2 the
Fig. 16.
J! and P2 about 0
= OA . R sin <£
= moment of R about 0.
This result can be immediately extended to any number of vectors
localized at a point.
It follows that, when a vector localized at a point (x1} y^) in
the plane of (x, y), or in a line
passing through this point, is
specified by its components
Xl and Y1 parallel to the axes
of x and y, its moment about
the origin is ^Fj — y1X1. See
Fig. 17. For example, the mo-
ment about the origin of the
. (dx dy\ . .
velocity [-jf, ~r I ot a particle
moving in the plane of (x, y) is
f. ~ y -rr) . where x and y are the coordinates of its position
Fig. 17.
at time t.
23. Acceleration. A point moving with a variable velocity,
relative to any frame, is said to have an acceleration relative to
that frame.
When the point is moving in such a way that its velocity
22 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
increases by equal amounts in equal intervals of time, however
short the intervals may be, it is said to have a uniform accelera-
tion, provided that the velocity acquired in every interval has the
same direction and sense.
Uniform acceleration is determined, as regards magnitude,
direction, and sense, by the velocity added in a unit of time.
When the acceleration is not uniform, the moving point is said
to have a variable acceleration.
The acceleration of a point moving in a straight line is the rate
of increase of its velocity per unit of time. This is a short way of
expressing the following definition : —
Let v be the velocity of the point at time t, and v its velocity
?) •—• 7)
at time t'y then its acceleration is the limit of the fraction — —
t ~~ t
when the interval t' — t is diminished indefinitely, or in words it is
the limit of the fraction
number of units of velocity added in an interval of time
number of units of time in the interval
when the interval is diminished indefinitely. The number v is
a function of the number t, and its differential coefficient with
respect to t is the acceleration, i.e. the acceleration is measured
. dv
^dt'
When the point is not moving in a straight line it will in general
have a variable velocity parallel to each of the lines of reference
(coordinate axes). Let u, v, w be component velocities parallel to
these axes at time t, and u, v', w' corresponding Components at
, , , „ . u' — u v'—v w'—io. .
time t , then the fractions —, — - , -; — - , -—, — - have limits when
t — t t — t t — t
the interval t' — t is diminished indefinitely, and these limits are
the differential coefficients -j , -tr . The vector which has these
at at at
components parallel to the axes is defined to be the acceleration of
the point, or in other words we define the acceleration of a moving
point to be the vector, localized in a line through the point, whose
resolved part in any direction is the rate of increase of the velocity
in that direction per unit of time.
23-25] ACCELERATION 23
24. Measurement of acceleration. The measure of any
particular acceleration is the number expressing the ratio of the
acceleration to the unit acceleration.
The unit acceleration is that uniform acceleration with which
a moving point gains a unit of velocity in a unit of time.
The number expressing an acceleration is the ratio of a number
expressing a velocity to a number expressing an interval of time.
It therefore varies inversely as the unit of length and directly as
the square of the unit of time.
Acceleration is accordingly said to be a quantity of one dimension
in length and of minus two dimensions in time, or its dimension
symbol is LT~2.
Accelerations are not measured directly. The quantities which are
measured directly are lengths and angles. By measuring angles we can
estimate intervals of time, using a clock or watch, for example. The values
of velocities are deduced from a knowledge of the distances described in
different intervals of time. The values of accelerations are deduced from a
knowledge of the values of velocities at different times.
25. Notation for velocities and accelerations. We have
so frequently to deal with differential coefficients of quantities with
regard to the time that it is convenient to use for them an abbre-
viated notation. We shall therefore denote the differential coefficient
of any quantity q with regard to the time t by placing a dot over
the q, thus q stands for -57 .
at
Now let x, y, z be the coordinates of a moving point at time t,
then its component velocities parallel to the axes are denoted by
4 y, z-
Again let u, v, w be the component velocities of a point parallel
to the axes, then its component accelerations are denoted by u, v, w.
0. dab dii dz .. .
Since u = -7- , v = -£ , w = -7- it is convenient to write for them
at dt dt
.... . rp,, , „ d2x d (dx\
x, y, z respectively. Ihen x stands for ~r- or -5- ( -y-j , and so on.
dv at \cut /
In the same way when we have to deal with any function of
the time, say q, we may write q for -rf , as we write qfor-~. Also,
at" at
24 DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
following the analogy of the case where q is x, y, or z, we may call
q the velocity with which q increases, and q the acceleration with
which q increases.
26. Angular velocity and acceleration. Let a line, for
example the line joining the positions at any time of two moving
points, move so as always to be in the same plane with reference
to any frame. To fix ideas we shall take the plane to be the
coordinate plane of (x, y). Suppose the line to make an angle 9
(measured in radians) with the axis x at time t, and an angle
9 + A# with the same axis at time t + A£. Then A0 is the measure
of the angle turned through by the line in the interval measured
by A£, and the limit of the ratio of these two numbers is 6, the
differential coefficient of 9 with respect to t. This number, 9, is
called the angular velocity of the line. In the same way 9 is called
the angular acceleration of the line.
27. Relative coordinates and relative motions. Let xlt
yi,2i be the coordinates of a point A at time t referred to axes with
origin at 0, #2> 2/2, ^2 the coordinates of a second point B at the
same time referred to the same axes, and £, 77, £ the coordinates of
B at the same time referred to parallel axes through A. Then £,
77, £ are called the coordinates of B relative to A .
We have #2 = xl + %, }
2/2 = 2/1 + *;, [ ........................... (1)
«•-* + £ J
Let accented letters denote at time t' the quantities that cor-
respond to unaccented letters at time t, thus let #/, ?//, z{ be the
coordinates of A', the position of A at time t'. Then as before
By subtraction we deduce
#2' - a?, = fa' - x,} 4- (f - f), |
2/.;-2/2 = (y1'-2A) + <y-'?), | ............... (2)
*,'-*, = (*i'-*i) + ((T-D. J
The terms on the left are the components parallel to the axes
of the displacement of B.
25-28] RELATIVE MOTION 25
The terms in the first brackets on the right are the components
parallel to the axes of the displacement of A.
The terms in the second brackets on the right are the com-
ponents of the displacement of B relative to parallel axes with
origin at A.
Thus we have the result : — The displacement of a point B
relative to axes at 0 is compounded of the displacement of a point
A relative to the same axes and the displacement of B relative to
parallel axes through A.
By dividing both members of each of the equations (2) by t' - t
and passing to the limit when t' — t is diminished indefinitely, or,
what is the same thing, by differentiating equations (1) with respect
to t, we find
#2 = #1 + £, 2/2 = y\ + *n, z2 = Zi + 1
and by differentiating again we find
These equations may be expressed in words as follows : —
rp,, f velocity )
ihe \ . /. }• of B relative to axes at 0 is compounded
I acceleration]
of the -I ' . [ of A relative to the same axes and the
(acceleration)
' . >• of B relative to parallel axes through A.
(acceleration)
28. Geometry of relative motion. The geometrical view of
relative motion is instructive, and leads easily to results of some importance.
For shortness we shall speak of displacement, velocity, and acceleration of a
point relative to a second point, meaning thereby displacement, velocity, and
acceleration of the point relative to axes drawn through the second point
parallel to the axes of reference.
Let A be the position at any time t of a point which moves relatively
to a frame having its origin at 0, and let A' be its position at time if.
From 0 draw OH equal and parallel to A A', and in the same sense ; the
vector represented by OH is the displacement of A.
Similarly let B be the position at time t of a second point referred to
the same frame, and B' its position at time if. From 0 draw OK equal and
parallel to BB', and in the same sense ; the vector represented by OK is the
displacement of B.
26
DISPLACEMENT, VELOCITY, ACCELERATION [CH. I
Then the displacement of B relative to A is the vector that must be
compounded with the displacement of A in order that the resultant may
be the displacement of B.
Join HK. Then the vector OK is compounded of OH, HK.
Hence HK represents the displacement of B relative to A in magnitude,
direction, and sense.
Now the vector HK is the resultant of HO, OK.
Fig. 18.
Hence to obtain the displacement of B relative to A we must compound
the displacement of B with the reversed displacement of A. The resultant
is the required relative displacement.
I velocity } . I velocity )
In the same wav the •{ . , . v of /> relative to A is the -{ . *. V
(acceleration) (acceleration)
which must be compounded with the -! . V of A in order that the
(acceleration)
f velocity )
resultant may be the -{ . V of B.
(acceleration)
Since the velocity of a point in any direction is the rate of increase of its
displacement in that direction per unit of time, and since its acceleration in
any direction is the rate of increase of its velocity in that direction per unit
of time, we have the rules : —
The { velo(:liJ I Of B relative to A is the resultant of the { velocit* I
(acceleration) (acceleration)
of B and the •! , . \ of A reversed,
(acceleration)
The compositions and resolutions described in this Article are to be
effected as if the vectors involved were not localized, but the velocity and
acceleration of B relative to A are to be regarded as localized in lines through B.
CHAPTER II
THE MOTION OF A FREE PARTICLE IN A FIELD OF FORCE
29. Gravity. An unsupported body near the Earth's surface
generally falls towards the Earth. The differences in the be-
haviour of "light" bodies and "heavy" bodies are to be traced to
the buoyancy and resistance of the air. When the effects due to
the presence of the air are eliminated, for instance, when bodies
fall in the exhausted receiver of an air pump, it is found that all
kinds of bodies fall to the Earth with the same acceleration. The
direction of this acceleration at any place is the " vertical at the
place." The magnitude of this acceleration depends to some extent
on latitude; but, in the neighbourhood of any place, it is practically
constant. We call it the " acceleration due to gravity," and we
denote it by the letter (7. When the centimetre is the unit of length,
the value of g in London is 981 '2, when the foot is the unit of
length the value is 32'2. The fact that bodies fall to the Earth
with a constant acceleration was discovered by Galileo.
30. Field of force. A region in which a free body moves with
a certain acceleration is called a " field of force." The magnitude
of the acceleration is the " intensity of the field," and the direction of
the acceleration is the "direction of the field." When the intensity
and direction of the field are the same at all points the field is said
to be "uniform."
For example, the neighbourhood of the Earth is a field of force
of which the intensity near the Earth is g. We call it the "field
of the Earth's gravity." If we confine our attention to a small part
of the Earth's surface we may regard the field as uniform.
31. Rectilinear motion in a uniform field. Let the direction
of the field be the axis of x, and let /be its intensity. A particle
moving in the field parallel to the axis of x has an acceleration/.
Let x0 be the value of x at the initial position of the particle, and
u its velocity (parallel to the axis of x) in this position.
Then we are given x =/
with the conditions x = #„ when t = 0, and x = u when t = 0
28 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
Writing v for x, so that v is the velocity at time t, we are given
•-/
with the condition v = u when t = 0.
Now one function of t having the constant f for its differential
coefficient is the function ft, and the most general expression for
a function having this differential coefficient is ft + C, where C is
an arbitrary constant. Hence v must be of the form ft + C.
Putting t = 0, we find u = C, so that the constant is determined.
Hence v = u -\-ft, or x = u +ft.
Again one function of t having the function u +ft for its
differential coefficient is ut + ^ft2, hence x must be of the form
C' + ut + ^ft'2, where C' is an arbitrary constant.
Putting £ = 0, we find x0—C', so that the constant is deter-
mined.
Hence x = ac0 + ut + \f&.
If s is the distance described in the interval t, s is x — x0, so
that
s = ut 4- \f&.
By elimination of t between this equation and the equation
v = u +ft, we find
vz - u- = 2/s.
In particular, the velocity acquired in moving from rest over a
distance s is \/(2/s). This is described as the " velocity due to falling
through s with an acceleration f,"
32. Examples.
1. Prove that, when the acceleration is uniform, the average velocity in
any interval of time is the velocity at the middle of the interval.
2. Obtain the formula v2-w2 = 2/s by multiplying both sides of the
equation x =f by x and integrating.
3. Let the distance s be divided into a great number of equal segments,
and the sum of the velocities after describing those segments divided by their
number, a velocity will be obtained which will have a limit when the number
of segments is increased indefinitely, and this limit may be called the average
velocity in the distance. Prove that, when the initial velocity is zero, this
average velocity is equal to | of the final velocity.
33. Parabolic motion under gravity. When a particle
moving in the field of the Earth's gravity, near a place on the Earth's
31-33]
MOTION UNDER GRAVITY
29
surface, does not move vertically, it has a component velocity in a
horizontal direction. We prove that the particle describes a para-
bola with a vertical axis.
Let the axis of y be drawn vertically upwards, and let the plane
(x, y} be the vertical plane through the initial direction of motion.
Since the acceleration parallel to the axis z is always zero, the
particle does not acquire velocity parallel to this axis; and, since
at time £ = 0 it has no velocity parallel to this axis, it undergoes
no displacement parallel to this axis ; thus the particle moves in
the plane (x, y).
At time t = 0 let the velocity of the particle be V in a direction
making an angle a with the axis x (see Fig. 10).
PUS. 19.
We have the equations x = 0,
y=-g>
with the conditions that when t = 0,
x — V cos a, y = Fsin a.
The equation x = 0 shows that x is constant ; and, since
x = Fcos a when t = 0,
the constant value of x is Fcos a. Thus the horizontal component
of the velocity is constant.
The equation y = — g shows that y must be of the form
— gt + const. ;
and, since if = V sin a when t = 0, y = Fsin a — gt.
Let the coordinates of the position of the particle at the instant
when t — 0 be x0, y0.
30 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
The equation x = V cos a shows that x must be of the form
Vt cos a + const.; and, since x = x0 when t = 0, we have
x = x0 + Vt cos or.
The equation y = V sin a — gt shows that y must be of the form
Vt sin a — £ gt* + const.; and, since y = y0 when t = 0, we have
y — Wo + Vt sin a — i gt2.
Thus the coordinates x and y are expressed in terms of t. Observing
that the formula for y can be written
F2sin2a 1 . .
we can eliminate t, and obtain the equation of the path of the
particle in the form
F2sin2a !-/«-. x -
y — y0= — rt-— — — ( V sin a — a •== —
2# 2/7 \ y Fcos «
showing that the path of the particle is a parabola with a vertical
axis*.
34. Examples.
1. Prove that the vertex of the path is reached at time Fsin a/g, that its
coordinates are x0+( Vz/g) sin a cos a, y0 + ( V2/2g) sin2 a, arid that, if the path
is referred to axes of #', y' with origin at the vertex and axis of y' drawn
vertically downwards (Fig. 19), the coordinates at time £', measured from the
instant of passing the vertex as initial instant, are given by the equations
2. Find the length of the latus rectum of the parabolic trajectory, and
determine its focus and directrix.
3. If v is the velocity at any point of the path, show that the point is at a
distance t>2/2</ below the directrix.
4. Prove that the time until the particle is again in the horizontal plane
through the point of projection is (2 Fsin a)/g. [This is called the time of flight
on the horizontal plane through the point of projection.]
5. Prove that the distance from the starting point of the point where the
particle strikes the horizontal through the starting point is ( F-' sin 2a)/g. [This
is called the range on the horizontal plane through the point of projection.]
6. To find the range and time of flight on an inclined plane through the
point of projection. Let 6 be the inclination of the plane to the horizon.
Resolve up the plane, and at right angles to it. The resolved accelerations
are
The result was discovered by Galileo.
33, 34] MOTION UNDER GRAVITY 31
the resolved initial velocities are
Fcos(a-<9), Fsin(a-<9);
the resolved velocities at time t are
Fcos (a - 0) - gt sin 6, Fsin (a - 6} - gt cos 6 ;
the distances described in time t parallel and perpendicular to the inclined
plane are
Vt cos (a - 6} - %gt- sin <9, Vt sin (a - <9) - \gP cos 6.
K
Fig. 20.
The time of flight is obtained by making the second of these equal to zero, it is
2 Fsin (a -0)
gcosd
The range is found by substituting this value for t in Vt cos (a - 0) — %gtz sin 6.
Prove that the range in question is
3— (tan a — tan 0),
gcosd
-and that this is the same as
F2
g cos2 0
[sin (2a - 6} - sin ff\.
7. Prove that, when the velocity of projection is given, the range on an
inclined plane is greatest when the direction of projection bisects the angle
between the plane and the vertical.
8. Show that, if a parabola is constructed having its focus at the point of
projection S, its axis vertical, and its vertex at a height F2/2<? above the point
of projection, then the parabolic path for which the range on a line through S
is greatest touches this parabola at the point where the line cuts it.
[From this it follows that all possible paths of particles moving with
uniform acceleration g downwards, and starting from a point S with given
velocity F, touch a paraboloid of revolution about the vertical through S
having its focus at S. This paraboloid is the envelope of the trajectories of
such particles.]
9. A particle is to be projected from the origin with a given velocity F
so as to pass through a given point (x, y), the axes of coordinates being the
same as in Art. 33. Prove that the direction of projection must make with
the axis x an angle a which satisfies the equation
gxz tan2 a - 2 V*x tan a + (2 V*y+ga?) = 0,
and hence show that there are, in general, two directions in which the particle
32 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
can be projected, with given velocity, from one given point, so as to pass
through another given point.
[Clearly the point (x, y] must lie within the parabola 2V'2y+gx* = V*/gf
which is the envelope considered in Ex. 8.]
10. Prove that, in the different trajectories possible under gravity between
two points A, B, the times of flight are inversely proportional to the velocities-
of the projectile when vertically over the middle point of AB.
11. A particle moves under gravity from the highest point of a sphere of
I'adius c. Prove that it cannot clear the sphere unless its initial velocity
exceeds J(\gc).
12. Prove that the greatest range on an inclined plane through the point
of projection is equal to the distance through which the particle would fall
during the time of flight.
35. Motion in a curved path. When the motion of a body,
treated as a particle, is observed, the things that can be observed
are the positions of the particle at different times. The aggregate
of these positions constitutes the path of the particle. For example,
the path may be a circle, and equal arcs may be described in equal
times. In such cases we have the mathematical problem of deducing
the acceleration of the particle from the observations, that is to say
the problem of determining the direction and intensity of the field
offeree. Conversely we may set before ourselves the problem : Given
the acceleration of the . particle, to determine its path and its
positions at different times. . The solutions of such problems are
facilitated by a theorem of kinematics to which we proceed.
36. Acceleration of a point describing a plane curve.
Let a particle move in the plane of (x, y}.
Let v be the velocity at any point P of the path, v the velocity
at a neighbouring point Q, and A$ the angle QTA between the
Fig. 21.
tangent at P and the tangent at Q. Also let A£ be the time taken
by the particle to move from P to Q, and let As be the length of
the arc PQ.
34-37] MOTION IN A CURVED PATH 33
The velocity at Q can be resolved into components v' cos A<£ in
the direction of the tangent at P and v' sin A<£ in the direction of
the normal at P.
Hence the acceleration in the direction of the tangent at P is
the limit of - -- when A£ is diminished indefinitely. Now
v' cos A<£ — v _v' — v ,1— cos A$
~~~ ~
A
The limits of the three factors of this expression are |, <£, zero.
/V-y
Hence the above limit is -^- or v. Since we have
at
dv _dv ds _ dv
dt ds ' dt ds'
we may write v -=- for the component acceleration parallel to the
as
tangent, and we may also write s for it, since v is s.
Again the acceleration in the direction of the normal at P is
the limit of ~^~f > an(^ this is the same as the limit of
, sin A<6 A<f> As
v -
A$ As A£ '
and the limits of these factors in order are v, 1, - , v, where p is the
P
radius of curvature of the curve at P. Thus_the_ acceleration in
the direction of the normal drawn towards the centre of curvature
. v2 s2
is - or - .
P P
37. Examples.
1. A particle describing a circle of radius a with velocity v has an accele-
ration v2la along the radius directed inwards.
If the radius vector drawn from the centre to the particle turns through
an angle 6 in time t, the acceleration of the particle has components a02 along
the radius (directed towards the centre) and ad along the tangent in the sense
of increase of 6.
2. Verify the result that, in parabolic motion of a projectile under gravity,
the value of v2/p at any point of the path is equal to the resolved part along
the normal to the path of an acceleration equal to g.
L. M. 3
34 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
3. Assuming this result, and that the horizontal component of the velocity
is constant, deduce the result that the path is a parabola.
4. Interpret the formula v2/p for the normal component acceleration so
as to show that the velocity, at any point P, of a particle describing a curved
path, in any field of force, is equal to that due to falling through one quarter
of the chord of curvature at P, drawn in the direction of the field, with an
acceleration equal to the intensity of the field at P.
38. Simple harmonic motion. A point moving in a straight
line in such a way that its displacement from a fixed point at time
t can be expressed in the form
a cos (nt + e),
where a, n, e are any real constants, is said to have a " simple
harmonic motion."
Let the straight line be the axis of x, and the fixed point the
origin. Then we have
x = a cos {nt + e),
and therefore x = — rilx.
We shall now show^ that, if the acceleration is connected with
the displacement by an equation of the form
X — — /JiX,
where /i is a positive constant, the motion is simple harmonic
motion.
Multiply both sides of the equation by x. Observe that
^ = ^ (&**)> and ^ = ^(K).
so that -,- (| a? -f \ /j,x2) = 0.
Hence a? + ^ is constant. Since x* + pji? is necessarily positive,
we may take the constant value of it to be pa2, where a is real, and
may be taken to be positive. Then
& = p (a2 - a-2).
Since x1 is necessarily positive, this equation shows that the value
of x cannot be greater than a or less than — a. We may therefore
introduce a real variable 6, in place of x, by the equation
x = a cos 0.
Then the equation x- = //. (a2 — x*) becomes
& = p,
which gives 6 = ± (t vV* + e),
37-39] SIMPLE HARMONIC MOTION 35
where e is an arbitrary constant. We have thus obtained the com-
plete primitive of the equation x — — px in the form
x = a cos (t \lfj, + e).
This equation represents a simple harmonic motion. The motion
is periodic, that is to say, it repeats itself after equal intervals of
time. The period is 2?r/\//*. In the formula for x the constant a is
called the " amplitude " of the motion, and the constant e deter-
mines the " phase " of the motion.
On putting
a cos e = A , - a sin e = B,
the formula for x becomes
x=A cos (t ^Jfi) + B sin (t vV)>
which is another form of the complete primitive of the equation x = —fix.
Let the moving point have at time t=0 a position denoted by x0 and a
velocity denoted by x0. In the formula last written put t = 0, then x0= A.
Again differentiate both sides of the formula with respect to t, and in the
result put ( = 0, then x0 =B x//x. Hence the formula may be written
x= XQ cos (t v//x) H — j- sin (t *Jfi).
Simple harmonic motion may be regarded as the type of to-aud-fro, or
oscillatory, motion. Oscillatory motions can generally be described either as
simple harmonic motions or as motions compounded of simple harmonic
motions in different directions.
39. Composition of simple harmonic motions. We consider
the case where the moving particle has a simple harmonic motion
2_
of period -j- parallel to each of the axes of x and y, the acceleration
Vr*
in each case being directed towards the origin.
We have the equations x = — fix,
y = -t*y,
and we deduce that x and y must be given by equations of the form
x = A cos (t V/A) + B sin
y=C cos (t V/A) + D sin
where A, B, C, D are arbitrary constants depending on the initial
conditions, viz. A and C are the coordinates, and B ^p, D -y/yu, the
resolved velocities at the instant t — 0.
Solving the above equations for cos (t vX) and sin (t V^)> we have
3—2
36 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
( AD - BC) cos (t vV) = Dx- By, (AD - BC) sin (t y» = Ay- Cue ;
eliminating t, we find
(Das - Byf + (Ay- Gxf = (AD- BC)2,
so that the path of the moving point is an ellipse whose centre is
the origin, and whose position with reference to the origin and axes
o
is fixed. The whole motion is clearly periodic with period -.- .
Y/u,
Let us change the axes to the principal axes of the ellipse, and
suppose the moving point to be at one extremity (x = a) of the major
axis at the instant t = 0, then at this instant x = a, y = 0, and, since
the point is moving at right angles to the major axis, x = 0. Let
y = b \//4 at this instant. Then we must have at time t
x = a cos (t V/*)' y = ^ sin (^ V/-0-
Thus 26 is the minor axis, and t vV is the eccentric angle at time t.
The point therefore moves so that its eccentric angle increases
uniformly with angular velocity ^/j,.
40. Examples.
1. Prove that, if a point N moves on a fixed diameter of a circle (Fig. 22)
so that its acceleration is given by
the equation x = -fix, a point P on
the circle, whose projection on the
fixed diameter ' is iV, describes the
circle with constant speed.
2. Prove that, when the equation
is x=f*x) where p is positive, and the
initial conditions are that .v=x0 and
x=x0 when £ = 0, then at any time t
3. Prove that when the accelera-
tion of a particle moving in a plane
is directed from the origin and is pro-
portional to the distance the path is
an hyperbola.
4. In the elliptic motion of Art. 39 prove that the velocity v at distance r
from the centre is given by
v2 + pr2 = const.,
and evaluate the constant.
5. In the hyperbolic motion of Ex. 2 prove that the velocity v at distance r
from the centre of the hyperbola is given by
v'2=fj.r- + const.,
and evaluate the constant.
39-42]
EQUABLE DESCRIPTION OF AREAS
37
41. Kepler's laws of planetary motion. From a long series
of observations of the Planets, and more especially of Mars, which
were made by Tycho Brahe, Kepler* concluded that the motions
of the Planets could be very precisely described by means of the
two laws : —
(i) Every planet describes an ellipse having the Sun at a
focus.
(ii) The radius drawn from the Sun to a Planet describes
equal areas in equal times.
42. Equable description of areas. We consider the second
of Kepler's laws, and suppose that a
particle describes a plane curve in
such a way that the radius vector
drawn to it from a fixed point in the
plane describes area uniformly. In c
Fig. 23 0 represents the fixed point,
B any fixed point on the curve, P
the position of the particle at time t,
r the radius vector OP, p the per-
pendicular from 0 on the tangent at
P, v the velocity of the particle at P.
Let P' be a point on the curve
near to P, A£ the time of moving
from P to P', As the arc PP', Ac
the chord PP', q the perpendicular
from 0 to this chord. The area of the triangle POP' is
Hence the rate of description of area is the limit of \
Ac As
\q — -r- : and this limit is ^ps or ^pv. If therefore we write
pv = h,
h is twice the rate of description of area, and the condition that
the radius vector describes area uniformly is expressed by saying
that h or pv is constant.
Now pv is the moment of the velocity about 0. If therefore
Ac.
Ac
—
Oat
or
* Joannes Kepler, Astronomia nova...tradita Commentariis de Motibus Stella
Martin, 1609.
38 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
we take 0 to be the origin of coordinates and draw the axes of x
and y in the plane of motion, we have (cf. Article 22)
pv = Xy - yx = h ;
and, since this is constant, we have -^ (xy — yx) = 0, or xy — yx = 0,
ctt
and therefore
It follows that the direction of the acceleration is that of the
radius vector, drawn from or towards the origin. We conclude that,
if a particle moves in a plane path, so that the radius vector drawn
to it from a fixed point describes area uniformly, it is in a field of
force, and the direction of the field at any point is either directly
towards or directly away from the fixed point. Such a field of force
is described as " central," the fixed point being the "centre offeree,"
and the path of the particle is a " central orbit."
In the motion discussed in Article 39 the ellipse is a central
orbit, and the centre of the ellipse is the centre of force.
Kepler's second law of planetary motion may be interpreted in
the statement that the Planets move in a central field of force, the
centre of force being in the Sun.
43. Radial and transverse components of velocity and
acceleration. Let a particle move in the plane of (x, y) and let
Fig. 24.
r, 6 be the polar coordinates of its position at time t. It is required
to express, in terms of r, 6 and their differential coefficients with
42-44] RADIAL AND TRANSVERSAL RESOLUTION 39
respect to t, the components of the velocity and acceleration in the
direction of the radius vector and at right angles to it. The senses
are to be those in which r and 9 increase, as in Fig. 24.
Let vl} v2 be the required components of velocity. Then x, y
are the components parallel to the axes of x, y of the same velocity.
We have therefore
vl cos 6 — v2 sin 0 = x = -j- (r cos 9) = r cos 9 — r9 sin 9,
vl sin 9 -f v2 cos 9 = y = -r (r sin 9) = r sin 9 + r9 cos 9.
Solving these equations, we find
^/a be the required components of acceleration. We have
in like manner
/, cos 9 -/2 sin 9 = x = T- (r cos 9}
= r cos 9 - 2f# sin 9 - r'9 sin 9 - r9'2 cos 0,
d?
/i sin 9 +/2 cos 9 = y = -7- (r sin 9}
= r sin 9 + 2r0 cos 0 + r9 cos 0 - r92 sin 0.
Solving these equations, we find
It is important to observe that the acceleration parallel to the radius
vector is the resolved part along the radius vector of the acceleration relative
to the frame Ox, Oy ; it is riot the acceleration with which the radius vector
increases.
44. Examples.
1 . Since the moment of the velocity about the origin is r . rft, we verify
the formulae of Differential Calculus
2. In a central orbit we have
h=r20.
3. A point P describes a curve C relatively to axes through 0. Prove
that, relatively to parallel axes through P, 0 describes a curve equal in all
respects to 6', and that any point dividing OP in a constant ratio describes,
relatively to either of these sets of axes, a curve similar to C.
40 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
45. Acceleration in central orbit. Let /be the magnitude
of the central acceleration at P, and let it be directed towards P.
Let r, p, p denote the radius vector OP drawn from the centre of
force 0, the perpendicular from 0 on the tangent at P, and the
radius of curvature of the path at P. (Cf. Fig. 23 in Art. 42.)
The resolved part of the acceleration parallel to the normal at
PiBf*.
J r
v"
But this resolved part of the acceleration is — .
P
TT V* fP
Hence - = / - .
p J r
From this equation and the equation vp = h \ve may eliminate
v, and obtain the equation
v*
Since p = r -7- , we may also write this equation
fr- d£
^ ps dr '
46. Examples.
1. Show that, when the orbit is an ellipse described about the centre, the
acceleration is proportional to the radius vector.
2. In the same case show that the velocity at any point is proportional
to the length of the diameter conjugate to the diameter through the point.
3. Points move from a position P with a velocity V in different directions
with an acceleration to a point C proportional to the distance. Prove that all
the elliptic trajectories described have the same director circle.
Let the tangent at P to one of the trajectories meet the director circle in
T, and let Q be the point of contact of the other tangent to this trajectory
drawn from T. Prove that the trajectory in question touches at Q an ellipse
having C as centre, and P as one focus, and that 2CT is the length of the
major axis of this ellipse.
[This ellipse is the envelope of the trajectories of points starting from P
with the given velocity and moving about C with the given central acceleration.]
4. Show that the central acceleration when a circle is described as a central
orbit about a point on the circumference is 8h2a?jr°, a being the radius of the
circle.
5. Show that the central acceleration when an equiangular spiral is de-
scribed as a central orbit about its pole is proportional to r~ 3.
45-47] ACCELERATION IN CENTRAL ORBIT 41
6. Show that, for an ellipse described as a central orbit about any point 0
in its plane, the central acceleration at any point P is proportional to rjq\
where r is the radius vector OP, and q is the perpendicular from P on the
polar of 0.
47. Elliptic motion about a focus. We consider now the
interpretation of the first of Kepler's laws (Art. 41). Let an ellipse
of semi-axes a, b be described as a central orbit about a focus S.
Let 8' be the second focus, e the eccentricity, 21 the latus rectum.
Let P be any point on the ellipse ; let r and r be the radii
vectores drawn from S and S' to P ; let p and p be the perpen-
diculars from S and S' on the tangent at P ; let C be the centre,
and CD the semi-diameter conjugate to OP.
Fig. 25.
Then
p = CD3/ab, rr'=CD\ pp'=b*, r + r'=2a, b* = al.
Also, since £ SPY = tS'PY', we have
f) t)' I T^f) b
- =*-T , and therefore each of these = A /•"r = 7^-rv •
r r V rr CD
Now the acceleration, /, is given by
AV
P3P
_ fcrab iCD\3 _h?a_h?_
~ CLP \br) ~7*b*~rH'
Thus the acceleration varies inversely as the square of the
distance r, and, if we write ji/r* for it, we have h? = pi.
Accordingly Kepler's first and second laws of planetary motion
may be interpreted in the statement that the field of force in which
42 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
the Planets move is directed radially towards the Sun, and the in-
tensity of the field varies inversely as the square of the distance from
the Sun. The field is described as that of the Sun's gravitation.
48. Examples.
1. Prove that, if any conic is described as a central orbit about a focus,
the acceleration is/u/r2 towards the focus, and /x = /<2/£.
Prove also that when the conic is a parabola v2 = 2/i/r, and when it i.s. an
hyperbola v2=/
/2 1
=n( ---
r «
2. Prove that the velocity v at any point of the ellipse is given by the
equation
2 1
r
3. Prove that in elliptic motion about a focus *S the velocity at any point
P is perpendicular and proportional to the radius vector from the other focus
to the point W, where SP produced meets a circle centre S and radius 2a.
[From the formula in Ex. 2, this circle isffe,lled the " circle of no velocity."]
4. Prove that the velocity at P can be resolved into two constant com-
ponents, one at right angles to the radius vector SP, and the other at right
angles to the major axis.
5. The periodic time in which the ellipse is described is
6. To find the time of describing any arc of the ellipse.
Draw the auxiliary circle A QA '.
Fig. 26.
Let </>, =LQCA in the figure, be the eccentric angle of P, and 0, =LASP>
the vectorial angle.
Then curvilinear area ASP= curvilinear area A NP - triangle SPN
= - (curvilinear area A NQ} — triangle SPN-
47, 48] ELLIPTIC MOTION ABOUT A FOCUS 43
Now curvilinear area ANQ= sector ACQ- triangle CQN
= 5 (tt20 — a2 sin <p cos 0),
and triangle SPjV= A b sin 0 (ae — a cos <f>).
Hence curvilinear area ASP=^ab ((f) — e sin <£).
Let t be the time from A to /*, then, since h is twice the area described per
unit of time,
lit = ab((f) — e sin <£).
3
rru 0* , • JN
Inus £= — (0 — esm<p).
The quantity vW«^ is known as the " mean motion " and is denoted by n,
so that the time in question is given by
nt = (f) — esin <f>.
By putting (f) — 2ir we find the periodic time, as in Ex. 5.
Prove that 6 is connected with </> by the equation
e + cos 6 , ,, , .„ .
cos m = , and that, it e is small,
1 + e cos Q
d = nt + 2e sin nt approximately.
7. (i) In an ellipse described about a focus S let (f>1, <f>2 be the eccentric
angles of two points P1} P2> an(l let <£i>02- Let & = £ (0i — </>2)> an(i le^ °" be
such that cos o- = e cos ^ (0j + {/>.j) and sin o- sin 8 is positive. Prove that, if?'i,
r2 are the focal radii SPi, SP2, d is the chord PiP2, and t the time of
describing the arc PiP2, then
r\ + r2 + d=2a{l -cos(o- + 8)}, r1 + r2-o?=2a (1 -cos(o--S)},
and nt = YI — YO
where sin iXi = i V{to + »z + <*)/«}, sin 1 X2 = 1 v'{(r! + r2 - d)ja}.
(ii) By taking in Ex. 7 (i) both ^j and <£2 to lie between - \tr and ^TT, prove
that the point of intersection of Pl P2 with the major axis lies on CS or CS
produced according as <r<8 or a->8; and, by passing to the limit when the
ellipse becomes a parabola, prove that the time t of describing the arc PI P2
of a parabolic orbit about S is given by
6t V/i = (T! + r2 + rf)* ± (rt + r2 - d)\
where the upper or lower sign is to be taken according as S lies within the
finite area bounded by P1 P2 and the parabola or not.
8. Two points describe ellipses of latera recta I and I' in different planes
about a common focus, and the accelerations to the focus are equal when the
distances are equal. Show that, when the relative velocity of the points is
along the line joining them, the tangents to the ellipses at the positions of the
points meet the line of intersection of the planes in the same point, and that
the focal distances, r and /, make with this line angles 6 and & such that
r sin 6 r' sin 6'
44 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
49. Inverse problem of central orbits. In regard to the
problem : Given the field of force to find the orbit — we prove a
general theorem as follows : — The path of a particle moving in a
central field of force is in a plane through the centre of force, and the
radius vector draivnfrom the centre of force to the particle describes
equal areas in equal times.
At any instant, chosen as initial instant, let a plane be drawn
through the tangent to the path of the particle and the centre of
force. Let this be the plane (#, y\ and let the centre of force be
the origin. Then at the initial instant z and z vanish.
Since the acceleration is directed along the radius vector we
have
? = y=f
x y z'
or yz — zy = 0, zx — xz = 0, xy — yx = 0.
Hence, by integration,
yz — zy = const., zx — xz = const., xy — yx = const.
The first two constants of integration vanish because z and z vanish
initially. If the third also vanishes, the velocity is directed along
the radius vector, and the particle moves in a straight line. We
omit, for the present, the case of rectilinear motion (see Art. 54).
We may consider the equations
xz — xz = 0, yz — yz = 0
as simultaneous equations to determine z and z. If xy — xy does
not vanish, these equations can only be satisfied by putting z and z
equal to zero. Hence z is always zero, and the particle moves in
the plane (x, y).
Since xy — yx, or the moment of the velocity, is constant, the
rate of description of area by the radius vector is constant ; for we
saw in Art. 42 that this rate, whether constant or not, is always
half the moment of the velocity about the origin.
50. Determination of central orbits in a given field. The
tangential component of the acceleration of a particle describing
dv
any path can be expressed as v -j- (Art. 36). When the acceleration
is of magnitude /, and is directed towards the origin, the tangential
49-5l] INVERSE PROBLEM OF CENTRAL ORBITS 45
cZ/* d')*
component is —f~r-> f°r ~T~ i§ the cosine of the angle between the
tangent and the radius vector drawn from the origin. We have
therefore the equation
dv _ ,dr
Vds~~fds'
When/ is a function of r, this equation can be integrated in the
form
fc (2)
where J. is a constant. Now, according to Art. 43, we have
and we have also, by Ex. 2 in Art. 44,
r*6 = h,
Hence we may write
**&*-*&•
and equation (2) becomes
h2 /dr\2 h2 n f , 7
— f^J + — =2A-2lfdr.
If u is written for 1/r, this equation becomes
2
in which f is supposed to be expressed as a function of u. By this
du
equation we can express -^ as a function of u, and then by inte-
gration we can find the polar equation of the path.
It is often more convenient to eliminate A from equation (3) by
differentiating with respect to 6. This process gives the equation
d*u f
51. Orbits described with a central acceleration vary-
ing inversely as the square of the distance. When /= ^w2
equation (4) of Art. 50 becomes
d*u t 1
46 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
where I is a constant. To integrate this equation we put
u
then w satisfies the equation
1
u — + iv,
The complete primitive of this equation is of the form (cf.
Art. 38)
w =*A cos (6 — e),
where A and e are arbitrary constants. We write e/l for A. Then
the most general possible form for u is
u = j {1 + ecos(0-e)}.
v
Hence all the orbits that can be described with central accele-
ration equal to //,/r2 are included in the equation
- = 1 + e cos (6 — e),
r
in which e and e are arbitrary constants, and / is equal to A2//x.
The possible orbits are conies having the origin as a focus, and
the latus rectum is equal to 21 or 2A,2/At-
According to the results of Examples 1 and 2 in Art. 48, the
conic is an ellipse, parabola or hyperbola according as the velocity
at a distance r is less than, equal to, or greater than ^(2/jb/r).
52. Additional Examples of the determination of central
orbits in given fields.
1. If / is any function of r, any circle described about the centre is a
possible orbit.
2. lff=nr equation (3) of Art. 50 gives
Hence prove that, when /x is positive, all the possible orbits are ellipses having
the centre of force as centre.
3. To find all the orbits which can be described with a central acceleration
varying inversely as the cube of the distance.
If f—fjiU3 equation (4) of Art. 50 gives
51-53] DETERMINATION OF CENTRAL ORBITS 47
There are three cases according as 7«2>, =, or <p..
(1) When 7i2>/x, 1 — j^ is positive, put it equal to n2.
Then all the possible orbits are of the form u = A cos (n6 + a).
(2) When A2=w, we have -r^r = 0, so that u = A6 + B, where A and B are
dd*
arbitrary constants. If ^1 = 0 the orbit is a circle, otherwise it is a hyperbolic
spiral, as we see by choosing the constant B so as to write the above
(3) When 7t2</x, 1 - ^ is negative, put it equal to -n2.
Then all the possible orbits are of the form
u = A cosh (nd + a) or u = ae'6 + be~"8.
Putting a or b equal to zero we have an equiangular spiral.
4. Deduce the equation
from the equation
5. From the equations
•which are obtained from the results of Art. 43, deduce the results
0=w, y^+»=—
53. Conic described about a focus. Focal chord of curva-
ture. Let a particle be projected from a point P, with a given velocity V, in
an assigned direction P77, and move
in a central field of force, directed
to S, and equal to p/r2 at a distance r
from S. It describes a conic with a
focus at S. Let PQ be the focal chord
of curvature of this conic at P. Then,
according to the result of Ex. 4 in
Art. 37,
and according to the results of Exx.
1 and 2 in Art. 48, the orbit is an
ellipse, parabola, or hyperbola accord-
ing as V2 is less than, equal to, or „.
greater than 2/*/»SrP. We conclude
that, when a point P, the tangent PT, a focus S, and the focal chord of
48 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
curvature PQ are given, one conic, and one only, can be described, and this conic
is an ellipse, parabola, or hyperbola according as SP> , =, or < ^PQ. This
may be proved directly as follows : —
Let U be the middle point of PQ. Draw PG at right angles to PT, and
UG parallel to PT; draw UO and GK at right angles to SP meeting PG and
SP in 0 and K respectively.
Then by similar triangles OPU, UPG, GPK we have
OP : PU=PU : PG=PG : PK.
PG3
Whence OP=-~.
rft.-
Now describe a conic with focus S and axis SG to touch PT at P, G
is the foot of the normal, and PK is half the latus rectum. Hence 0 is the
centre of curvature.
Since SG : SP= eccentricity, the conic is determinate and unique.
Since a semicircle on PU as diameter passes through G, we have when
SP>%PU, SG<SP; when SP<$PU, SG>SP; when SP=\PU, SG=SP.
Thus the conic is an ellipse, parabola, or hyperbola according as
SP>, =,or
54. Law of inverse square. Rectilinear motion. Let a
particle move in a straight line, taken as axis of x, with an accelera-
tion directed to a fixed point of the line, taken as origin, and equal
to ft/a? at distance x. We have the equation
Multiplying both sides of this equation by x, we have an equation
which may be written
and, on integrating this equation, we have
where G is an arbitrary constant.
Let the particle start from rest at the point specified by x = 2af
where a is positive, at the instant when t = 0. Then we have
C=-/i/(2a), and
/2 IN
a?= n --- =
\oc aj
ax
This equation shows that, so long as x is positive, it is less than 2a,
53-55]
LAW OF INVERSE SQUARE
49
and we may introduce a new variable 6, in place of x, by the
equation
x = 2a cos2 6,
where Q = 0 when t = 0. The equation for x becomes
i6a3 cos4 e . er- = ^
giving 4a^ cos2 0 . 6 = + «//*•
Since x diminishes as t increases, a; is negative and 0 is positive,
so the upper sign must be taken. Then the equation gives
where no constant is added because 9 = 0 when t = 0.
Here x is not expressed in terms of t but x and t are both
expressed in terms of a parameter 0.
It is to be observed that, if the law of the acceleration could remain
unchanged until x vanishes, or the particle reaches the origin, the velocity
would become infinite. In a physically possible system, by which the
acceleration could be produced, either the particle could not reach the origin
or the law of acceleration would have to change. See Art. 150 and Ex. 3 in
Art. 177 infra.
55. Examples.
1. The results obtained in Art. 54 may also be found as follows : — Let 0 be
the point to which the acceleration
is directed, A the starting point. On
OA as diameter describe a circle,
and let a point P describe this
circle under a central force directed
to 0. Let N be the foot of the per-
pendicular from P to OA (Fig. 28).
The acceleration of P is 8haa*/OP*t
where h is twice the rate at which
OP describes areas about 0. (See
Ex. 4 in Art. 46.) The acceleration
of N is the resolved part of this in
the direction A 0. Prove that it is
A2/(« . ON2). Observing that ht=
twice the curvilinear area AOP,
and taking the angle AOP to be
0, deduce the results Fig. 28.
2. Prove that, if the law of acceleration remains unchanged until #=0,
the time of moving from A to 0 is rra
L. M.
50 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
56. Field of the Earth's gravitation. It is consonant with
observations of falling bodies to state that the field of force around
the Earth is central, and the acceleration of a free body in this
field is directed towards the centre of the Earth. The Moon
describes a nearly circular orbit about the Earth, in a period of
about 27| days; this motion is nearly uniform, and the distance of
the Moon from the Earth is about 60 times the radius of the
Earth. Now the central acceleration of a particle describing a
circular orbit of radius R uniformly in time T is — ^— ; and, if the
radius is 60 times the Earth's radius (3980 miles), and the period
is 27 J days, this acceleration, when expressed in foot-second units,
32'1
is equal to approximately. Thus the Moon moves around the
Earth in nearly the same way as if it were under gravity diminished
in the ratio 1 : (60)2.
From this result we conclude that the field of force around the
Earth extends to the Moon, and that the intensity of this field, like
that of the field around the Sun, varies inversely as the square of
the distance.
For bodies in the neighbourhood of the Earth there is a cor-
rection of gravity due to height above the Earth's surface. If g is
the acceleration due to gravity at the surface, and a the Earth's
radius, the acceleration due to gravity at a height h above the
surface is
ga?/(a + h)\
There are other corrections of gravity at least as important as
that here mentioned. The most important, depending upon the
Earth's rotation, will occupy us in Chapter X.
57. Examples.
1. The envelope of the elliptic orbits described by particles, which start
from a point P with velocity F, and move with an acceleration directed
towards a point S and varying inversely as the square of the distance, is
an ellipse, which has S and P as foci, and touches any of the trajectories
at the point where the line drawn from P to the second focus of the trajectory
meets it.
2. Show that a gun at the s.ea level can command l/n- of the Earth's
surface if the greatest height to which it can send a shot is l/n of the
Earth's radius, variations of gravity due to altitude being taken into
account.
56, 57] MISCELLANEOUS EXAMPLES 5 1
3. Prove that the time in which a particle falls to the Earth's surface
from a height k is ( — ) (l+jr: 1 approximately, a being the Earth's radius
\ y / \ /
and (A/a)2 being neglected.
MISCELLANEOUS EXAMPLES
1. Prove that the time in which it is possible to cross a road of breadth c,
in a straight line, with the least uniform velocity, between a stream of
omnibuses of breadth 6, following at intervals a, moving with velocity F, is
2. A straight line AB turns with uniform angular velocity about a point
A, retaining a constant length, and a second straight line BC, also of constant
length, moves so that C is always in a certain straight line through A. Prove
that the velocity of C is proportional to the intercept which BC makes on the
line through A at right angles to AC.
3. A point C describes a circle of radius r with angular velocity w' about
the centre 0, and a point P moves so that CP is always equal to a and turns
with angular velocity o> in the plane of the circle described by C. Prove that
the angular velocity of OP is
where R is the length of OP.
4. Two particles start simultaneously from the same point and move
along two straight lines, one with uniform velocity, the other with uniform
acceleration. Prove that the line joining the particles at any time touches a
fixed parabola.
5. A body is projected vertically upwards with velocity v ; after a time t
a second body is projected vertically with velocity v' (<v). If they meet as
soon as possible after the instant when the first was projected
6. Two particles describe the same parabola under gravity. Prove that
the intersection of the tangents at their positions at any instant describes a
coaxial parabola as if under gravity. Prove also that, if T is the interval
between the instants when they pass through the vertex, the distance
between the vertices of the two parabolas is J^r2.
7. Prove that the angular velocity of a projectile about the focus of its
path varies inversely as its distance from the focus.
8. A particle is projected from a platform with velocity V and elevation
P. On the platform is a telescope fixed at elevation a. The platform moves
horizontally in the plane of the particle's motion, so as to keep the particle
always in the centre of the field of view of the telescope. Show that the
original velocity of the telescope must be Fsin (a — 0) cosec a, and its accelera-
tion g cot a.
4—2
52 MOTION OF A FflEE PARTICLE IN A FIELD OF FORCE [CH. II
9. A cricketer iii the long field has to judge a catch which he can secure
with equal ease at any height from the ground between kl and kz ; show that
he must estimate his position within a length
/zy(i-*8/A)-V(i-*i/A)},
where 2R is the range on the horizontal and h the greatest height the ball
attains.
10. A heavy particle is projected from a point A so as to pass through
another point B ; show that the least velocity with which this is possible is
*J(2gl) cos £a, and that the highest point of the path is at a height I cos* \a
above A, where AB=l and makes an angle a with the vertical.
11. A man travelling round a circle of radius a with speed r throws a
ball from his hand at a height h above the ground, with a relative velocity F,
so that it alights at the centre of the circle. Show that the least possible
value of V is given by F2= v*+g {^(a2 + A2) - h}.
12. If A and B are two given points, and C any given point on the line
joining them, prove that, in the different trajectories possible under gravity
between A and B, the time of flight varies as x/((7/>), where D is the point
in which the trajectory meets the vertical through C.
13. A gun is placed on a fort situated on a hill side of inclination a to
the horizon. Show that the area commanded by it is 4nh (h + dcos a) sec3 a,
where >J(^gh} is the muzzle-velocity of the shot, and d the perpendicular
distance of the gun from the hill side.
14. It is required to throw a ball from a given point with a given velocity
V so as to strike a vertical wall above a horizontal line on the wall. When
the ball is projected in the vertical plane at right angles to the wall, the
elevation must lie between 6± and #2. Prove that the points on the wall
towards which the ball may be directly projected lie within a circle of radius
V* sin (0j - 6$l{g sin (6l + 02)}.
15. Water issues from a fountain jet in such a manner that the velocity
of emission in a direction making an angle 6 with the vertical is >J(ga cosec 0),
the jet being at the height h above the centre of a circular basin. Prove that,
if all the water is to fall into the basin, its radius must not be less than
ESo{a-lV(«*+#)}J*.
16. Prove that, if the sole effect of a wind on the motion of a projectile is
to produce an acceleration f in a horizontal direction, the locus of points in a
horizontal plane which can just be reached with a given velocity v of projec-
tion is an ellipse of eccentricity //%/(/2+^2) and area TTV* \/(/2+#2)/<73.
17. A man standing on the edge of a cliff throws a stone with given
velocity «, at a given inclination to the horizon, in a plane perpendicular to
the edge of the cliff ; after an interval r he throws another stone from the
same spot with given velocity v at an angle i?r — 6 with the line of discharge
of the first stone and in the same plane. Find T so that the stones may strike
each other, and show that the maximum value of T for different values of 6 is
7, and occurs when sin 6 — v\u, w being the vertical component of v.
MISCELLANEOUS EXAMPLES 53
18. Two particles describe the same ellipse in the same time as a central
orbit about the centre. Prove that the point of intersection of their directions
of motion describes a concentric ellipse as a central orbit about the centre.
19. Particles are projected from points on a sphere of radius a with
velocity \f(gb) and move with an acceleration to the centre equal to gr/a
at distance r. Prove that the part of the surface on which they fall is the
smaller of the two segments into which the sphere is divided by a small circle
of radius b.
20. A particle P describes a rectangular hyperbola with an acceleration
P.CP from the centre (7; a point Fis taken in CP so that CP . CF=a2; prove
that the rate at which P and Y separate is
where 2a is the transverse axis.
21. If the acceleration of a particle is directed to a point S and varies
inversely as the square of the distance, prove that there are two directions,
if any, in which it can be projected from a point P with given velocity so as
to pass through a point Q, and that the velocity of arrival at Q is the same
for both. Prove also that the angle between one of the directions of pro-
jection and PQ is the same as the angle between the other and PS.
22. Prove that two parabolic orbits can be described about the same
focus so as to pass through two given points, and that the focus lies
within the finite area bounded by the lin« joining the given points and one
parabola, and outside the finite area bounded by the same line and the other
parabola.
23. Prove that the greatest radial velocity of a particle describing an
ellipse about a focus is
where 2« is the major axis, e the eccentricity, and T the periodic time.
24. A particle describes an ellipse as a central orbit about a focus, and a
second particle describes the same ellipse in the same time with uniform
angular velocity about the same focus. The particles start together from
the farther apse. Prove that the angle which the line joining the particles
subtends at the focus is greatest when the angle described by the first
particle is cos"1 (1 — (1 — e2)*}/e, e being the eccentricity.
25. Prove that the central orbit described with acceleration /i/(distance)2,
by a particle projected with velocity V from a point where the distance is R,
is a rectangular hyperbola if the angle of projection is
-1
54 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
26. Prove that the focal radius and rectorial angle of a particle describing
an ellipse of small eccentricity e at time t after passing the nearer apse are
given approximately by the equations
r = a 1 — ecos nt
where 2a is the major axis and %irjn is the periodic time.
Prove also that if e2 is neglected the angular velocity about the other focus
is constant.
27. Two particles describe the same ellipse in the same periodic time,
starting together from one end of the major axis. One of them has an
acceleration directed to a focus, and the other an acceleration directed to the
centre. Prove that, if <£t and $2 are their eccentric angles at any instant, then
fa — (f)o=e sin 0X.
28. If the perihelion distance of a comet is - th of the radius of the
Earth's orbit, supposed circular, show that the comet will remain within the
Earth's orbit for
(V2/3ir) (1 + 2/») v/(l - l/») years,
the comet's orbit being parabolic.
29. If the parabolic orbits of two comets intersect the orbit of the Earth,
supposed circular, in the same two points, and if ti , t2 are the times in which
the comets move from one of these points to the other, prove that
+(ti - <j)*—(4r/fcr)*, where T7 is a year.
30. Three focal radii SP, &$, SR of an elliptic orbit about a focus S are
determined, and the angles between them. Show that the ellipticity may be
found from the equation &A=aA', where A is the area of the triangle PQR,
and A' is the area of a triangle whose sides are
and two similar expressions.
31. A particle is projected from A with velocity ^/(^p)lOA- and moves
with an acceleration ^/(distance)5 directed to 0, the direction of projection
making an angle a with OA. Prove that the particle will arrive at 0 after a
time
OA3 a -sin a cos a
V(2/i) sin3 a
32. Prove that the acceleration with which a particle P can describe a
circle as a central orbit about a point S is inversely proportional to SP2 . PP3,
where PP is the chord through S.
If points are taken on the orbit such that the squares of their distances
from S are in arithmetic progression, the corresponding velocities are in
harmonic progression.
MISCELLANEOUS EXAMPLES 55
33. Prove that any conic can be described by a particle with an
acceleration always at right angles to the transverse axis and varying
inversely as the cube of the distance from it.
34. A particle moves with an acceleration \iy~* towards the axis #,
starting from the point (0, k] with velocities U, V parallel to the axes of x, y.
Prove that it will not strike the axis x unless /i > V2k2, and that, in this case,
it strikes it at a distance 6T£2/(v//x - Vk) from the origin, U, V, k being
positive.
35. Prove that the acceleration towards the centre of the fixed circle
with which a particle can describe an epicycloid is proportional to r/p*, where
»• is the radius vector and p the perpendicular from the centre to the
tanent.
36. Prove that the curve r=a (1 +^^f6 cos 6} is a central orbit about the
origin for acceleration proportional to r~* + ^ar~&.
37. A series of particles are describing the same curve as a central orbit
about a point 0 with an acceleration whose tangential component is hz/p'2(f>' (p}.
Prove that, if the line density at any time is constant and =p0, the line
density p at any subsequent time t is given by
\h being the rate of description of areas about 0, and p the perpendicular
from 0 on the tangent.
38. If inverse curves with respect to 0 can be described as central
orbits about 0 with accelerations /, /', prove that
i*f r'3f _ 2
where h and h' are constants, r and / are corresponding radii vectores, and
leTangle r or r' makes with the tangent.
\,
39. If / is the acceleration and %h the areal velocity in a central orbit
about a point 0, prove that the angular acceleration a about 0 satisfies the
equation
du u
where u is the reciprocal of the distance from 0.
40. Prove that a body ejected from the Earth with velocity exceeding
seven miles per second will not in general return to the Earth, and may leave
the solar system.
41. Prove that the least velocity with which a body could be projected
from the North Pole so as to meet the Earth's surface at the Equator is
nearly 4^ miles per second, and that the angle of elevation is 22|°.
56 MOTION OF A FREE PARTICLE IN A FIELD OF FORCE [CH. II
42. A stream of particles originally moving in a straight line K with
velocity V is under the influence of a gravitating sphere of radius R, whose
centre moves with velocity v in a straight line intersecting the line K and
making with it an angle a. Prove that, if the distance of the sphere from the
line is originally very great, a length
(ZR/v) cosec a x' ( T- - 2 Vr cos a + v2 + ZgR)
of the line of particles will fall upon the sphere, g being the force per unit
mass at the surface of the sphere.
CHAPTER III
FORCES ACTING ON A PARTICLE
58. The force of gravity. Consider a heavy body supported
near the Earth's surface. The body may, for example, rest upon a
horizontal plane, which is then the plane surface of some other
body, or it may be supported by a rope or a spiral spring. In
either case we should say that there was a force acting upon it
and counteracting the force of the Earth's field. When the body
is supported by a spring, the spring is stretched ; if the body is
supported even by a steel bar, the bar is stretched a little*, and
the stretching of the bar can be observed by means of suitable
instruments. If the body is supported by a man carrying it, his
muscles are thrown into a state of strain, analogous to the stretch-
ing of the steel bar, and the man has a sensation of muscular
effort. We should say that he exerted "force."
The operation of weighing a body in a common balance deter-
mines a certain quantity : the number of pounds or grammes
which the body weighs. The number so determined is independent
of the latitude and longitude of the place where the operation is
performed ; and it is independent also, so far as observation can
tell, of the altitude of the place above, or its depth below, the
mean surface of the Earth.
The stretching of a spring supporting a body can be measured;
and, when the weight of the body, as determined by the common
balance, is not too great, the stretching of the spring, at any
definite place on the Earth's surface (e.g. in London), is propor-
tional to the weight so determined. We may therefore use this
stretching to determine the weight of the body, and . then the
body is said to be "weighed by a spring balance." The weight of
the body, determined by the spring balance, is different in different
* A steel bar, of sectional area one square inch, banging vertically, and sup-
porting a load of 1 ton, is extended by the fraction 0-00007 of its length, approxi-
mately.
58 FORCES ACTING ON A PARTICLE [CH. Ill
latitudes and at different altitudes. It is found to be proportional
to the local value of g (the acceleration of a free falling body).
The primitive notion of " force " is based upon the muscular
sensations of a man supporting a heavy body. The measure of
force which is suggested by the above considerations is the
stretching of an ideal spring supporting a heavy body*. This
stretching is always proportional (i) to the weight, as determined
by a common balance, (ii) to the local value of g. We are thus led
to measure the force of the Earth's gravity as proportional to each
of these factors.
The operation of weighing a body in a common balance teaches
us how to assign to any body of sufficiently small bulk a definite
constant quantity : the number of pounds or grammes which the
body weighs. This quantity, or any suitable constant multiple of
it, will be called the mass of the body. For a body which cannot
be weighed in a common balance, e.g. a battleship, the mass may
be determined by adding the masses of the several parts, each
being determined by weighing in a common balance or by some
equivalent method. The definition of "mass" does not cover such
cases as the mass of the Earth, or Sun, or Moon. A more general
definition will be given in Chapter VI. We denote the mass of a
body by the letter ra.
The force of the Earth's gravity acting upon a bodyf is
measured by the product of the number of units of mass in the
mass of the body and the number of units of acceleration in the
local value of g. We denote this force by W, and write
W = mg.
59. Measure of force. Force may be defined as a certain
measure of the action which one body exerts upon another. In the
particular case of a oody supported upon, a horizontal plane, the
force counteracting the force of the Earth's gravity is traced to an
action of the body having the horizontal plane for part of its
surface ; this force is called the pressure of the plane upon the
supported body. In the case of a body supported by a rope or
* The spring is "ideal" in as much as the extension is supposed to be pro-
portional to the weight, however great the weight may be. An actual spring would
be damaged by a sufficiently heavy weight, and it would not measure that weight
correctly.
t This force is sometimes called the "weight" of the body.
58-60] MEASURE OF FORCE 59
spring, the force counteracting the force of the Earth's gravity is
traced to an action of the rope or spring ; this force is called the
tension of the rope or spring. The force of the Earth's gravity
acting upon a body is, in like manner, traced to a supposed action
of the Earth upon the body.
In this last case we know that the effect of the action, if not
counteracted, is to produce in the body a certain acceleration ; and
the measure of the force is the product, as explained above, of the
mass of the body and the acceleration which it produces.
In like manner, we may say that the effect of any force on a
body, when not counteracted by other forces, is to produce in the
body an acceleration, and the measure of the force is the product
of the measures of the mass and the acceleration. If a force P acts
upon a body of mass m, it produces in it an acceleration f, and we
have the formula
P = mf.
60. Units of mass and force. In the "C.G.s. system" of
units, the gramme is the unit of mass. It is the one-thousandth
part of the mass of a certain lump of platinum known as the
" Kilogramme des Archives," made by Borda, and kept in Paris.
The unit of force is called the "dyne." It is the force which, acting
upon a body of mass one gramme, produces in it an acceleration of
one centimetre per second per second.
In the "foot-pound-second system," the pound is the unit of
mass. It is the mass of a certain lump of platinum kept in the Royal
Exchequer in London. The unit of force is called the "poundal." It
is the force which, acting upon a body of mass one pound, produces
in it an acceleration of one foot per second per second.
In the "British engineers' system" the unit of force is the
force of the Earth's gravity acting in London upon a body which
weighs a pound, when weighed in a common balance. It is called
a "force of one pound." The unit of mass is the mass of a body
which weighs 32'2 pounds in a common balance. The mass of
a body which weighs one pound in a common balance is ^^
oZ'Z
units of mass. In this system, as in the others, the unit force,
acting upon the unit mass, produces in it an acceleration of one
unit of length (one foot) per second per second.
60 FORCES ACTING ON A PARTICLE [CH. Ill
In any system of units, force is a quantity of one dimension in
mass, one dimension in length, and — 2 dimensions in time. The
dimension symbol is MLT~2.
61. Vectorial character of force. In the cases which we
have examined so far, either there has been a single force acting
upon a body, which for definiteness we thought of as a " particle,"
or else the forces acting upon the body have exactly counteracted
each other. In the former case, the body moves with a certain
acceleration. In the latter case, it remains at rest. In the case of
a heavy body supported by the tension of a cord, we may regard
the Earth's gravity as producing in it the acceleration g down-
wards, and the tension of the cord as producing in it the accelera-
tion g upwards. If we do this we are able to maintain in both
cases the measure of force as the product of the mass and the
acceleration that is produced by the force.
Consider a body supported upon a plane horizontal surface.
Let the surface be gradually tilted so that the plane becomes an
inclined plane. It is found that the body will begin to slide*
down the plane when the plane is tilted at an angle which exceeds
a certain limiting angle. If the surfaces in contact are highly
polished the angle at which sliding begins is small. We might
imagine the surfaces to be so smooth that sliding would take place
at any inclination however small. The acceleration with which the
body slides down the plane is the resultant of the acceleration g in
the direction of the downward vertical and some other acceleration,
/. Let a be the inclination of the plane ; then the acceleration g
can be resolved into two components, viz. : g sin a in the direction
of a line of slope drawn down the plane, and g cos a at right angles
to the plane. See Fig. 29. If the accelera-
tion/"is directed at right angles to the plane
its amount must be g cos a, in the sense
opposite to one of the two components of g,
as shown in Fig. 29, since the body moves
9 cos a on the plane, and so has no acceleration at
right angles to the plane. In this case, the
Fig. 29.
acceleration with which the body slides
* The body should have a flat base. A solid sphere, or any body with a curved
surface, placed on an inclined plane, will generally roll. We avoid for the present
the complication of rolling.
60-62] COMPONENT FORCES AND RESULTANT FORCE 6 1
down the plane is g sin a*, and the pressure of the plane on the
body is of amount mg cos a, the mass of the body being m. This
state of things cannot be exactly realised in practice, but it can be
approximately realised when the surfaces are very smooth.
In any actual case the acceleration with which the body slides
down the plane is less than g sin a, and the motion is said to be
resisted by "friction." For the present we shall suppose that the
surfaces are so smooth that the effect of friction is negligible. We
have learnt that the effect of the Earth's gravity on the body is
the same as that of two forces : one mg sin a producing acceleration
down the line of slope, and the other mg cos a producing accelera-
tion at right angles to the plane.
This result leads us to the conclusion that force, as a mathe-
matical quantity, is to be regarded as a vector quantity, equivalent
to "component forces" in the same way as any other vector quantity
is equivalent to components.
In particular, we see that force acting on a particle ought to
be regarded as what we have called a " vector localized at a point "
(Art. 17), the point at which the vector is localized being the
position of the particle. The line, drawn through the point, by
which the vector is determined, is the "line of action" of the force.
The line of action of the force and the sense of the force are the
direction and sense of the acceleration which the force produces.
According to this statement any forces acting on a particle are
equivalent to a single force, to be determined from the separate
forces by the rules for the composition of vectors. This single force
is called the "resultant" of the forces acting on the particle.
62. Examples +.
1. Find the time of descent of a particle down an inclined tube when
friction is neglected and the particle starts from rest at a given point of
the tube.
2. Prove that the time of descent down all chords of a vertical circle,
starting at the highest point of the circle, or terminated at its lowest point,
is the same.
3. Prove that the line of quickest descent from a point A to a curve,
which is in a vertical plane containing A, is the line from A to the point of
* This result was used by Galileo for the determination of g.
f The results in Examples 2 and 3 were noted by Galileo.
62 FORCES ACTING ON A PARTICLE [CH. Ill
contact with the curve of a circle described to have A as its highest point and
to touch the curve. Prove also that the line of quickest descent from a curve
to a point A is the line to A from the point of contact with the curve of a
circle described to have A as its lowest point and to touch the curve.
4. Prove that each of the lines of quickest descent in Ex. 3 bisects the
angle between the vertical and the normal to the curve at the point where
it meets the curve. Hence show that the line of quickest descent from one
given curve to another in the same vertical plane bisects the angle between
the normal at either end and the vertical.
5. Prove that a particle projected in any manner on an inclined plane,
and moving on the plane without friction, describes a parabola.
63. Definitions of momentum and kinetic reaction.
The momentum of a particle of mass m, moving with a velocity
v, is a vector, localized in the line of the velocity, of which the
sense is the same as that of the velocity, and the magnitude is the
product mv.
The kinetic reaction of a particle of mass m, moving with an
acceleration /, is a vector, localized in the line of the acceleration,
of which the sense is the same as that of the acceleration, and the
magnitude is the product mf.
The kinetic reaction of a particle is the same quantity as the
rate of change of momentum of the particle per unit of time.
64. Equations of motion. The discussion of the nature of
force in Articles 58 — 61 leads to the following statement : —
The kinetic reaction of a particle has the same magnitude,
direction and sense as the resultant force acting on the particle.
This statement is to be regarded as a general principle which
is suggested by the facts stated in the previous discussion arid
other facts of like nature. In other words it is an induction from
experience. From the nature of the case it is not capable of
mathematical proof. The truth of it is only to be tested by the
comparison of results deduced from it with results of experiment.
The statement is expressed analytically by certain equations,
which are called the "equations of motion" of the particle. They
are obtained by equating the resolved part of the kinetic reaction
in any direction to the sum of the resolved parts of the forces in
that direction.
62-65] EQUATION OF MOTION 63
Let X, Y, Z be the components parallel to the axes of x, y, z
of the resultant force acting on the particle, or, what comes to the
same thing, the sums of the resolved parts of the forces in the
directions of these axes. Let m be the mass of the particle, and
a, y, z the coordinates of its position at time t. The equations of
motion are
mix = X, my = Y, mz = Z.
We have had several examples already of equations which are really
equations of motion.
For example, the equations
£ = 0, y = -ff
in Art. 33 are really equations of motion.
As a further example, consider the motion of a particle in a central field of
force. If / is the intensity of the field, and the centre of force is the origin,
and if the force is an attraction, it is of amount mf and is directed towards
the origin ; and the equations of motion are
,x .. ,y .. ,z
mx = - mf - , my = —mf - , mz = - mf - ,
where r denotes distance from the origin. Just as in Art. 49, these equations
show that the motion takes place in a fixed plane. By means of the result of
Art. 43 the equations of motion, expressed in terms of polar coordinates in
the plane, can be written
m (f — r92) = — mf, m — =- (r2d) = 0.
r dt ^
EQUATIONS OF MOTION IN SIMPLE CASES
65. Motion on a smooth guiding curve under gravity.
The motion of a small ring on a very smooth wire, or of a small
spherical shot in a very smooth tube, can be discussed by treating
the ring or shot as a particle constrained to describe a given curve,
and supposing that the particle is subject not only to the force of
the field, but also to a force — the pressure of the curve — directed
along the normal at any point of the curve. We take the case
where the curve is a plane curve in a vertical plane, and the field
is that of the Earth's gravity at a place. We draw the axis of y
vertically upwards, and denote by s the arc of the curve measured
from some fixed point of it up to the position of the particle at
the instant t, and by v the velocity of the particle in the direction
of increase of s. We denote the pressure of the curve by R, and
suppose that its sense is towards the centre of curvature. If the
64 FORCES ACTING ON A PARTICLE [CH. Ill
pressure really acts outwards, the value found for R will be
negative.
In the left-hand figure (Fig. 30) are shown the components of
the kinetic reaction along the tangent and normal. In the right-
hand figure are shown the forces acting on the particle. The
equations of motion, obtained by resolving the forces along the
tangent and normal, are ^
dv . v2 r.
mv -r = — mg sin <p, m — = R — mg cos <p.
Now sin </> = ~ , and the first of these equations becomes
dv dy
mV.ds = -m^d-s'
This equation can be integrated in the form
where C is a constant. Let v0 be the velocity at some point (,v0, y0)
of the curve. Then C = ^mv<? + mgy0, and the equation can be
written
\tnv- - $mv<? = mg (y0 - y).
This equation can be partially interpreted in the statement
that the velocity of a particle moving under gravity without
friction is always the same when it comes back to the same level*.
If the particle starts with an assigned velocity from a given
point of the curve, this equation determines the velocity in any
position; the equation mv2/p = R — mg cos </> determines the pressure
at any point of the curve.
* This result was found by Galileo.
65-67]
65
66. Examples.
1. When the curve is a circle, the angle <£ of Fig. 30 is the angle which
the radius of the circle drawn from the centre to the particle makes with the
vertical drawn downwards. Prove that, if the particle starts from rest in a
position in which $ = o, the velocity v in any position is given by the equation
v2 = Zga (cos (f> - cos a),
where a is the radius of the circle.
Find the pressure in any position.
2. Find the greatest angle through which a person can oscillate in a
swing, the ropes of which can support a tension equal to twice the person's
weight.
3. When a particle moves on a smooth
cycloid under gravity, the vertex of the
cycloid being at the lowest point, the equa-
tion of motion, by resolution along the
tangent in direction QP, may be written
s= —gsin 6,
s being the arc measured from the vertex
to P, and 6 the angle which the normal OP
makes with the vertical. Now, by a known
property of the cycloid, s = 4asin#, where a £• •
is the radius of the generating circle, and thus the above equation becomes
showing that the motion in s is simple harmonic with period
Thus the time taken to fall to the vertex from any point on the curve is
independent of the starting-point, and in fact is ir*f(a/g).
[This property is known as the " Isochronism of the cycloid."]
4. Show that the time a train, if unresisted, takes to pass through a
tunnel under a river in the form of an arc of an inverted cycloid of length 2s
and height h cut off by a horizontal line is
where v is the velocity with which the train enters and leaves the tunnel.
67. Kinetie energy and work. The quantity obtained by
multiplying the-number of units of mass in the mass of a particle
by half the square of the number of units of velocity in the velocity
of the particle is called the "kinetic energy" of the particle.
The " work done" by a constant force acting on a particle is a
quantity which is defined in terms of the force and the displace-
ment of the particle. We resolve the displacement into components
66 FORCES ACTING ON A PARTICLE [CH. Ill
parallel and perpendicular to the line of action of the force. The
component parallel to this line (taken in the sense of the force)
has a certain magnitude, which is a number of units of length,
and a certain sign. We multiply this number, with this sign, by
the number of units of force in the measure of the force. The
product so obtained is the work done.
In the case of a particle moving under gravity, the work done
by the force of gravity is the product of the force, mg, and the
distance through which the particle descends, y0 — y. The equation
\ mv2 - % mv02 = mg (y0 - y}
can be expressed in words in the statement : —
The increment of kinetic energy in any displacement is equal to
the work done by the force of gravity in that displacement.
68. Units of energy and work. The unit of work is the
work done by the unit force in a displacement of one unit of length
in the direction of the force. The unit of kinetic energy is the
kinetic energy acquired by a free body on which one unit of work
is done.
In the c.G.s. system of units the unit of work is called the
erg. It is the work done by a force of one dyne acting over one
centimetre.
In the foot-pound-second system the unit of work is the "foot-
poundal." It is the work done by a force of one poundal acting
over one foot.
In the British engineers' system the unit of work is the "foot-
pound." It is the work done by a "force of one pound" acting
over one foot. It is equal to the work done in the latitude of
London in raising through one foot a body which weighs one pound
in a common balance.
In any system of units, work and kinetic energy are quantities
of 1 dimension in mass, 2 dimensions in length, and — 2 dimensions
in time. The dimension symbol is ML'2T~2. +
69. Power. An agent which does one unit of work per unit
of time is said to be working up to a unit of power. If 550 foot-
pounds of work are done per second the power is one horse-poiuer.
Power is a quantity having the dimensions ML2T~3. For a more
extended discussion see Chapter VI.
67-7l] KINETIC ENERGY AND WORK 67
70. Friction. Consider a body sliding down an inclined
plane. Let a be the inclination of the plane. The acceleration of
the body down the lines of slope is less than <7sin a. Let /be the
acceleration up the lines of slope which must be compounded with
the acceleration g sin a down the lines of slope in order that the
resultant may be the actual acceleration of the body. The forces
acting on the body are the force of gravity mg vertically down-
wards, the pressure mgcosa. at right angles to the plane, and a
third force which is of magnitude ?n/and acts up the lines of slope.
This force is called the "friction."
The body will not slide down the plane unless the inclination
a exceeds a certain angle i. When a = i, the friction just prevents
motion. In this case g sin a =f, or f= g sin i, and the friction
= mg sin i. In the same case the pressure = mg cos i. Hence the
ratio of the friction to the pressure when motion is just about to
take place is tan i. We write /j, for tan i, so that when the body is
about to slide the friction is equal to the product of /* and the
pressure.
It is found that, when motion takes place, the ratio of the
friction to the pressure remains constant. This ratio (equal to tan i
or /JL) is called the " coefficient of friction." The angle i is called
the " angle of friction."
The angle of friction, and the coefficient of friction, depend
upon the materials of the bodies in contact and the degree of polish
of the surfaces.
Whether the body moves up or down the plane, the friction
acts in the sense opposite to that of the velocity, and is equal to
the product of p and the pressure.
71. Motion on a rough plane. We shall take the plane to
be inclined at an angle a to the horizontal, and treat the body
sliding on it as a particle moving down a line of slope. Draw the
axis of x down a line of slope. The equations of motion are
mx = mgsma — F, 0 = mgcosa—R,
where F is the friction and R the pressure. Also we have
F=fiR.
Hence the particle moves down the line of slope with acceleration
g (sin a.— /ji cos a).
5—2
68 FORCES ACTING ON A PARTICLE [CH. Ill
When the particle moves up a line of slope, the friction acts
down the line, and the acceleration is equal to
g (sin a + /JL cos a)
down the line.
When the body slides on a horizontal plane the pressure is
equal to mg vertically upwards, and the friction is equal to p,mg
in the sense opposite to that of the velocity.
This last result is generally taken to be applicable to the
motion of a train on level rails. The resistance to the motion is
taken to be proportional to the mass. The force by which the
train is set in motion and kept in motion against the resistance
is called the "pull of the engine." We shall consider this force
further in Chapter VIII.
When there is friction the increment of kinetic energy in any
displacement is less than the work done by gravity in that dis-
placement.
72. Examples.
1. A particle is projected with a given velocity up a line of slope of a
rough inclined plane. Find the height above the point of projection of the
point at which it comes to rest. Supposing the inclination of the plane to be
greater than the angle of friction, find the velocity with which the particle
returns to the point of projection.
2. A carriage is slipped from an express train, going at full speed, at a
distance I from a station, and comes to rest at the station. Prove that the
rest of the train will then be at a distance Ml/(M— m) beyond the station,
Jfand m being the masses of the whole train and of the carriage slipped, and
the pull of the engine being constant.
3. Prove that the extra work required to take a train from rest at one
station to stop at the next at a distance I in an interval t is
21 1 / f/1 1\ /l 1 T
times the work required to run through both without stopping, where the
incline of the road is 1 in m, and the resistance of the road and the brake
power per unit mass are equal to the components of gravity down uniform
inclines of 1 in n and 1 in k respectively.
73. Atwood's machine*. Another simple example of equa-
tions of motion is afforded by the problem of two bodies attached
* G. Atwood, A treatise on the rectilinear motion and rotation of bodies, Cam-
bridge, 1784.
71-73]
ATWOOD'S MACHINE
69
Fig. 32.
to a string or chain which passes over a vertical pulley. This
arrangement constitutes in principle the in-
strument called "Atwood's machine." We
shall assume that the tension of the chain
is the same throughout. This amounts to
assuming that there is no friction between
the pulley and the chain, and that the mass
of the chain is negligible in comparison with
the masses of the bodies (see Chapter VI).
Let ra, ra' be the masses of the bodies,
x the distance through which m has de-
scended at time t. Then a; is also the distance
through which mf has ascended at time t. If
m has ascended and ra' descended, x is negative. Let T be the
tension of the chain. The forces acting on m are mg vertically
downwards, and T vertically upwards. The kinetic reaction of m
is m'x vertically downwards. The equation of motion of m is
therefore
m'x = ing — T.
The forces acting on m' are m'g vertically downwards, and T
vertically upwards. The kinetic reaction of ra' is m'x vertically
upwards. The equation of motion of m' is therefore
m'x = T — m'g.
By adding the left-hand, and also the right-hand, members of
these equations, we find
(m + ra') x = (m — ra') g,
It follows that the heavier body descends, and the lighter
ascends, with an acceleration
ra + m' '
The value of g is sometimes determined by means of Atwood's
machine. Various corrections have to be applied to the result.
Generally the pulley turns with the motion of the chain, and the
most important correction is on account of the mass of the pulley.
(See Chapter VIII.)
70 FORCES ACTING ON A PARTICLE [CH. Ill
74. Examples.
1. The kinetic energy of the two bodies in the case of the simple
Atwood's machine, in which the friction and the masses of the chain and
pulley are neglected, is
The work done by gravity is mgx — m'gx. Assuming that the increment of
kinetic energy in any displacement is equal to the work done by gravity,
deduce the acceleration of either body.
2. Prove that the tension of the chain is
2mm'
.a,
m + m
3. In Atwood's machine the smaller mass m' is rigid, the mass m consists
of a rigid portion of mass m' and a small additional piece resting lightly
uptfn it. As m descends it passes through a ring by which the additional
piece is lifted off. Prove that, if m starts from a height h above the ring, and
if after passing through the ring it falls a distance k in the time £, then
m + m' k'2
the friction and the masses of the pulley and chain being neglected.
75. Simple circular pendulum executing small oscilla-
tions. A particle constrained to describe a circle in a vertical
plane, without friction, is called a " simple circular pendulum."
An ordinary pendulum consists of a massive body, called the
"bob," suspended by a bar which can turn about a horizontal axis.
When the bob is small and massive, and the bar thin, the motion
of the bob, treated as a particle, approximates to that of a simple
circular pendulum.
We denote the radius of the circle by I. When the radius of
the circle which passes through the particle
makes an angle 0 with the vertical as in
Fig. 33, the acceleration along the tangent to
the circle is 10 (Ex. 1 of Art. 37). We may
write down one equation of motion in the
same way as in Art. 65 in the form
1*8,10 = —«tg sin 0.
If 0 is very small throughout the motion,
sin 0 may be replaced by 0, and we have the approximate equation
10 = - gO.
74-76] SMALL OSCILLATIONS OF PENDULUM 7 1
This equation shows that the motion in 6 is simple harmonic
motion of period 27r^(l/g). (Of. Art. 38.)
The pendulum swings from side to side of the vertical. If it
starts from rest, in a position slightly different from the position
of equilibrium, it falls to this position in the time ^7r^/(l/g),
passes through it, and proceeds to move away from it on the other
side until its displacement is numerically equal to that at starting,
and comes to rest after an interval ^7r\/(l/g) from the equilibrium
position. The motion is then reversed. The time from rest to rest
is 7r\f(l/g). This is known as the time of a "beat," the period
2?r \/(lfg} is the time of a "complete oscillation."
A pendulum which beats seconds is known as a "seconds'
pendulum"; the time of a complete oscillation of such a pendulum
is two seconds. The length of the seconds' pendulum at a place
is given by the equation
Pendulum experiments afford the most exact method of deter-
mining the value of g.
76. Examples.
1. Prove that, if in London # = 98] '17, the units being the centimetre
and the second, then the length of the seconds' pendulum there is 99'413
centimetres.
2. A balloon ascends with constant acceleration and reaches a height of
900 ft. in one minute. Show that a pendulum clock carried with it will gain
at the rate of 27'8 seconds per hour, approximately.
3. If li is the length of a slightly defective seconds' pendulum which
gains n seconds in an hour, and 12 the length of another such pendulum
which loses n seconds in an hour, n being small, prove that the square root
of the true length of the seconds' pendulum is the harmonic mean between
v'?i and ^/12.
4. The bob of a pendulum which is hung close to the face of a cliff is
attracted to the cliff with a horizontal force of intensity/. Show that the
time of a beat is
where I is the length of the pendulum.
5. A bead slides on a smooth circular wire of radius a, whose plane is
inclined at an angle a to the vertical. Find the period of its small oscillations
about the lowest point.
72 FORCES ACTING ON A PARTICLE [CH. Ill
77. One-sided constraint. A particle may be constrained to
describe a circle by means of a thread of constant length attached
to the centre of the circle ; or it may be inside a smooth circular
cylinder. More generally a particle may be constrained to describe
a curve in a vertical plane by being inside a cylinder, of which the
normal section is the curve and the generators are horizontal, and
not too far from the lowest generator. Or it may be outside such
a cylinder, and not too far from the highest generator. In either
case the constraint is " one-sided," and the particle may leave the
curve. This will happen if the pressure vanishes. The particle
then describes a parabola under gravity until it strikes the curve
again.
Now the pressure is given, according to Art. 65, by the
equation
vz
R = m — f- mg cos <f>,
P *
where <£ is the angle which the tangent, drawn in a definite sense,
makes with the horizontal. To make R vanish, we must have
v-
cos <p = ,
gp
where v* is known. This equation determines the point at which
the particle leaves the curve.
78. Examples.
1. The bob of a simple circular pendulum is projected horizontally from
its equilibrium position with a velocity V. Find limits between, which V
must lie in order that the suspending fibre may become slack, and determine
the position of the bob at the instant when the fibre becomes slack.
2. A cylinder whose section is a parabola is placed with its generators
horizontal, the axis of a normal section vertical, and the vertex upwards, and
a particle is projected along it in a vertical plane. Prove that if it leaves the
parabola anywhere it does so at the point of projection.
3. A particle is projected from the lowest point of a vertical section of a
smooth hollow circular cylinder, whose axis is horizontal, so as to move round
inside the cylinder. Prove that, if the velocity is that due to falling from
the highest point, the particle leaves the circle when the radius through it
makes with the vertical an angle cos-1§.
Find the least velocity of projection in order that the particle may describe
the complete circle.
77-80]
CONICAL PENDULUM
73
4. A particle is constrained to describe a circle by means of an inex-
tensible thread, and leaves the circle when the thread makes an angle £ with
the vertical drawn upwards. Prove that when it strikes the circle again the
thread makes an angle 3/3 with the same vertical.
79. Conical pendulum. A particle can be constrained to
describe a horizontal circle uniformly by the tension of a string
or thread, attached to a fixed point on the vertical straight line
which passes through the centre of the circle. In any position of
the particle the string lies along a generator of a right circular
cone having its vertex at the fixed point.
Let 2a be the vertical angle of the cone and I the length of the
string. The radius of the circle is
/ sin a. Let v be the velocity of the
particle and T the tension of the
string. The kinetic reaction of the
Til/ V^
particle is 7—: — directed alone: the
I sin a -
radius of the circle towards its
centre. The forces acting on the
particle are the force of gravity,
mg vertically downwards, and the
tension T of the string, directed
along the generator of the cone
towards the fixed point. We form
equations of motion by resolving vertically, horizontally along the
radius of the circle, and horizontally along the tangent of the circle.
Neither the kinetic reaction nor the forces have any components in
the third of these directions; and we therefore have the two equations
mv2
T-. — = Tsina, 0 = mq—Tcosa.
Ism a.
By eliminating T we find the equation
sin2 a
.
cos a
This equation determines the velocity with which the circle
can be described when I and a are given, or the angle a when
v and I are given.
80. Examples.
1. A train rounds a curve, of which the radius of curvature is p, with
velocity v. Prove that to prevent the train from leaving the metals the
74 FORCES ACTING OX A PARTICLE [CH. Ill
outer rail ought to be raised a height equal to bv-jpg above the inner, b being
the distance between the rails.
[The train may be treated as a conical pendulum, in which the pressure
of the rails, directed at right angles to the plane of the rails, takes the place
of the tension of the string.]
2. The point of suspension of a simple pendulum of length I is carried
round in a horizontal circle of radius c with uniform angular velocity o>.
Prove that, when the motion is steady, the inclination a of the suspending
thread to the vertical is given by the equation
w2 (c + 1 sin a) =g tan a.
Prove also that, if (gju>2)^ <l* — cs, the inclination can be inwards towards
the axis of the circle.
THEORY OF MOMENTUM
81. Impulse. Let the equations of motion of a particle be
written in the forms
mx = X, my = Y, mz = Z,
and let both members of each of these equations be integrated
with respect to t over an interval from t0 to tt. Let xl, yt, zl be
the components of velocity at the instant tf1} and x0, y0, z0 the
components of velocity at the instant t0. The result is
rt, rti rt,
mx-L — mx0 = Xdt, my^ — my0 = Ydt, mz-^ — mz0 = Zdt.
'to -to J«t
The quantities in the left-hand members of these equations are
the components of a vector, which is the change of momentum of
the particle during the interval. The quantities in the right-hand
members of the same equations are the components of another
vector which is called the " impulse of the force " acting on the
particle during the interval. The equations can be expressed in
words in the statement: — The change of momentum of a particle
in any interval is equal to the impulse of the force acting on the
particle during the interval.
82. Sudden changes of motion. Changes of motion of bodies
sometimes take place so rapidly that it is difficult to observe the
gradual transition from one state of motion to another. We may
allow for the possibility of sudden changes of motion by supposing
that the force acting on a particle becomes very great during a very
short interval of time, in such a way that the impulse of the force
80-84] IMPULSE AND MOMENTUM 75
has a finite limit when the interval is diminished indefinitely. Let
t' denote the instant at which the sudden change of motion takes
place. In the equations of the type
ff,
mx^ — mx0 = Xdt,
J to
the right-hand members have finite limits when tQ = t' — \T, and
£j = t' + \r, and r is diminished indefinitely. We write
rt'+hr rt'+fr rf+lr
Lt Xdt = X, Lt I Ydt = Y, Lt Zdt = Z.
T = OJt'-%T T=0 t'-\r T = QJt'-$T
Then the equations are
mx-L — mx0 = X, my \ — my0 = Y, mzi — mz0 = Z.
We define the vector, localized at the position of the particle,
of which the components parallel to the axes are X, Y, Z, to be
the " impulse exerted on the particle " at the instant t', at which
the sudden change of motion takes place.
83. Constancy of momentum. The equations of motion of
the form mx = X
may also be written -^ (mac) = X,
and this equation may be expressed in words in the statement : —
" The rate of increase of the momentum of a particle in any
direction is equal to the sum of the resolved parts in that direction
of all the forces which act upon the particle."
If the line of action of the resultant force acting on the
particle is at right angles to a fixed line, the resolved part of the
momentum in the direction of this line is constant.
We had an example of this in the parabolic motion of pro-
jectiles (Art. 33).
If the velocity of a particle undergoes a sudden change, the
resolved part of the momentum in any direction at right angles to
the direction of the resultant impulse is unaltered.
84. Moment of force, momentum and kinetic reaction
about an axis. Let the axis be the axis of z, and consider a force
applied at the point (x, y ', z). Let F be the force, and X, Y, Z
its components parallel to the axes. Let a plane pass through the
point (of, y', z'} and cut the axis of s at right angles in the point P.
76 FORCES ACTING ON A PARTICLE [CH. Ill
Resolve the force F into components : Z parallel to the axis of z,
and F' at right angles to this axis. Then the moment of F about
the axis of z is denned to be the same as the moment of F' about
P. The rule of signs is that when the axis of z and the direction
of F' are related like the directions of translation and rotation in
an ordinary right-handed screw the sign is +. Otherwise the sign
is — .
The theorem of Art. 22 gives for the moment of F about the
axis of z the expression
x'Y-y'X.
Let O, y, z) be any point on the line, r its distance from
(x, y, z'\ I, m, n the cosines of the angles which the line, supposed
drawn from the point (a?', y', z'} to the point (as, y, z}, makes with
the axes of x, y, z, drawn in positive senses. Then
x — x' = Ir, y — y' = mr, z — z' = nr.
Now the sense from (x, y', z') to (or, y, z) is either the sense of the
force F or the opposite sense, and we have therefore
either X = IF, Y=mF, Z = n F,
or X = - IF, Y = - mF, Z = - nF.
In both cases we have the equations
~X~ Y ~Z~ '
and therefore xY — yX = x'Y— y'X.
It follows that, so long as the magnitude, line of action and sense
of the force remain the same, the moment is independent of the
point of application.
Now let the force be supposed to be applied at that point
in its line of action at which the common perpendicular to the
line of action and the axis of z meets the line of action. Then
the force F' is at right angles to this common perpendicular.
Hence the moment is the product, with a certain sign, of the
length of the common perpendicular and the resolved part of the
force at right angles to the axis. The rule of signs is, as before, the
rule of the right-handed screw.
This result leads to a general definition of the moment of a
localized vector about an axis : — Let the axis be a line L to which
84, 85]
MOMENT OF LOCALIZED VECTOR
77
a certain sense is assigned, and let the vector be localized in a
line L', or be localized at a
point in L' and have for di-
rection the direction of L'.
Resolve the vector into com-
ponents parallel to L and at
right angles to L. The mo-
ment of the vector about the
axis L is the product, with
a certain sign, of the resolved
part of the vector at right
angles to L, and the length
of the common perpendicular
to L and L'. The rule of signs is the rule of the right-handed
screw.
From what precedes it is clear that, if the vector is resolved
into any components, or is the resultant of given component
vectors, the moment of the resultant about any axis is the sum
of the moments of the components.
The moments of a force (X, Y, Z), applied at a point (x, y, z\
about the axes of x, y, z, are respectively
yZ-zY, zX-xZ, xY-yX.
The moments of the momentum of a particle about the axes are
m (yz — zy), m (zx — xz), m (xy — yx),
where x, y, z are the coordinates of the position of the particle at
time t. The moments of the kinetic reaction of the particle about
the axes are
m (yz — zy), m (zx — x'z), m (xy — yx).
85 Constancy of moment of momentum.
Let x, y, z be the coordinates at time t of a particle which is
subject to any forces, and let X, Y, Z be the components of the
resultant force parallel to the axes. We have the equations
m'x = X, my = Y, m'z = Z.
Multiply both members of the second of these equations by x,
and both members of the first by y, and subtract the results. We
have
m (xy — yx) = xY — yX.
78 FORCES ACTING ON A PARTICLE [CH. Ill
This equation may be expressed in words in the statement :—
" The moment of the kinetic reaction of a particle about an axis
is equal to the sum of the moments about the same axis of all the
forces acting on the particle:"
The equation may also be written
jt {m (asy - yx)\ =xY-yX;
and now the left-hand member may be read as "The rate of
increase of the moment of momentum of the particle about the
axis."
If the line of action of the resultant force acting on the particle
meets a fixed axis, or is parallel to such an axis, the moment of
momentum of the particle about the axis is constant.
We have had an example of this in central orbits.
If the velocity of a particle undergoes a sudden change the
moment of the momentum about any line which meets, or is
parallel to, the line of the resultant impulse is unaltered.
85 A. Note. In Art. 85 the formula
•jt {™> (xy -y%)} = m (x'y - y$}
is interpreted in the statement : — " The moment of the kinetic reaction of a
particle about & fixed axis is equal to the rate of increase, per unit of time, of
the moment of momentum of the particle about the same axis." It is some-
times convenient to take moments about an axis defined by means of a body
which is in motion relatively to the axes of reference, or, as it would usually
be described, a "moving axis." Let the moving axis be parallel to the axis
of z, and meet the plane of (x, y] in the point (£, 77). Then the moment
of the kinetic reaction of the particle about the instantaneous position of the
moving axis is
m{(x-£)y-(y-r,)x},
and the moment of momentum about the same line is
j
Now [m {(x - £) $ - (y -
and thus the moment of the kinetic reaction is not in general equal to the
rate of increase of the moment of momentum when the axis is in motion.
85-87] WORK DONE BY A FORCE 79
WORK AND ENERGY
86. Work done by a variable force. Let a particle move
along a curved path, of which the arc measured from a fixed
point to a variable point is denoted by s, and let F be a force
acting on the particle, 6 the angle which the line of action of F at
any point of the curve makes with the tangent to the curve at
the point. We suppose this tangent to be drawn in the sense in
which the curve is described.
Let the arc between any two points A and B of the curve be
replaced by a polygon of n sides, sl, s2, ... sn, having all its vertices
on the curve. If the force F were the same at all points of any
of these sides, and, at any point on the side sK(tc — l, 2, ... n), its
magnitude were FK and the angle which its line of action makes
with the side were 0K, the work done by the force, as the particle
describes the polygon, would be
Fl . s1 cos 9l + F.2 . s2 cos #2 + . . . + Fn . sn cos 0n.
When the number of sides of the polygon is increased inde-
finitely, and the lengths of all of them are diminished indefinitely,
this expression tends to a limit, Called " the line-integral of the
tangential component of F" along the arc of the curve between
the points A and B. It is expressed by
•B
Fcosdds.
If X , Y, Z are the components of the force at any point (x, y, z),
this expression is the same as the line-integral
{( X ^ + Y ^ + Zd/\ ds, or ((Xdx + Ydy + Zdz),
J \ ds ds as) J
taken along the curve from the point A to the point B.
This expression represents the work done by the force upon
the particle in the displacement from A to B along the curve.
It is clear from the form of the expression that the work done
by the resultant of any forces acting on a particle is equal to the
sum of the works done by the separate forces.
87. Calculation of work. For the actual calculation of
the work it would in general be necessary to know how to express
the coordinates of a point of the curve in terms of some parameter,
80 FORCES ACTING OX A PARTICLE [CH. Ill
say 0, and also to know the values of the components of the force
in terms of the position of the particle. Then at any point on the
curve we could express X, Y, Z in terms of x, y, z, and therefore
dx dii dz . c a A 4-u
of 0, and we could also express ^ , ^ , ^ m terms of 0, and thus
we should have to integrate an expression of the form
dd+ de^" d6
between two fixed values of 0, corresponding to the points A and
B. In this expression X, . . . and -^ , • • • would be expressed in
terms of 0.
It is clear that the result, if it could be obtained, would
depend in general upon the curve; that is to say it would be
different for different curves joining the same two points.
In the case where the force is a central attractive force, mf(r\
which is a function of the distance r from a fixed point, the
dr
tangential component of the force is — mf(r) -j- , and the work
done is
rn
— mf(r} dr,
where r0 and rl are the distances of A and B from the fixed point.
Now let <f> (r) be the indefinite integral of /(r), so that
then the work done is m [<£ (r0) — <j> (?*j)]. It depends on r0 and i\ ,
but is the same for any two curves joining the points A and B.
Another example in which the work is independent of the curve
is afforded by a constant force as we saw in Art. 67.
88. Work function. When the work is independent of the
path, we may choose arbitrarily a fixed point A, and take the
integral
along any path drawn from the point A to a point P. The result
is a function of the coordinates of P. This function is the work
87-89] WORK FUNCTION AND POTENTIAL 81
function. The value of the work function at any point P is equal
to the work done by the forces upon the particle as the particle
moves along any path from the chosen fixed point A to the
assigned point P.
When the work is independent of the path, so that a work
function exists, the forces are said to be " conservative."
89. Potential function. In the case of a particle moving in ^J\
a field of force, we denote by f the intensity of the field at any
point. Let A be an arbitrary fixed point in the field, s the arc of
a curve measured from A, 6 the angle which the direction of the
field at any point makes with the tangent to the curve at the
point, the sense of the tangent being that in which the curve
would be described by a particle starting from A. The work done
by the force of the field in the displacement of a particle of mass in
along the curve from the chosen point A to a variable point P is
f
f. cos 6 . ds.
m
J A
If the force of the field is conservative, this expression is equal
to the value of the work function at P ', we write it
mV(P] or mV.
Then V(P) is defined to be the value of the potential function
at the point P, and the function V is called the " potential " at
a point. It is the line-integral of the tangential component of the
force of the field (estimated by its intensity) taken along any curve
joining the chosen point A to the variable point P.
The potential function vanishes at the point A.
If we replace the point A by any other fixed point B, the
potential function is increased by a constant, which is the value
of the integral
<-B
f. cos 6 . ds.
A
In the case of a central field, of which the intensity at a
distance r from the centre of force is — . we take the point A at an
f1
infinite distance. The potential function is then given by the
equation
F=f-.(-f
J oo r- \ ds
L. M.
82 FORCES ACTING ON A PARTICLE [CH. Ill
or the potential at any point is the product of the constant /u, and
the reciprocal of the distance of the point from the centre of force.
In the case of a uniform field of intensity g, we may draw the
axis z in the direction opposite to that of the field, then the
potential at a point is — gz.
90. Forces derived from a potential. Let mX, mY, mZ
be the components of the force of a field acting on a particle of
mass m, so that the direction of the vector (X, Y, Z) is the
direction of the field, and the resultant of (X, Y, Z) is the in-
tensity of the field. Let V be the potential of the field, supposed
conservative.
Let P be any point (x, y, z), and P' any neighbouring point
(x+Sx, y+Sy, z + Sz). The difference V(P') - V(P) is the
value of
[ (Xdx + Ydy + Zdz)- t (Xdx+Ydy + Zdz),
•> A J A
and this is the same as the value of the integral
.p,
(Xdx + Ydy + Zdz)
taken along the straight line drawn from P to P.
Now there exist some values X', Y, Z', intermediate between
the greatest and least values of X, Y, Z that occur on the line
PP, which are such that
tP
J (Xdx + Ydy + Zdz) = X'8x + Y'Sy + Z'Sz.
This is, of course, a fundamental theorem of Integral Calculus.
Hence we have
X'Bx + Y'Sy + Z'Sz = V(x + 8x,y + Sy,z + Sz) - V (x, y, z\
Let By and 8z be zero, so that the line PP' is parallel to the
axis of x. Then we have
x/ = V(x + 8x,y,z) - V(x, y, z) .
8x ~;
and therefore, in the limit, when P' moves up to P,
X = d~
dx '
89-9l] WORK FUNCTION AND POTENTIAL 83
In like manner we should find
y W y_W
JL = -5— , & = Tf- •
oy dz
The result may be interpreted in the statement : — The force of
the field (estimated per unit of mass), in any direction, is equal to the
rate of increase of the potential per unit of length in that direction.
If, adopting a different notation, we denote by X, Y, Z the
components parallel to the axes of the force acting on a particle,
and if a work function U exists, we have
v dU dU dU
-A — „ - , J: = ^r— , A = ^r— .
ox oy oz
When the components of force are, as here, the partial
differential coefficients of a function of the coordinates, the force is
said to be " derived from a potential."
91. Energy equation. Multiply the left-hand and right-
hand members of the equations of motion
mx = X, my = Y, mz = Z
by x, if, z respectively, and add the results. The sum of the left-
hand members, viz.
m (xx + yy + zz),
is ^[2W(^2+ f + zz)],
where the quantity differentiated is the kinetic energy of the
particle at time t. The sum of the right-hand members is
Xx + Yy+Zz,
and this expression represents the rate at which work is done by
the forces.
Hence we have the equation
jt [im (^ + y2 + z*)] = Xx + Yy + Zz;
and this equation can be expressed in words in the statement : —
The rate of increase of the kinetic energy of a particle is equal to
the rate at which work is done by the forces acting on the particle.
Let s denote the arc of the path measured from a fixed point
6—2
84 FORCES ACTING ON A PARTICLE [CH. Ill
A of it to a variable point P of it. We multiply both sides of the
equation just written by -r- . It becomes
and we hence find the equation
rp
£mv> - %mv* = (Xdx + Ydy + Zdz\
J A
where v and v0 are the values of the velocity of the particle at P
and A, and the integral is a line-integral taken along the path.
The equation can be expressed in words in the statement :—
The increment of kinetic energy in any displacement is equal to the
work done by the forces in that displacement.
When the forces are conservative, and U denotes the work
function, the right-hand member of the equation last written is
U(P) - U (A), and we have
Jmw*- U(P} = const.
We call this equation the " energy equation."
We have already had several examples of energy equations. In the para-
bolic motion of projectiles we have the result in Art. 34, Ex. 3, in the case of
simple harmonic motion we have a result used in Art. 38, in the case of
central orbits we have the result in equation (2) of Art. 50 and the special
results in Art. 40, Ex. 4, and Art. 48, Exx. 1 and 2.
92. Potential energy of a particle in a field of force.
The work function at a point P, with its sign changed, is the
work that would be done by the force of the field upon a particle
which moves from the point P by any path to the chosen fixed
point A.
This quantity is called the " potential energy of the particle
in the field."
The energy equation can be written
" Kinetic Energy + Potential Energy = const."
The potential energy of a body, treated as a particle, in the
field of the Earth's gravity is mgz, where z is the height of the
particle above some chosen fixed level, and m is the mass of the
body.
93. Forces which do no work. When a particle moves on
a fixed curve or surface, forming part of the surface of a body, the
91-94] ENERGY EQUATION 85
pressure of the curve or surface does no work ; for it is always
directed at right angles to the path.
Forces which do no work are frequently called " constraints."
In forming the energy equation we may always omit such
forces from the calculation.
94. Conservative and non-conservative fields. All fields
of force which are found in nature are conservative.
It is easy to invent analytical expressions for non-conservative
fields. For example, let the force at a distance r from a fixed point
be always directed at right angles to the radius vector drawn from
the point, and be equal to /*?• ; and let a particle be guided by a
"constraint" to describe, under the action of the force, a plane
closed curve containing the point.
The work done can be shown easily to be equal to the product
of 2/u, and the area of the curve. Hence every time that the
particle moves round the curve it acquires an increment of kinetic
energy expressed by this product.
If such a system could be devised it could be used to drive
a machine. We should then have a "perpetual motion." The
statement that natural fields of force are conservative is included
in the statement that there cannot be a perpetual motion.
By a " perpetual motion " is meant a self-acting machine which continu-
ally performs work. In the above example the particle, after each circuit of
the curve, might yield up its increment of kinetic energy by striking against
an external body. It would then start always from the same initial position
with the same initial velocity. Its motion would be periodic, and yet it
would transfer kinetic energy to an external body. In natural systems,
when periodic motions are performed without friction, there can be no incre-
ment of kinetic energy available for transfer to an external body. In general
there are forces of the nature of friction which have the effect that, when the
initial position is recovered, the kinetic energy is diminished. For this reason
an ordinary machine, once started, and subject to natural forces, does not go
on for ever, but gradually comes to rest.
It is to be observed that a function U may exist which is such
that the force (X, Y, Z) satisfies the equations
Y dU dU 7_dU
A — — , 1 — -^— , & — -= — ,
ox oy oz
and yet the field of force may not be conservative. In a con-
servative field the work done in displacing a particle round any
closed curve whatever vanishes. Now if U were of the form
86 FORCES ACTING ON A PARTICLE [CH. Ill
A tan-1 (ylx) an amount of work equal to Z-rrA would be done in
displacing a particle round any curve surrounding the axis of z.
We may express the restriction to which this example points by
saying that, in a conservative field, not only is the force derived
from a potential, but also the potential is a one-valued function.
MISCELLANEOUS EXAMPLES
1. Prove that the time of quickest descent along a straight line from
a point on one vertical circle to another in the same plane is
where c is the distance between their centres, a is the sum of the radii, and k
the vertical height of the centre of the former circle above that of the latter.
2. A parabola of latus rectum 4a is placed in a vertical plane with its
vertex downwards and its axis inclined to the vertical at an angle /3. Prove
that the time down the chord of quickest descent from the focus to the
curve is >/(2a^r~1 sec3 J/3).
3. A train of mass m runs from rest at one station to stop at the next
at a distance I. The full speed is F, and the average speed is v. The
resistance of the rails when the brake is not applied is u Vflg of the weight of
the train, and when the brake is applied it is u' Vjlg of the weight of the train.
The pull of the engine has one constant value while the train is getting up
speed, and another constant value while it is running at full speed ; prove that
the average rate at which the engine works in starting the train is
F2 ( 1
4. Two equal bodies, each of mass J/, are attached to the chain of an
Atwood's machine, and oscillate up and down through two fixed horizontal
rings so that each time one of them passes up through a ring it lifts a bar of
mass m, while at the same instant the other passes down through its ring
and deposits on it a bar of equal mass. Prove, neglecting friction, that the
period of an excursion of amplitude a is
and that the successive amplitudes form a diminishing geometric progression
of which the ratio is
where p. is a mass which distributed over the circumference of the pulley will
produce the same effect on the motion as the inertia of the actual mechanism.
5. A particle is projected along the circumference of a smooth vertical
circle of radius a. It starts from the lowest point and leaves the circle before
reaching the highest point. Prove that, if the coefficient of restitution
between the circle and the particle is unity, and if the initial velocity is
V[a<7 {2 +fv/(3-v/3)}],
the particle after striking the circle will retrace its former path.
MISCELLANEOUS EXAMPLES 87
6. A particle moves on the outside of a smooth elliptic cylinder whose
generators are horizontal, starting from rest on the highest generator, which
passes through extremities of major axes of the normal sections. Prove that
it will leave the cylinder at a point whose eccentric angle (f> is given by the
equation
e2 cos3 <f) = 3 cos <£ — 2,
where e is the eccentricity of the normal sections.
7. Two cycloids are placed in the same vertical plane, with their axes
vertical, and their vertices downwards a.nd at the same level. Two particles
start to describe the cycloids from points at the same level. Show that they
will next be at the same level after a time 2n- J(aa')j{(Ja + */a!') *Jg}, and next
after that at time 4n- *J(aa'}/{(*/a + *Ja') *Jg] or 2ir v/(aa')/{(v/a~v/a') %/#}, which-
ever is less, a and a' being the radii of the generating circles.
8. A railway carriage is travelling on a curve of radius r with velocity v,
2a is the distance between the rails and h is the height of the centre of
gravity of the carriage above the rails. Show that the weight of the carriage
is divided between the rails in the ratio gra — v2k : gra + vzh, and hence that
the carriage will upset if
v > \f(ffra/h).
9. A train starts from rest on a level uniform curve, and moves round
the curve so 'that its speed increases at a constant rate /. The outer rail is
raised so that the floor of a carriage is inclined at an angle a to the horizon.
Show that a body cannot rest on the floor of the carriage unless the coefficient
of friction between the body and the floor exceeds
v/C/2 +#2 sm2 a)lff cos «•
10. Prove that the impulse necessary to make a particle of unit mass,
moving in an equiangular spiral of angle a under the action of a force to the
pole, describe a circle under the action of the same force, is
r being the distance from the pole, and F the force at the moment of impact.
•
11. A particle is describing an ellipse of eccentricity e about a focus and
when its radius vector is half the latus rectum it receives a blow which makes
it move towards the other focus with a momentum equal to that of the blow.
Find the position of the axis of the new orbit and show that its eccentricity
is \(e~l — e).
12. A particle is describing an ellipse about a centre of force in one focus
S, and when it is at the end E of the further latus rectum it receives a blow
in direction SE which makes it move at right angles to SE. Find the
momentum generated by the blow, and prove that the particle will proceed
to describe an ellipse of eccentricity {2e2/(l +e2)}.
MOTION OF A PARTICLE UNDEE GIVEN FORCES
95. THE application of the principles which have been laid
down in previous Chapters to the discussion of the motions of
particles in particular circumstances is the part of our subject
usually described as " Dynamics of a Particle." We shall devote
to it the two following Chapters. This part of our subject divides
itself into two main branches, referring respectively to motions
under given forces, and to constrained and resisted motions taking
place under forces which are not all given. We confine our atten-
tion in the present Chapter to motions under given forces.
96. Formation of equations of motion. The method of
formation of the equations of motion has been described in Article
64. It consists in equating the product of the mass of the particle
and its resolved acceleration in any direction to the resolved part
of the force acting upon it in that direction. The equations thus
arrived at are differential equations. The left-hand member of any
equation contains differential coefficients of geometrical quantities
with respect to the time. The right-hand member is, in general,
a given function of geometrical quantities ; in special cases it may
be a given function of the time. Although there are many cases in
which equations of this kind can be solved, there exists no general
method for solving them.
Diversity can arise, in regard to the formation of the equations,
only from the choice of different directions in which to resolve.
Thus we may resolve parallel to the axes of reference, or we may
resolve along the radius vector from the origin to a particle, and
in directions at right angles thereto, or again we may resolve along
the tangent to the path of a particle and in directions at right
angles thereto. The most suitable directions to choose in particular
cases are determined by the circumstances.
t Articles in this Chapter which are marked with an asterisk (*) may be omitted
in a first reading.
95-97] EXPRESSIONS FOR COMPONENT ACCELERATIONS 89
Methods by which the components of acceleration in chosen directions
can be expressed in terms of suitable geometrical quantities have been
exemplified in Arts. 36 and 43. Further illustrations are given in the next
two Articles.
*97. Acceleration of a point describing a tortuous curve.
We recall the facts that, if x, y, z are the rectangular coordinates of a point
of a curve and s the arc measured from some particular point of the curve to
the point (x, y, z), the direction cosines of the tangent, in the sense in which s
dx du dz . „ . , , . (dx\2 /dy\2 idz\2
increases, are -=- , ~ , -j- , satisfying the relation ^- +~r -H ~r 1 "" 1 5
ds ds ' ds' \ds/ \dsj \dsj
the direction cosines of the principal normal directed towards the centre of
curvature are p -j-, , p -j4[ , p -T-JJ , satisfying the relation
where p is the radius of circular curvature ; and the direction cosines of the
/d2ydz d2zdy\ /d2z dx d2xdz\ fd2x dy d2u dx\
bmormal are p I -^ -5- — -T-» --r I , P ( -rs -5 -7-3 ~r , p ( ~r^, -,- - ~r» -y- •
r \ds2 ds ds2 ds / r \ds2 ds ds2 ds) r \dsL ds as2 ds)
..... dx d2x dy d2y dz d2z
We recall also the relation -= — =-5- + rf- -y~ + -=- -5-5 »0.
OS ds2 ds ds* ds as-
In the expressions ,i:, y, z for the component accelerations parallel to the
axes we change the independent variable from t to s.
We have, writing v for the speed, so that v stands for s,
d2x d /dx\ ds d (ds dx\ d ( dx
dt2 dt \dt) dt ds \dt ds) V ds \ ds
dv dx „ d2x
so that x = v -. — j- + v2 —j-s .
ds as ds*
dv du Ud2y
Sim ilarly y = v -=- -j- + v* -~ ,
ds ds ds2
dv dz „ d2z
and z = v — [- v — .
If we multiply these component accelerations in order by the direction
cosines of the tangent and add, we obtain the component acceleration parallel
to the tangent to the curve in the sense in which s increases ; we thus find
for this component the expression
dvT/dxY2 fdy\z A^\2~] % fdx d2x dy d2y dz d2z\ dv
Vds[_\d~s) +\Js) WJ+* W* ~d# + d* ^? + ds ds2 ) ' ° ^' ds '
Again, if we multiply by the direction cosines of the principal normal and
add, we obtain the component acceleration parallel to the principal normal
directed towards the centre of curvature ; we thus find for this component
the expression
dv [dx d^x dy d2^ dz d*z~\ 2 [id2x\2 (d2y\ fd%
Vdsp\_ds ds2 + ds ds2+dsds2_\"Vl}[\ds2) +\ds2) +\ds2
or — .
P
90 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
Finally, if we multiply by the direction cosines of the binormal and add,
we find no component acceleration parallel to the binormal.
Thus the acceleration of a point describing a tortuous curve is in the
osculating plane of the curve, and its resolved parts parallel to the tangent
and principal normal are v ^ and -, exactly as in the case of a point describing
a plane curve. As in that case, the expression for the former component may
be replaced by v, or by ».
*98. Polar coordinates in three dimensions. The co-
ordinates are r the distance from the origin, 6 the angle between the radius
vector and the axis 2, <£ the angle between the plane containing the radius
vector and the axis z and a fixed plane passing through the axis z.
The plane containing the radius vector and the axis z will be called the
"meridian plane," and the circle in which this plane cuts a sphere r= const,
the "meridian."
We denote distance from the axis z by -or, so that w=r sin 6.
In a plane parallel to the plane (#, y\ -at and (f> are plane polar co-
ordinates ; in the meridian plane z and or are Cartesian coordinates, and r and
6 are plane polar coordinates.
Hence the velocity (ir, y) parallel to the plane (x, y} is equivalent to ra at
right angles to the axis z in the meridian plane, and tzr<£ at right angles to
this plane ; and the velocity (£, y, z) is equivalent to (z, w] in the meridian
plane and n:0 at right angles to this plane. Also the velocity (z, -si) in this
plane is equivalent to r along the radius vector and rd along the tangent to
the meridian. The components of velocity are therefore
r along the radius vector;
r6 along the tangent to the meridian,
r sin 0<j> at right angles to the meridian plane.
The accelerations x, y parallel to the axes #, y are equivalent to w — rzr<^2
and — r (w2d>) in and perpendicular to the meridian plane. Hence the
or at ^
acceleration is equivalent to z parallel to the axis z, w — w<p? at right angles
to the axis z and in the meridian plane, - -j- (ro2</>) at right angles to the
w dt
meridian plane.
Taking the components 2, or, which are in the meridian plane and are
parallel and perpendicular to the axis z, we see that these are equivalent to
r-r62 along the radius vector and - -=- (r2#) along the tangent to the
meridian.
We resolve the acceleration — S702, which is in the meridian plane and at
right angles to the axis 2, into components parallel to the radius vector and
97-101] EXPRESSIONS FOR COMPONENT ACCELERATIONS 91
to the tangent to the meridian. These components are — tap'sind and
— t<7</>2 cos B. Hence the components of acceleration are
r -r62-r sin2 #$2 along the radius vector,
- -7- (r2#) — r sin 6 cos $<£>2 along the tangent to the meridian,
-=- (r2sin2#0) at right angles to the meridian plane.
99. Integration of the equations of motion. Whenever
there is an energy equation (Art. 91) it is an integral of the equa-
tions of motion.
When the particle moves in a straight line under conservative
forces the energy equation expresses the velocity in terms of the
position ; and the position at any time, or the time of reaching
any position, is determined by integration. For an example see
Art. 54.
When the particle does not move in a straight line other
integrals of the equations are requisite before the position at any
time can be determined. If there is an equation of constancy of
momentum (Art. 83), or of moment of momentum (Art. 85), these
also are integrals of the equations of motion. These, combined
with the energy equation, are sometimes sufficient to determine
the position at any time. Examples are afforded by the parabolic
motion of projectiles and by elliptic motion about a focus.
100. Example.
Deduce the result that the path of a particle moving freely under gravity
is a parabola from the equation expressing the constancy of the horizontal
component of momentum and the energy equation.
101. Motion of a body attached to a string or spring.
Simple examples of Dynamics of a Particle are afforded by prob-
lems of the motion of a body attached to an extensible string or
spring. We consider cases in which the particle moves in the line
of the string or spring (supposed to be a straight line).
When the mass of the string is neglected *f*, and there is no
friction acting upon it, the tension is constant throughout it
(Chapter VI).
When the length of a string can change there is a particular
length which corresponds to a state of zero tension. This state
t A string of which the mass is neglected is often called a "thread."
92 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
is called the " natural state," and the corresponding length the
" natural length."
Let 10 be the natural length, I the length in any state. The
quantity (I — 10)/10 is called the " extension."
The law connecting the tension and the extension is that the
tension is proportional to the extension. If e is the extension, the
tension is equal to the product of e and a certain constant. This
constant is called the " modulus of elasticity " of the string.
If, in the course of any motion of an extensible string, the
string recovers its natural length, the tension becomes zero, and
the string becomes " slack." A particle attached to the string is
then free from force exerted by the string until the length again
comes to exceed the natural length.
A string which exerts tension, but is never sensibly extended,
must be thought of as an ideal limit to which an extensible string
approaches when the extension e tends to zero, and the modulus X
tends to become infinite, in such a way that the product Xe is the
finite tension of the string. Such a string would be described as
"inextensible."
A spring, when extended, exerts tension in the same way as
an extensible string ; when contracted, it exerts pressure which is
the same multiple of the contraction (/0 — 1)/10 as the tension is of
the extension.
A body attached to a spring, of which one end is fixed, and
moveable in the line of the spring, is subject to a force equal to
fjix, where /A is a constant called the " strength of the spring," and
x is the displacement of the body from the position in which the
spring has its natural length. When the length is increased by x
the force is tension ; when it is diminished by x the force is
pressure. The equation of motion of the body, considered as a
particle of mass m, is mx = — fix.
It follows that the motion of the particle is simple harmonic
motion of period 2?r \/(ra//i).
This result may also be obtained by forming the energy equation. For
the work done by the force in the displacement x is
101-102A] FORCES PRODUCING SIMPLE HARMONIC MOTION 93
or it is - 1/xA'2 ; and the kinetic energy of the body, treated as a particle, is
^mx2. Hence the energy equation is
tynx2 + |/ivi'2 = const.,
and the result that x is of the form a cos {t ^(n/m) + a} can be obtained by
integrating this equation.
102. Examples.
1. A particle of mass m is attached to the middle point of an elastic
thread, of natural length a and modulus X, which is stretched between two
fixed points. Prove that, if no forces act on the particle other than the
tensions in the parts of the thread, it can oscillate in the line of the thread
with a simple harmonic motion of period «• *J(ma/\).
2. A particle of mass m is attached to one end of an elastic thread, of
natural length a and modulus X, the other end of which is fixed. The
particle is displaced until the thread is of length a + b, and is then let go.
Prove that, if no forces act on the particle except the tension of the thread, it
will return to the starting point after a time 2 ( «• + 2 7- ) A /~\~ •
\ o/ \f \
3. Prove that, if a body is suddenly attached to an unstretched vertical
elastic thread and let fall under gravity, the greatest subsequent extension is
twice the statical extension of the thread when supporting the body.
4. Prove that, if a spring is held compressed by a given force and the
force is suddenly reversed, the greatest subsequent extension is three times
the initial contraction.
5. An elastic thread of natural length a has one end fixed, and a particle
is attached to the other end, the modulus of elasticity being n times the
weight of the particle. The particle is at first held with the thread hanging
vertically and of length a', and is then let go from rest. Show that the time
until it returns to its initial position is
2 (TT - S + ff + tan 6 - tan 6') J(ajng),
where 6, & are acute angles given by
sec 6 = na'/a — n-l, sec2 & = sec2 6 - 4ra,
and a' is so great that real values of these angles exist.
102 A. Force of simple harmonic type. The most im-
portant case of forces, which are given in terms of the time, is
that where simple harmonic motion is disturbed by a force pro-
portional at time t to a function of the form cos ( pt + a), in which
p and a are constants. Such forces have definite periods, and are
often loosely described as "periodic forces"; they may be described
strictly as " forces of simple harmonic type."
Let a particle of mass m be attached to a spring of such
strength that, if free from the action of all forces except that
94 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
exerted by the spring, it would have a simple harmonic motion of
period 2ir/n ; and let it be subject also to a force of magnitude
mP cos(pt + a) in the line of the spring. The equation of motion is
x + iC-x = P cos ( pt + a).
A solution of this equation would be found by putting x equal to
Q cos(pt + a) if (w2 — p-) Q = P. To obtain a more general solution
we put
x = (p/(n* - p*)} cos ( pt + a) + £
and then find that £ satisfies the equation £ + n?£ = 0, so that
f must be of the form A cos nt + B sin nt, where A and B are
arbitrary constants.
The motion of the particle is compounded of two simple
harmonic motions in the same straight line, one having the period
that the motion in the absence of the disturbing force would have,
and the other having the period of the disturbing force. These
are called the " free oscillation " and the " forced oscillation." The
phase of the forced oscillation is the same as that of the force
producing it if n > p, that is if the period of the force is longer
than the period of free oscillation. When the period of the force is
shorter than the period of free oscillation, or p > n, the phase of
the forced oscillation is opposite to that of the force producing it,
or the displacement in the forced oscillation always has the opposite
sign to the force.
The amplitude of the forced oscillation becomes very great
when p is nearly equal to n. When p = n the equation of motion
is satisfied by putting x = (P/2n) t sin (nt + a), as may be verified
easily, and the complete primitive of the equation is then
a; = A cos nt + B sin nt + (P/2n) t sin (nt + a).
In this case the forced oscillation may be described as a simple
harmonic motion of variable amplitude, which increases continu-
ally with the time. The phase of this motion is always one quarter
of a period behind that of the force producing it.
The above is an example of a principle of wide application, to
the effect that a system, which can oscillate in a definite period,
can be thrown into a state of violent oscillation by the action of
forces, which are of simple harmonic type and the same period.
On account of its importance in the Theory of Sound this result
is known as the " principle of resonance."
102A-104] THE PROBLEM OF CENTRAL ORBITS 95
103. The problem of central orbits. We have already in-
vestigated this problem in some detail in Arts. 49 — 52. We found that
a particle moving under a central force directed to a fixed point, moves in
a fixed plane which contains the centre of force and the tangent to the path
at any chosen instant. We found that the equations of motion could be
expressed in the form
m(r-r^=-mf, mi|(rM)-0,
where m is the mass of the particle, and / is the intensity of the field of force,
taken to be an attraction. We suppose that f is given as a function of r.
The energy equation is
im (r2 + r*6-} = const. — m I fdr.
J
and the equation of constancy of moment of momentum about an axis through
the centre of force at right angles to the plane of motion is
mrzB = mh,
where h is a constant which represents twice the rate of description of area
by the radius vector.
We found that these equations lead to the equation
/du\2 2 A 2 f f ,
— \ +ut=——-\-— i •'—du*
\dQ h- i* ] u*
\ / j
where J. is a constant, u is written for I/r, and / is now supposed to be ex-
pressed in terms of «. This equation determines the path of the particle.
When / is given, and the particle starts from a point at a distance a
from the centre of force, with a velocity T, in a direction making an angle a
with the radius vector, the value of h is Va sin a. The initial value of
( -jT. } + w2 is I/a2 sin2 a, for it is the reciprocal of the square of the perpendicular
\atij
from the origin of r upon the tangent to the path. Hence the equation of
the path takes the form
MA2, 2_ 1 2 fu f_j
\dd) "" ' a2 sin2 a " F2a2 sin2 a J i ^ '
a
When the path is known, so that u becomes a known function of 6, the
time of describing any arc of the path is the value of the integral
I
Va sin a '
taken between limits for 6 which correspond to the ends of the arc.
104. Apses. An apse is a point of a central orbit at which
the tangent is at right angles to the radius vector.
There is a theory concerning the distribution of the apses when
the central acceleration is a single-valued function of the distance,
96 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
O
Fig. 36.
i.e. for the case where the acceleration depends only on the distance
and is always the same at the same distance.
Let A be an apse on a central orbit described about a point 0,
/ the central acceleration, supposed a
single- valued function of distance, TAT'
a line through A at right angles to AO.
Then a point starting from A at right
angles to AO with a certain velocity
would describe the orbit. Let V be this
velocity.
If a point starts from A with velocity
V in direction AT or AT', and has the
acceleration f towards 0, it describes the
orbit; so that two points starting from A
in these two directions with the same velocity V and the same
acceleration f describe the same orbit. Since the two points have
the same acceleration at the same distance, the curves they de-
scribe are clearly equal and similar, and are symmetrically placed
with respect to the line AO. Thus the orbit is symmetrical with
respect to AO in such a way that chords drawn across it at right
angles to AO are bisected by AO. The parts of the orbit on either
side of AO are therefore optical images in the line AO.
Now let the point start from A in direction AT, and let B
be the next apse of the orbit that it passes
through, also let A' be the next apse after B
that it passes through. Then the parts A OB,
BOA of the orbit are optical images in the
line OB, and the angle A OB is equal to the
angle A'OB, and the line AO is equal to
the line A'O. In the same way the next
apse the point passes through will be at a
distance from 0 equal to OB, and thus all
the apses are at distances from 0 equal to
either OA or OB ; these are called the apsidal distances, and the
angle between consecutive apses in the order in which the moving
point passes through them is always equal to AOB; this is called
the apsidal angle.
The theory just explained is usually stated in the form:—
There are two apsidal distances and one apsidal angle.
- 37-
104-106] APSES OF CENTRAL ORBITS 97
It is clear that the radius vector is a periodic function of the
vectorial angle with period twice the apsidal angle.
105. Examples.
1. If the apsidal distances are equal the orbit is a circle described about
its centre.
2. Write down the lengths of the apsidal distances and the apsidal angle
for (1) elliptic motion about the centre, (2) elliptic motion about a focus,
(3) all the orbits that can be described with a central acceleration varying
inversely as the cube of the distance.
3. Explain the following paradox :— Four real normals can be drawn to
an ellipse from a point within its evolute, and in Ex. 6 of Art. 46 we found
the central acceleration to any point requisite for the description of an ellipse ;
there are apparently in this case four apsidal distances and four apsidal
angles.
106. Apsidal angle in nearly circular orbit. Let the
central acceleration be f(r) at distance r, then a circle of radius c V^
described about its centre is a possible orbit with ^h for rate of
describing area provided that
or A2 = c3/(c).
Let us suppose the point to be at some instant near to the
circle, and to be describing an orbit about the origin with moment
of momentum specified by this h.
The equation of its path is
#u _f(r)
d0* * h*u* '
At the instant in question u is nearly equal to 1/c ; if it was
precisely 1/c, and if the point was moving at right angles to the
radius vector, the point would describe the circle of radius c. We
assume that it is always so near to the circle that the difference
u — 1/c is so small that we may neglect its square ; the investigation
we give will determine under what condition this assumption is
justifiable.
Write <£ (u) for/(r) and a for 1/c, and put u = a + x, so that
h? = $ (a)/a3.
L. M. 7
98 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
Then
d*x a*d> (a + x) 1
-J5- + X + a = - 7^— , — TT,
d(f- 9 (a) (a + xy
.
i(a) a2 da
»+-{»r^r
d6* ( <j> (a)
if x2 is neglected.
Now if 3 — aft (a)/<f> (a) is positive we may put it 'equal to /c2,
and then the solution of the above equation is of the form
x = A cos (icQ 4- a),
so that the greatest value of x is A, and by taking A small enough
x will be as small as Ave please and the neglect of x2 will be
justified.
In this case u, and therefore r, will be a periodic function of d
with period 2-7r/\/{3 — a<£'(a)/</> (a)}, the orbit is nearly circular and
its apsidal angle is 7r/V{3 — «</>' («)/<£ («)}•
Again, if 3 — a<£'(a)/</> (a) is negative we may put it equal to
— «2, and then the solution of the above equation is of the form
x = Ae*e + Be-*e,
and it is clear that one of the terms increases in geometrical pro-
gression whether 6 increases or diminishes, so that x will very soon
be so great that its square can no longer be neglected, whatever
the number we agree to neglect may be. In this case the orbit
tends to depart widely from the circular form.
In the former of these cases the circular motion is said to be
stable, in the latter unstable.
107. Examples.
1. If f(r) = r~n or </>(W) = M", prove that the possible circular orbits are
stable when n < 3 and unstable when n > 3.
2. For n = 3 prove that the circular orbit is unstable, and find the orbit
described by a point moving with the moment of momentum required for
circular motion in a circle of radius c through a point near the circle.
3- If f(r) = r~ 4, prove that the curve described with the moment of
momentum required for circular motion in a circle of radius c, when the
point of projection is near to or on this circle, is either the circle r=c or one
of the curves
r _ cosh 0 + 1 r _ cosh 0 — 1
c cosh tf - 2 ' c ~ cosh 6 + 2 '
106-109] RADIAL AND TRANSVERSAL RESOLUTION 99
108. Examples of equations of motion expressed in terms
of polar coordinates.
1. When the radial and transverse components of force acting on a
particle which moves in one plane are /?, T, the equations of motion are
m(r-rfa = R, - ~ (rW) = T.
r at
2. When the forces are derived from a potential V we have
9F mdV
R=m^> r=7M'
and there is an energy equation
im (ja + ,.202) = m Y+ const>
3. Put r26 = k, u = r~l ; in general h is variable. The equation of the
path can he found by eliminating h between the equations
d T ^'
4. When the forces are derived from a potential, as in Ex. 2, the equation
of the path can be written in the form
5F= 3d_ V
d d du d
where -T^ stands tor ^ + -^ ~- .
dQ c6 dd cu
109. Examples of motion under several central forces.
1. A particle of mass m moves under the action of forces to two fixed
points A, A' of magnitudes m/j./r2, mp.'/r'2 respectively, where r and / are the
distances of the particle from A and A', and \i and p.' are constants. The
equations of motion possess an integral of the form
rV W = a (p. cos 6 - p.' cos &} + const.,
where a is the distance AA'.
7—2
100 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
Resolving at right angles to the radius vector r, we have
m- j- (r20) = m ^sin x, where x *s the angle A PA',
T dt T
so that r'2 j- (r2f>) =p?r sin x =/*'« si" 0',
similarly ?-2 -3- (/2£') = - /*/ sin ^ = - pa sin 0.
dt
Multiplying by 6' and 0, adding, and integrating, we have an equation of
the given form.
This equation with the energy equation determines the motion.
2. A particle of mass m moves under the action of forces to two fixed
points of magnitudes mp.r, mpfr'. Prove, with the notation of Ex. 1, that
there is an integral equation of the form
p.r'2d + p.'r"20' =• const.
3. A given plane curve can be described by a particle under central forces
to each of n given points, when the forces act separately. Prove that it can
be described under the action of all the forces, provided that the particle is
properly projected.
Let/* be the acceleration produced in the particle by the force to the <th
centre 0K, VK the velocity of the particle at any point when the curve is
described under this force, rK the distance of the point from 0K, andjoK the
perpendicular from 0K on the tangent to the .curve at the point, p the radius
of curvature and ds the element of arc of the curve at the point. Then we
are given that
* ds J* ds ' p ~^K rK '
Now the curve can be described under all the forces if there exists a
velocity V satisfying the two equations
d V n dr V2 n n
ds ~ i ok ' p ~ i ?'« '
and it is clear that these are satisfied by
H
1
Thus the condition is that the kinetic energy when all the forces act must
be the sum of the kinetic energies when they act separately.
4. Prove that a lemniscate rr'=c2, where 2c is the distance between the
points from which r and r' are measured, can be described under the action
of forces m^r and wi/*// directed to those points, and that the velocity is
constant and equal to § x/(»V)-
5. A particle describes a plane orbit under the action of two central
forces each varying inversely as the square of the distance, directed towards
two points symmetrically situated in a line perpendicular to the plane of the
orbit. Show that the general (p, r) equation of the orbit, referred to the
109-lll] SEVERAL CENTRAL FORCES 101
point where the line joining the centres of force meets the plane as origin, is
of the form
(l-a2/p2)2 = 62/(c2 + r2),
where c is the distance of either centre of force from the plane, and a and b
are constants.
6. A point describes a semi-ellipse, bounded by the axis minor, and its
velocity, at a distance r from the nearer focus, is a^/{f(a-r)/rC2a-r)},
2a being the axis major, and / a constant. Prove that its acceleration is
compounded of two, each varying inversely as the square of the distance, one
tending to the nearer focus, and the other from the further focus.
110. Disturbed elliptic motion. The motion of the Planets
about the Sun does not take place exactly in accordance with
Kepler's Laws (Art. 41). Although the Sun's gravitational attraction
preponderates very greatly over the attractions between the Planets,
these attractions are not entirely negligible. The theory of the
motion of the Planets presents us with the problem of determining
a motion which, apart from relatively small forces, would be elliptic
motion about a focus.
We shall consider here some examples of elliptic motion dis-
turbed by small impulses in lines which lie in the plane of the
orbit. The ellipse described after the impulse is a little different
from that described before. The ellipses, having a given focus, are
determined by the lengths of the major axes, the eccentricities, and
the angles which the apse lines make with some fixed line in the
plane of the orbit. We denote the major axis by a, the eccentricity
by e, and the angle in question by OT.
111. Tangential impulse. Let a particle P, describing an
elliptic orbit about a focus S, receive a small tangential impulse
increasing its velocity by &v. Let R be the distance of the particle
from S at the instant, fjb/r2 the acceleration to S when the distance
is r, a + Sa the semi-axis major of the orbit immediately after the
impulse.
We have, by Ex. 2 of Art. 48,
2 1
2
a
JJ
giving — = — — approximately.
102 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
Again, if h is the moment of the velocity about S before the
impulse, h+Bh afterwards, since the tangent to the path is un-
altered, we have
h + Bh h
v + Bv ~ v '
giving
Fig. 39.
Hence if I is the semi-latus rectum before the impulse, I +
afterwards, we have
, with A2 = pi,
ts
giving $1 = 11— approximately.
Now l = a(\- e*\ and if e becomes e + Be,
Bv
(1 - e-) Ba - 2eaBe = 2a (1 - e2) — ,
(l-e2)f vBv Bv
giving &r»^T^[
l-e22Sw /I 1\
or oe = -- a I "D -- •
e v \R aj
The equation of the orbit will be changed from
Ifr = 1 + e cos (0 - w) to (I + Bl)/r = 1 + (e + Se) cos \0-(ia + B&)}.
Taking TO- = 0 we find
... SI Be (I \
esm^..6CT= •= -- ^-1 .
R e \R )
If the particle is subject to a disturbing force producing a small
tangential acceleration f we shall have
2aV 2/Z/l 1\
a = - *-~ «= — - D~- K
fj, v e\R a)
21 f e/l
e sin -J- - -
21 f e/l _\
= -=J- - - I ^ - 1 .
Jc w e VR /
111-113] DISTURBED ELLIPTIC MOTION 103
112. Normal impulse. Suppose the particle .to receive an
impulse imparting to it a velocity 8v in the direction of the normal
inwards. Then the resultant velocity is, to the first order, un-
altered, and consequently a is unaltered, or 8a = 0.
If p is the perpendicular from the focus 8 on the tangent at P,
meeting it in Y, then the value of h is increased by PYSv, or we
have
Bh = VCR2 - p') Bv.
Hence ^l = 2h8h = ZpvSv V(#2 - p*) :
also SI = - 2aeSe, so that
5. pvSv D.,
be = — - — V(^ — p\
The equation giving Stn- is
- 2aeSe/R = (^ - l) ^ + e sin 6 . Svr.
If the particle is subject to a disturbing force producing a small
normal acceleration / we have
• n • P/v I/-DI -a • /2a<? , l-R\
a = 0, e = - ^— *J(R2 - p2), esin.0.'sr = — e( —^ -\ -- ~- .
\ R
113. Examples.
1. For a small tangential impulse prove that
be = 28v (e + cos d)/v, SOT = 28v sin 6 lev.
2. For a small normal impulse prove that
8e = - r8v sin 6/av, 8& = 8v (2ae + r cos d)/aev.
3. For a small radial impulse prove that
8a = 2a?e8v sin 0/k, 8e=h8v sin d/p, 8or = — h8v cos d/e^.
4. For a small transversal impulse prove that
e cos 6)/h, 8e = 8v {r(e + cos 6) + 1 cos 6}//i, 8&=8v sin 6 (l + r)jeh.
MISCELLANEOUS EXAMPLES
1. Eelatively to a certain frame a point 0 describes a straight line
uniformly with velocity V, and a second point P describes a curve in such
a way that the line OP describes areas uniformly ; prove that the resolved
part perpendicular to OP of the acceleration of P is 2 Vv sin (j>/OP, where v
is the velocity of P, and 0 the angle which the tangent to its path makes with
that of 0.
104 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
2 Relatively to a certain frame, a point A describes a circle (centre 0)
uniformly and a point B moves with an acceleration always directed to A
? thTar - covered by the line AB is described uniformly, prove that he
resolved part parallel to OA of the velocity of B is proportional to the
perpendicular from B on 0 A produced.
3 Prove that, if the acceleration of a point describing a tortuous curve
p dv
makes an angle * with the principal normal, then ten + = - ^ .
In the case of a plane curve the condition that the acceleration is always
d p cos \//- _ _ ,
directed to the same point is that the equation sin^+^ j— ^ = '
be satisfied at every point.
4 The position of a point is given by x, y, r, where *, y, z, r have their
usual signification relative to rectangular axes ; show that the component
accelerations are
«, i\ w being component velocities in the directions x, y, r.
5. If x, y are the coordinates of a point referred to rectangular axes
turning with angular velocity a, prove that the accelerations in the directions
of the axes are
and
6. The radii vectores from two fixed points distant c apart to the position
of a particle are rl5 rt, and the velocities in these directions are ult u2 ; prove
that the accelerations in the same directions are
7. The radii vectores from three fixed points to the position of a particle
are r,, r2, r3 and the velocities in these directions are ult u2, u3; prove that
the accelerations in these directions are
- ^ (U.2 cos du + uz cos 013),
rt
and the two similar expressions, in which 023, 6Z\, 6n are the angles contained
by the directions of (r2, r3), (r3, r,) and (rlf r2).
8. A particle is suspended from a point by an elastic thread and oscillates
in the vertical line through the point of suspension. Prove that the period
is the same as that of a simple pendulum of length equal to the excess of the
length of the thread in the position of equilibrium above its natural length.
9. A particle is attached to a fixed point by means of an elastic thread
of natural length 3a, whose coefficient of elasticity is six times the weight
of the particle. When the thread is at its natural length, and the particle
MISCELLANEOUS EXAMPLES 105
is vertically above the point of attachment, the particle is projected hori-
zontally with a velocity 3 >J(%ag) ; verify that the angular velocity of the
thread can be constant, and that the particle can describe the curve
r=a (4 — costf).
10. A particle moves in a nearly circular orbit with an acceleration
(j. + v(r — a), a being the mean radius; show that the apsidal angle is
•jr<a/^f(3a>2 + v), where co is the mean angular velocity.
11. A particle describes a central orbit with acceleration /*/(?• - of towards
the origin, starting with the velocity from infinity at a distance c (which is
greater than a and less than 2«) at an angle 2 cos"1 \/(a/c). Prove that the
path is given by the equation
£0= tanh - l ,/{('>• - a) I a} - tan - l v'{(r - a) /a}.
12. A particle is projected with velocity less than that from infinity under
a force tending to a fixed point and varying inversely as the nth power of the
distance. Prove that if n is not < 3 the particle will ultimately fall into the
centre of force.
13. A particle moves under a central force varying inversely as the »th
power of the distance (n > 1 ), the velocity of projection is that due to a fall
from rest at infinity, and the direction of projection makes an angle /3 with the
radius vector of length R. Prove that the maximum distance is R (cosec /3)2/("~3)
when n > 3, and that the particle goes to infinity if n= or < 3.
14. Prove that, if a possible orbit under a central force <£(/•) is known, a
possible orbit under a central force </>(?•) +Xr~3 can be found. In particular
prove that a particle projected from an apse at distance a with velocity
^/(X + /*)/«, under an attraction
%u.(n-\)an-*r-n + \r- 3, O>3),
will arrive at the centre in time
15. A particle is describing a circular orbit of radius a under a force to
the centre producing an acceleration f(r) at distance r, and a small increment
of velocity Aw is given to it in the direction of motion. Prove that the
apsidal distances of the disturbed orbit are
a and a + 4Aw «/{a/(a)}/{3/(a) + a/' (a)}.
Prove also that, if the increment of velocity imparted to the particle is
directed radially, the apsidal distances are approximately
16. A particle moves in a plane under a radial force P and a transverse
force T7, where
106 MOTION OF A PARTICLE UNDER GIVEN FORCES [CH. IV
prove that a first integral of the differential equation of the path can be
expressed in the form
where A02 and C are constants.
17. A particle moves under the action of a central force P and a transverse
disturbing force -/(*)• Prove that
d&+ ~
where F(t) = \f(t)dt.
18. A particle describes a circle under the action of forces, tending to the
extremities of a fixed chord, which are to each other at any point inversely as •
the distances r, r' from the point to the ends of the chord. Determine the
forces, and prove that the product of the component velocities along r and r'
varies inversely as the length of the perpendicular from the position of the
particle to the chord ; also show that the time from one end of the chord to
the other is
a (TT — O) COS a + sin a
V ~~cosTii~~
where V is the velocity of the particle when moving parallel to the chord,
a the radius of the circle, and a the angle between r and /.
19. A particle is projected from an apse of Bernoulli's Lemniscate
(r/ = c2) along the tangent with velocity v//*/2c and moves under the action of
forces
r'js* I" (3rr'-/2)3'
to the nearer and further poles respectively, r being the distance from the
nearer pole, and r' from the further pole. Show that it describes the lemniscate.
20. A particle P moves under the action of two fixed centres of force
Si, $2 producing accelerations pilrf and /i2/?*22 towards S\ and S-2, where i\, r2
are the distances S}P, SZP. Prove that, if the motion does not take place in
a fixed plane, there is an integral equation of the form
(J'i20i ) (r22^2) + ^2 c°t #1 cot #2 = c (MI cos #1 + M2 cos #2) + const. ,
where 01} 02 are the angles S^Sj^P and SiS2P, c is the distance S^, and h is
the moment of the velocity about the line of centres.
21. An ellipse of eccentricity e and latus rectum 21 is described freely
about a focus, with moment of momentum equal to h. When the particle is
at the nearer apse it receives a small radial impulse /*. Prove that the apse
line is turned through the angle
MISCELLANEOUS EXAMPLES 107
22. A particle of mass in describes an ellipse about a focus, pm being the
force at unit distance ; when the particle is at an extremity of the minor axis
it receives a small impulse m V in a direction perpendicular to the plane of
the orbit ; prove that the eccentricity of the orbit will be diminished by
£ V2ae/n, and that the angle which the axis major of the orbit makes with the
distance from the focus will be increased by
V*a (2 -e*)/{2pej(l -<?)},
where 2a is the axis major, and e the eccentricity of the orbit.
23. A comet describes about the Sun an ellipse of eccentricity e nearly
equal to unity. At a point where the radius vector makes an angle 6 with
the apse line, the cornet is instantaneously affected by a planet so that its
velocity is increased in the ratio n + 1 : n, where n is great, without altering its
direction. Show that, if the new orbit is a parabola,
e= I — (4/ri) cos2 %0 nearly.
24. A particle is describing an ellipse under a force to a focus S, and,
when the particle is at P, the centre of force is suddenly moved a short
distance x parallel to the tangent at P. Prove that the axis major is turned
through the angle (xjSO] sin </>sin (#-$), where G is the foot of the normal,
6 the angle which the normal makes with SG, and <£ the angle which the
tangent makes with SP.
25. Denning the instantaneous orbit under a central force varying as the
distance as that orbit which would be described if the resistance ceased to act,
show that, if at any point the resistance produces a retardation /, the rates
of variation of the principal semi-axes are given by the equations
d b f
a (a2 - r2) b (r2 - 62) v (a2 - &*) '
where v is the velocity and r the radius vector at the instant.
26. In the last Example there is a disturbance which produces a normal
acceleration g instead of the resistance. Show that the maxima of the rates
of variation of the principal semi-axes of the instantaneous ellipse are given
by the equations
d b + g
b~ a~
where p. is the central force on unit mass at unit distance.
CHAPTER Vt
MOTION UNDER CONSTRAINTS AND RESISTANCES
114. THE second main subdivision of "Dynamics of a Particle"
relates to motion of a particle in a given field of force when the
force of the field is not the only force acting on the particle, but
there are other, unknown, forces acting upon it.
Such forces may be constraints, that is to say they may do no
work. Another class of forces to be included in the discussion are
known as resistances. We had an example in the friction between
an inclined plane and a body placed upon it (Art. 71). The
characteristics of a resistance are that its line of action is always
the line of the velocity of the particle on which it acts, and its
sense is always opposed to the sense of the velocity. It follows
that the work done by a resistance is always negative. This work,
with its sign changed, is called the " work done against the
resistance."
When a particle moves in a given field of force, and is at the
same time subject to resistances, the increment of the kinetic
energy in any displacement is less than the work done by the force
of the field by the work done against the resistances.
115. Motion on a smooth plane curve under any forces.
Let a particle of mass m be constrained to move on a given smooth
plane curve under the action of given forces in the plane. Let s
be the arc of the curve measured from some point of the curve up
to the position of the particle at time t. Let S be the tangential
component of the forces in the direction in which s increases, and
N the component along the normal inwards. Let v be the velocity
of the particle in the direction in which s increases, and R the
pressure of the curve on the particle. We shall write down the
equations for the case where the particle is on the inside of the
curve, and R accordingly acts inwards. The equations for the case
in which R acts outwards can be obtained by changing the sign
ofE.
t Articles in this Chapter which are marked with an asterisk (*) may be omitted
in a first reading.
114-118] MOTION ON A SMOOTH CURVE 109
By resolving along the tangent and normal we obtain the
equations of motion
dv
mv -,- = 8.
as
v2
m-=N+R
P
When the forces are conservative, the first of these equations
has an integral, which is identical with the energy equation. It
may be written
m y2 = Sds + const.
When v is known from this equation, the second of the equations
of motion determines the pressure R.
In the case of one-sided constraint (Art. 77) the particle may
leave the curve. This happens when R vanishes.
116. Examples.
1. Prove that, when the particle leaves the curve, the velocity is that
due to falling under the force kept constant through one quarter of the chord
of curvature in the direction of the force.
2. Prove that, when the curve is a free path under the given forces for
proper velocity of projection, then, for any other velocity of projection, the
pressure varies as the curvature.
117. Motion of two bodies connected by an inextensible
string. We shall suppose that the bodies may be treated as
particles, that the mass and extension of the string can be neglected,
and that the tension of the string is the same throughout. (See
Chapter VI.) When this is the case the tension of the string does
no work, for the sum of the rates at which it does work on the two
particles vanishes. The equations of motion of the bodies can be
formed in the manner explained in Art. 73. In forming the
equations of motion we take account of the condition that the
length of the string is constant. For example, if the string is in
two portions, separated by a ring or a peg, the sum of the lengths
of the two portions is constant. If there is an energy equation, or
an equation of constancy of momentum, or of moment of momentum,
it is an integral of the equations of motion.
118. Examples.
1. Two particles of masses M, m are connected by an inextensible thread
of negligible mass which passes through a small smooth ring on a smooth
110 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
fixed horizontal table. When the thread is just stretched, so that J/is at a
distance c from the ring, and the particles are at rest, J/is projected on the
table at right angles to the thread. Prove that until m reaches the ring M
describes a curve whose polar equation is of the form
r = o sec [6 x/{ Mj( M+ m)}].
2. Two particles of masses M, m are connected by an inextensible thread
of negligible mass ; M describes on a smooth table a curve which is nearly a
circle with centre at a point 0, and the thread passes through a small smooth
hole at 0 and supports m. Prove that the apsidal angle of M' s orbit is
*119. Oscillating pendulum. The motion of a simple circular
pendulum, whether it executes small oscillations (Art. 75) or not,
M can be determined by the energy equation.
Let 0 be the angle which the radius of the circle drawn through
the position of the particle at time t makes with the vertical drawn
downwards. The kinetic energy is ^ml^d2, where m is the mass of
the particle, and I the radius of the circle, or length of the pendu-
lum. The potential energy of the particle in the field of the Earth's
gravity (Art. 92) is mgl (1 — cos 6), if the chosen fixed level from
which it is measured is that of the lowest point. Hence the energy
equation can be written
^162 = g cos 6 + const.
If the pendulum is displaced initially so that 0 = a, and is let
go from this position, the energy equation is
^16- = g (cos 6 — cos a),
~ I V°A" 2
showing that the pendulum oscillates between two positions in
which it is inclined to the vertical at an angle a on the right and
left sides of the vertical.
To express the position of the pendulum in terms of the time
t, since it was in the equilibrium position, we introduce a new
variable ^r defined by the equation
sin- i ^- • -
with the further conditions that as 0 increases from 0 to a, -fy in-
creases from 0 to £?r; as 6 diminishes from a to 0, i/r increases
from £TT to TT ; as 6 diminishes from 0 to - a, i/r increases from TT
118,119] OSCILLATING PENDULUM 111
to |TT ; and as 6 increases from — a to 0, ^ increases from |TT to 2?r.
With these conventions there is one value of \^ corresponding to
every instant in a complete period.
Now we have
^ 6 cos •= = ty sin - cos \lr,
_ £
in2 ^ — sin2 — = sin2 ^ cos2 y ,
CL .
— sm2 ^ sin2 •
Hence the time t from the instant when the particle was passing
through the lowest point in the direction in which 6 increases is
given by the equation
(- •''
where the square root is always to be taken positively. The com-
plete period is
n •
v ^. 0/77
. / 1
/77 • 2a • o , \'
. / 1 — sin2 77 sin- y-
V V 2 r/
With the above relation between t and i|r, sin -^ is said to be an
Elliptic Function of t /-, , and the relation is written
sin-\Jr = sn ( t A -, } (modsin^).
V V I) \ 2/
The function has a real period, and the integral
is one quarter of this period.
The position of the pendulum at any time t is determined by
the equation
sm-
• « /.. /9\ ( j • a\
= sm 7, sn (t . / T mod sm ^ .
2 V V t/ V 2/
112 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
*120. Complete Revolution. If the constant in the energy
equation of Art. 119 is such that 0 never vanishes, it must be
greater than g, and the velocity at the lowest point is greater than
that due to falling from the highest point. Hence there will be
some velocity at the highest point. Let us suppose the velocity at
the highest point to be that due to falling through a height h ;
then, when 0 — TT
I26*=2gh,
and for any other value of 6
giving sin | - sn ( £ Jfy (mod k\ where & = Zl/(h +
The period of a complete revolution is
lit
V g}«
*121. Limiting case. In the case where the pendulum is
projected from the position of equilibrium with velocity equal to
that due to falling from the highest point the equation can be
integrated by logarithms.
The constant in the energy equation of Art. 119 must then
be chosen so that 6 vanishes when 6 = TT, and the equation there-
fore is
which may be written
The time of describing an angle 6 is therefore t, where
II f* das II , / 0 0^
*=VaJnC^ = ValogVSeC2 + tan2,
V JfJrO V'OS.*' V y \ •& &t
It is to be noted that the particle approaches the highest point
indefinitely, but does not reach it in any finite time.
The same equations may be used to describe the motion of a
120-123]
FINITE MOTION OF A PENDULUM
113
particle which starts from a position indefinitely close to the un-
stable position of equilibrium at the highest point of the circle.
*122. Examples.
1. Prove that the time of a finite oscillation when the fourth power of a,
the angle of oscillation, is neglected, is 2?r (1 -t-yg-a2) >J(llg)-
2. Prove that, in the limiting case of Art. 121,
<9 = 2 tan-1 sinh {£
3. Prove that, if a seconds' pendulum makes a complete finite oscillation
in four seconds, the angle a is about 160°.
*123. Smooth plane tube rotating in its plane.
a particle of mass ra move in a
smooth plane tube, and let the
tube rotate in its plane about a
point 0 rigidly connected with
it. Let OA be any particular
radius vector of the tube, and
</> the angle which OA makes
with a fixed line in the plane of
the tube. Then <£ is the angular
velocity of the tube. We shall
write ft> for <.
Let
- 40-
Let P be the position of the particle in the tube at time t.
Let OP = r, and Z AOP=8. Then r arid 6 are polar coordinates
of P referred to OA as initial line, and r and 6 + <f> are polar
coordinates of P referred to a fixed initial line. Let p be the
radius of curvature of the tube at P.
Let v be the velocity of the particle relative to the tube. Then,
if arc AP = s, v is s, the direction of v is that of the tangent to the
tube, and the resolved parts of v along OP and at right angles to
OP are r and ru,
Now the resolved accelerations of the particle along OP and at
right angles to OP are
and
L. M.
114 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
These may be written
- ra>,
dt,. v, , 2ro,
Ci/V
Of these the terms independent of co are equivalent to v -y-
along the tangent to the tube at P and w2/P inwards along the
normal to the tube.
The terms containing 2&> as a factor are equivalent to 2&>u
inwards along the normal to the tube. This can be seen by con-
sidering that r along OP and rd transverse to OP are equivalent
to v along the tangent in the direction in which s increases, and
that we have, as multipliers of 2«o, the components of this resultant
turned through a right angle.
Now we can resolve a vector in the direction OP into com-
ponents along the tangent at P to the tube and inwards along
the normal by multiplying by -j- and *-, where p is the perpen-
dicular from 0 on the tangent; similarly for a vector transverse
to OP.
Hence finally the accelerations resolved along the tangent and
normal to the tube are
dv dr
'&**& + **>>
v* dr\
P
Now let the particle move in the tube under the action of
forces in the plane of the tube whose resolved parts along the
tangent and normal to the tube are S and N, and let M be the
pressure of the tube on the particle. Then the equations of motion
are
[dv dr ~]
^ 3 w r -r + wn \ = S
as - as ^J
m
1? dr~\
- + 2«wf c^p + Ar~ =
*124. Newton's Revolving Orbit. Suppose that the form
of the tube in Art. 123 is a free path under a central force to 0.
123-125] REVOLVING ORBIT 115
Let the tube turn about 0 with an angular velocity (f> which is
always equal to nd, where n is constant, and 0 is the angular
velocity of the radius vector in the free path when the particle is
at (r, 0). Then the path traced out by the particle is a free path
under the original central force and an additional central force
which varies inversely as the cube o*f the distance.
Let / be the central acceleration in the free path, and ^h the
rate of description of areas. Then we are given
Now, in the tube <j> = nd, so that
r"- (8 + <j>)'= h (l+n),
and r-r0 + <2 = --r02
Hence the path traced out by the particle in the revolving
tube is a free path with a central acceleration to 0 made up of two
terms, one of them being f, and the other being inversely propor-
tional to t-3.
This result may be stated in another form as follows : — Rela-
tively to a certain frame a particle describes a central orbit about
the origin with central acceleration/; if a second frame with the
same origin rotates about the origin relatively to the first frame,
with an angular velocity always the same multiple of that of the
radius vector in the said central orbit, the path of the particle
relatively to the second frame is again a central orbit with the
central acceleration increased by an amount inversely proportional
to the cube of the distance.
*125. Examples.
1. A particle moves iu a tube in the form of an equiangular spiral which
rotates uniformly about the pole, and is under the action of a central force
to the pole of the spiral. Prove that, if there is no pressure on the tube, the
central force at distance r must be of the form Ar + £r~3, where A and B
are constants.
2. Prove that motion which, relatively to any frame, can be described as
motion in a central orbit with acceleration ^/(distance)3 towards the origin
and moment of velocity h may be described, relatively to a different frame
with the same origin, as uniform motion iu a straight line, provided A2>/i.
8—2
116 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
3. A particle moves in a smooth plane tube, and is under a central force
to a fixed point about which the tube rotates uniformly. Prove that, if the
pressure is always zero, the central force is
m [tta2 + 2ro> (h - r^/p2 + (h- r2a>)2p ~ 3 dp\dr\
where m is the mass of the particle, mh is its moment of momentum about
the fixed point, w is the angular velocity of the tube, r is the radius vector,
and p the perpendicular from the lixed point on the tangent to the tube at
the position of the particle.
*126. Motion on a rough plane curve under gravity.
When a particle is constrained to describe a plane curve in a
vertical plane under gravity, but there is
frictional resistance to the motion as well
as pressure on the curve, we assume that
the friction is ytt times the pressure, where
H is the coefficient of friction. The friction
acts along the tangent to the curve in the
sense opposite to that of the velocity.
The equations of motion take different
forms in different circumstances. We shall
choose for investigation the case where the
particle is on the outside of the curve, and
is descending.
Let the arc s of the curve be measured from some point of the
curve so that it increases in the sense of the velocity, and let 9 be
the angle contained between the inwards normal and the down-
wards vertical. Then 9 increases with s, and ds/d<f>(=p) is the
length of the radius of curvature.
Let v be the velocity of the particle, m its mass, R the pressure
of the curve on the particle. The equations of motion are
dv . >,
mv -r = mg sm 9 — ftH,
v* I '
in — = mg cos <f> — R
Eliminating R we obtain the equation
Fig. 41.
dv v* . .
v~lj'~=9 (sm 4> ~
cos
or
dv
dd>
~ M COS ^
125-128] MOTION ON A ROUGH CURVE 117
This equation can be integrated after multiplication by the
factor e'2^, in fact it becomes
(sin <f> - /u, cos </>),
so that v2e~-^ = Zg pe~2^ (sin </> — //, cos ^>) d<j) + const.,
an equation which determines v as a function of (/>, and therefore
gives the velocity at any point of the curve. The velocity being
determined, the second of the equations of motion gives the
pressure, and, just as in the case of a smooth curve, if R vanishes
the particle leaves the curve.
The equations of motion take different forms according as the
particle is inside or outside the curve, and according as it is ascend-
ing or descending. But in each case the equations can be integrated
by the above method. There is accordingly no definite expression
for the velocity at any point of the curve in terms of the position,
but the expressions obtained are different in the different cases.
••••127. Examples.
1. Write down the equations of motion in the three cases not investigated
in Art. 126 and the integrating factor in each case.
2. A particle is projected horizontally from the lowest point of a rough
sphere of radius a, and returns to this point after describing an arc aa,
(a < ^?r), coming to rest at the lowest point. Prove that the initial velocity
is sin a \f{2ffa (1 +^2)/(l — 2/x2)}, where p is the coefficient of friction.
3. A particle slides down a rough cycloid, whose base is horizontal and
vertex downwards, starting from rest at a cusp and coming to rest at the
vertex. Prove that, if p is the coefficient of friction, y? e'x7r = 1.
4. A ring moves on a rough cycloidal wire whose base is horizontal and
vertex downwards ; prove that during the ascent the direction of motion at
time t makes with the horizontal an angle $, given by the equation
j=-|Lsec2
where e is the angle of friction.
*128. Motion on a curve in general. When a particle
moves on a given curve under any forces, we take m for the mass
of the particle, S for the tangential component of the resultant
force of the field, N for the component along the principal normal,
and B for the component along the binormal. Also we take Rl for
1 1 8 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
the component of the pressure along the principal normal towards
the centre of curvature, and R2 for the component of the pressure
along the binormal in the same sense as B. Further if the curve
is rough we take F for the friction.
We take s to be the arc of the curve from some point to the
position of the particle at time t, p to be the radius of curvature,
and v to be the velocity, and we suppose the sense in which s
increases to be that of v. Then the equations of motion are
dv ~ v
mV -r- = O — £ ,
ds
When the curve is smooth F is zero, and we can integrate the
first equation, in the same way as in Art. 115, in the form
^mv* = I Sds + const.,
and this result can be expressed in the form
change of kinetic energy = work done,
so that the velocity is determined in terms of the position. The
other two equations then determine the pressure.
When the curve is rough we have to eliminate F, Rl} R2 by
means of the equation
F^ = ^(R^ + R,Z),
which expresses that the friction is proportional to the resultant
pressure. There results a differential equation for tf, and, if we
can integrate this equation, we shall obtain an equation giving the
velocity in terms of the position. As in Art. 126 the velocity in
any position depends partly on the way in which that position has
been reached.
*129. Motion on a smooth surface of revolution with a
vertical axis.
Let the axis of revolution be the axis x (x being measured
upwards), and let the particle at time t be at distance y from the
axis, and be on a meridian curve of the surface in an axial plane
making an angle <£ with a given axial plane, and let <r be the arc
128-130] MOTION ON A SURFACE OF REVOLUTION 119
of the meridian from some particular circular section to the position
of the particle.
Then it is clear that the velocity along the tangent to the
meridian is a, and the velocity along the tangent to the circular
section is y$. Thus the energy equation is
| (cr- + 2/2<£2) + gx = const.
Fig. 42.
Again, since the pressure of the surface on the particle acts
along the normal to the surface, and the normal meets the axis of
revolution, while the force of gravity acts in a line parallel to this
axis, the forces acting on the particle have no moment about this
axis. Hence the moment of the momentum about the axis is con-
stant, or we have
y2(j> = const.
The equations which have been written down determine <7 and
<£, that is they determine the two components of velocity (a- and
yfy in two directions, at right angles to each other, which lie in
the tangent plane to the surface.
*130. Examples.
1. If the particle is projected properly it can describe a circle. If y is
the radius of the circle, and /3 the angle which the normal to the surface at
any point on the circle makes with the vertical, the required velocity of
projection is (gry tan/3)*.
In this case the pressure of the surface is equal to mg sec /3, where m is the
mass of the particle.
1 20 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
2. Prove that, if l/u is put for y, and #=/(«) is the equation of the
meridian curve of the surface, the projection of the path of the particle on a
horizontal plane is given by an equation of the form
where h is a constant.
*131. Motion on a surface in general. Let a particle move
on a fixed surface under the action of given forces and the pressure
and friction of the surface.
We may imagine the surface to be covered with a network of curves
belonging to distinct families, in such a way that at each point of the surface
one curve of one family meets one curve of the other family, and we may
suppose the curves that meet in any point to cut at right angles. At any
point we may resolve the force of the field into components along the tangents
to the curves that meet in that point, and along the normal to the surface.
We may resolve the acceleration along the same lines.
For a particle moving on a smooth surface in a conservative field there
will be an energy equation expressing the velocity in terms of the position.
We shall see presently that the pressure is determinate as soon as the velocity
is known.
When the surface is rough there will be two components of friction in the
directions of the tangents to the two curves that meet at any point, and the
resultant friction has the same direction as the velocity but the opposite
sense. Also the resultant friction is equal in magnitude to the product of the
coefficient of friction and the pressure.
We have thus the means of writing down equations of motion of the
particle, but the process can in general be simplified by using methods
of Kinematics and Analytical Dynamics which are beyond the scope of the
present work. We shall therefore confine ourselves to the simplest cases.
We proceed to investigate a general expression for the resolved
part of the acceleration along the normal to the surface.
Let v be the velocity of the particle, p the radius of curvature
of its path. The tangent to the path touches the surface, and we
suppose a normal section of the surface drawn through it. This
section is not, in general, the osculating plane of the path ; we
suppose that it makes an angle <£ with this osculating plane. We
take p' to be the radius of curvature of the normal section of the
surface through the tangent to the path.
Since the normal to the surface is at right angles to the tangent
to the path the resolved part of the acceleration along the normal
130-132]
MOTION ON A SURFACE
121
to the surface is the resolved part in that direction of the accelera-
tion along the principal normal to the path, it is therefore
v2
— COS 0.
P
Also by a well-known theorem we have p = p' cos <f>.
Hence the acceleration along the normal to the surface is tf/p,
and the pressure is determined by resolving along the normal.
*132. Osculating plane of path. In Ex. 1 of Art. 130 it is
stated that a particle may be projected along a horizontal tangent
of a smooth surface of revolution whose axis is vertical with such
velocity that it describes the circular section under the action of
Fig. 43.
gravity and the pressure of the surface. It is almost obvious that
if the velocity exceeds that requisite for description of the circle
the path of the particle rises above the circle, otherwise it falls
below the circle. We may use the result of Art. 131 to find the
position of the osculating plane of the path for any velocity of pro-
jection.
Let P be the point of projection, PG the normal to the surface
at P, PN = y the ordinate of P at right angles to the axis of
revolution, Q the point where the osculating plane of the path
meets the axis. Let Z GPN= a, and Z GPQ = </>.
When the particle is projected along the tangent to the circular
section with velocity V there is initially no acceleration along a
line in the meridian plane at right angles to PQ.
122 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
Hence resolving along this line we have
R sin <j> - mg cos (a — </>) = 0,
where m is the mass of the particle, and R is the pressure.
Again, resolving along PN, we have
y*
m — cos (a — </>) = -R cos a,
P
where p is the radius of curvature of the path.
Now, with the notation of Art. 131,
p' = PG, p = PG cos <j>.
Also y = PN=PGcosa.
Hence tan <£ = gy( F2.
This equation determines the position of the osculating plane
of the path.
Now if tan <f> > tan a, or F2 < gy cot a, the osculating plane of
the path initially lies below the horizontal plane through the point
of projection, and if tan</><tana, or Vz>gycota, it lies above
that plane.
*133. Examples.
1. A particle moving on a surface (smooth or rough) under no forces
but the reaction of the surface describes a geodesic.
2. A particle moves on a rough cylinder of radius a under no forces but
the reaction of the surface, starting with velocity F in a direction making an
angle a with the generators ; prove that in time t it moves over an arc
a/x~ J cosec2 a log (1 + p, Via ~ l sin2 a),
p. being the coefficient of friction.
3. A hollow circular cylinder of radius a is rough on the inside, and is
made to rotate uniformly with angular velocity a> about its axis which makes
an angle a with the vertical. Show that a particle can slide down a fixed line
parallel to the axis with uniform velocity
where p. is the coefficient of friction, and p, > cot a.
4. An ellipsoidal shell whose principal semi-axes are a, b, c(a>b>c) is
placed with the greatest axis vertical, and a particle is projected from one of
the lower umbilics with velocity v along the tangent to the horizontal section
within the ellipsoid. Show that the osculating plane of the path is initially
above or below this section according as
(«2-c2) (a2-62)}.
132-135] RESISTING MEDIUM 123
134. Motion in Resisting Medium. We consider cases of the
motion of a particle in a known field of force when, in addition to
the force of the field, there is exerted on the particle a force pro-
portional to a power of its velocity having the same direction as
the velocity and the opposite sense.
Problems of this kind are related to facts of observation in
regard to the motions of bodies in the air and in other fluid media.
In many cases it is found that the observed facts can be approxi-
mately represented by the supposition that the resistance is pro-
portional to the velocity, this is true for instance for the motion of
a pendulum swinging in air.
135. Resistance proportional to the Velocity. Since the
velocity of a particle is a vector whose direction and sense are de-
termined by the resolved parts x, y, z, the resistance has resolved
parts — KX, — icy, — KZ, where K is a constant.
Let the motion take place under gravity parallel to the negative
direction of the axis y, and first suppose the particle to move
vertically. The equation of motion is
my = -mg- Ky}
or y + \y + g = 0,
where \ is written for K/m. Multiplying by e* and integrating,
we have
A
where (7 is a constant of integration. Hence
y = Ce~Ktlm — mg/K.
If the particle continues to fall for a sufficiently long time the
value of y will ultimately differ very little from —gmjic, or the
particle falls with a practically constant velocity when it has been
falling for some seconds.
This velocity is called the terminal velocity in the medium.
The equation last written can be integrated again so as to
express y as a function of t.
Again suppose that the particle is projected in any other than
a vertical direction ; then the vertical motion is the same as before,
but for the horizontal motion we have an equation
mx = — KX,
124 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
giving x = A e-Kt/m,
where A is a constant of integration. This equation can be inte-
grated again so as to express x as a function of t.
Since x and y are known, as functions of t, the path can be
determined.
136. Damped Harmonic Motion. Consider the case where,
apart from the resistance, the motion would be simple harmonic in
period 2-jr/n, and the resistance is proportional to the velocity.
We have the equation
tux = — mri*x — KX,
or x + \x + nzx = 0,
where X is written for x/m. The complete primitive of this equation
takes different forms according as n2 > or < £X*. In the former
case, which is practically the more important, it is
x = e~±M [A cos {t V(w2 - £X2)} + B sin {t V(?i2 - |X2)}].
The motion may be described roughly as simple harmonic
motion with period 27r/v/(n2 — |X2), and with amplitude diminishing
according to the exponential function e~*M. It will be observed
that the period is lengthened by the resistance, and that the
amplitude falls off in geometric progression as the time increases
in arithmetic progression. Thus the motion rapidly dies away.
136 A. Effect of damping on forced oscillation. Let a
particle, which can move with a damped harmonic motion, be dis-
turbed by a force, which at time t is proportional to cos (pt + a),
where p and a are constants. The equation of motion is of the form
x + Xar + v?x = P cos (pt + a).
A solution of this equation can be found by putting x equal to a
function of the form Q cos (pt + a — e) provided
(n2 -p2) Q cos e + \p Q sin e = P,
-\pQcose + (nz-p°) Q sin e = 0,
whence Q {(n2 - p2)2 + X2j92] * = P, tan e = Xp/(?i2 - p*).
When n > £X the complete primitive is
x — e'W (A cos n't + B sin n't) + Q cos (pt + a - e),
where ?z/2 = n2 - ^X2, and the motion is compounded of a damped
harmonic motion, the free oscillation, and a simple harmonic motion,
135-137] RESISTING MEDIUM 125
the forced oscillation. The amplitude of the free oscillation dimin-
ishes in geometric progression as the time increases in arithmetic
progression, and thus after a time the motion is practically simple
harmonic. The period of the forced oscillation is the same as the
period of the force producing it ; its amplitude never becomes
infinite, even when the period of the force is the same as that of
the free oscillation ; its phase is behind or in advance of that of the
force according as the period of the free oscillation, when there is
no resistance, is less or greater than that of the force.
When n < |X the complete primitive is
x = e~^M (Aenf>t + Be-11"1} + Q cos (pt + a. - e),
where n"* = ±'\?— ri?. The term with coefficient Q represents, as
before, a forced oscillation.
137. Examples.
1. A particle is projected vertically upwards with velocity v in a medium
in which the resistance is proportional to the velocity. It rises to a height k
and returns to the point of projection with velocity w. Prove that
ghjv* =£-i(»/7) +i (W^)2 -i(*/r)3 +.-,
where F is the terminal velocity in the medium.
2. A particle moves under gravity in a medium whose resistance varies
as the velocity, starting with horizontal and vertical component velocities u0>
v0, and returning to the horizontal plane through the point of projection
with component velocities Ui, vx ; show that the range R and time of flight t
are given by the equations
Prove also that R — u0Vt/(V+v0), where Fis the terminal velocity in the
medium.
3. A body performs rectilinear vibrations under an attractive force to a
fixed centre proportional to the distance in a medium whose resistance is
proportional to the velocity. Prove that, if T is the period, and a, b, c are
the coordinates of the extremities of three consecutive semi-vibrations, then
the coordinate of the position of equilibrium and the time of vibration if
there were no resistance are respectively
ac-b* r 1 / a-6\*"l~*
— -r and T \\-\-— „ ( log - r)
a+c-26 L 7T2 \ ° c-bj J
4. If in the problem considered in Art. 136, A>2«, and the particle
starts from rest in any displaced position, it creeps asymptotically towards its
position of equilibrium, according to the formula
x=a(ae-f*t-He-at}l(a-$\
where a and /3 are the roots of the quadratic £2-X£ + ?i2 = 0.
126 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
5. A particle of unit mass is fastened to one end of an elastic thread
of natural length a and modulus an2, in a medium the resistance of which
to the motion of the particle is 2* (velocity). The other end of the thread
is fixed and the particle is held at a distance b (> a) below the fixed point.
Prove that, when set free, (i) it will begin to rise or fall according as
?t2(6-a)> or <<7, (ii) in its subsequent motion it will oscillate about a
point 0 which is at a distance a + g/ri* below the fixed point, (iii) the distances
from 0 of successive positions of rest form a geometric series of ratio e-«/»»,
(iv) the interval between any two positions of rest is ir/m, where m2 = n2 — «2.
6. A particle moves on a smooth cycloid whose axis is vertical and vertex
downwards under gravity and a resistance varying as the velocity. Prove that
the time of falling from any point to the vertex is independent of the starting
point.
7. A particle moves under a central force tf> (r) in a medium of which the
resistance varies as the velocity. Investigate the equatiqns
where h and p. are constants.
*138. Motion in a vertical plane under gravity. For
any law of resistance we can make some progress with the
equations of motion of a particle moving in a vertical plane under
gravity.
Let mf(v) be the magnitude of the resistance when the velocity
is v, m being the mass of the
particle, then resolving horizontally
we have
ii = -f(v)cos<f>,
where <£ is the angle which the
T~ direction of motion at time t makes
with the horizontal and u is the
horizontal velocity, so that u = v cos <£.
Again resolving along the normal to the path, since the resist-
ance is directed along the tangent, we have
- = gcos<f>,
where p is the radius of curvature. Since (j> diminishes as s
increases, p is — ds{d(f>, and the above equation may be written
v$ = — g cos <f>,
RESISTING MEDIUM 127
and thus, eliminating t, we get
du vf(v) ,
-T-T = ^-^ . where v = u sec ©.
d(f> g
This equation can be integrated when f(v) = rcvn, and we have
1 UK f dd>
t
un g Jcos7l+1</>
an equation giving u, and therefore also ?;, in terms of </>.
Now the equation
d<j> ,
„_=-£, COS <£
Cv
gives t = — I - sec <f)d(f> + const.,
so that t is found in terms of 0. Also the equations
dx dy . ds
-j- = cos rf>, -£• = sm <i, -y- = v
ds ds dt
[v2 fvz
give us x=— ~dd> + const., y = — I — tan d>dd> + const.,
*9 h
and thus the time and the position of the particle are determined
in terms of a single parameter $.
It is not generally possible to integrate the equation for
vertical rectilinear motion even for the case here described where
f(v) — KVn. In the special case, however, where the resistance is
proportional to the square of the velocity the velocity can be found
in any position. We have, when the particle is ascending,
y = -g- Ky\
y being measured upwards. Now
v = ^=Aa?/n-
y~ydy dy(*J)>
hence (£y») + ^s = _ ^
Multiplying by e2>cy and integrating, we have
%ip&*v =-~- &Ky + const.,
Z.K t
giving y2 - Ce~2KV - g/ic.
Again, when the particle is descending we have, measuring y
downwards,
128 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
or
giving y = £-
As in the case of resistance proportional to the velocity, there
is a terminal velocity, \%/«)> which is practically attained when
the particle has fallen through a considerable height.
*139. Examples.
1. A particle is projected vertically upwards in a medium whose resist-
ance varies as the square of the velocity. Prove that the interval that
elapses before it returns to the point of projection is less than it would be if
there were no resistance.
Prove also that, if the particle is let fall from rest, then in time t it
acquires a velocity Ui&nla (gt/U) and falls a distance U*g~l log cosh (gtjLT),
where U is the terminal velocity in the medium.
2. A particle of weight W moves in a medium whose resistance varies
as the «th power of the velocity. Prove that, if F is the resistance when
the direction of motion makes an angle 0 with the horizon, then
W
-= = 71 COS
," (f> I
3. If v is the velocity of a projectile when the inclination of its path
to the horizontal is <f>, a point whose polar coordinates are v and <£ traces out
a curve called the "hodograph" of the trajectory. Prove (i) that, when the
resistance is proportional to the velocity, the hodograph is a straight line ;
(ii) that, whatever the law of resistance may be, the sectorial area bounded
by an arc of the hodograph and two of its radii vectores is the product
of %g and the difference of the values of x in the corresponding arc of
the trajectory.
4. A particle of unit mass moves in a straight line under an attraction
H (distance) to a point in the line, and a resistance K (velocity)2. Prove that,
if it starts from rest at a distance a from the centre of force, it will first
come to rest at a distance 6, where
5. The bob of a simple pendulum moves under gravity in a medium of
which the resistance per unit of mass is K (velocity)2, and starts from the
lowest point with such velocity that if it were unresisted the angle of oscilla-
tion would be a. Prove that it comes to rest after describing an angle 6 which
satisfies the equation
(1 + 4*2£2) cos « = 4*2^ - 2*1 sin 6eZltl8+cos tie2"16,
where I is the length of the pendulum.
MISCELLANEOUS EXAMPLES 129
MISCELLANEOUS EXAMPLES
1. A particle is projected horizontally from the lowest point of a smooth
elliptic arc, whose major axis 2« is vertical, and moves under gravity along
the concave side. Prove that it will leave the curve if the velocity of projection
lies between sJ(Zga} and »J{ga (5 - e2)}.
2. A particle moves on a smooth curve in a vertical plane, the form
of the curve being such that the pressure on the curve is always m times
the weight of the particle. Prove that the time of a complete revolution is
2irm v/a/{v/#. (m2- 1)*}, and that the length of the vertical axis of the curve
is 2ma/(»z2- 1)2, the whole length of the curve being na (2m2 + l)/(m2- 1)^.
3. A bead moves on a smooth circular wire in a vertical plane its velocity
being that due to falling from a horizontal line HK above the circle. Prove
that, if / is the internal limiting point of the co-axal system of which the
circle and the line HK are members, then any chord through / divides the
wire into two parts which are described in equal times.
4. Prove that the time of a beat of a circular pendulum of length a
oscillating through an angle 2a is equal to the time of complete revolution of
a pendulum of length a cosec2 £a, the height of the line of zero velocity above
the lowest point being 2a cosecHa.
5. The point of support of a simple pendulum of length I and weight
w is attached to a massless spring so that it can move to and fro in a
horizontal line ; prove that the time of a small oscillation is
where W is the weight required to stretch the spring a length I.
6. A platform is sliding down a smooth spherical hill from rest at the
summit. From a point fixed on it a plumb-line is suspended in a tube which
is always held perpendicular to the surface of the hill at the point of contact
of the platform. Prove that the tension of the cord, when the platform has
descended a distance x measured vertically, is w(a—3x)/a, where a is the
radius of the sphere, and w is the weight of the lead.
7. Prove that, if the suspending fibre of a simple pendulum is slightly
extensible, the period of small oscillation is that due to the stretched length
of the fibre in the position of equilibrium.
8. A particle moves in a smooth tube in the form of a catenary being
attracted to the directrix with a force proportional to the distance from the
directrix. Prove that the period of oscillation is independent of the amplitude.
9. Two particles of masses P arid Q lie near to each other on a smooth
horizontal table, being connected by a thread on which is a ring of mass R
hanging just over the edge of the table. Prove that it falls with acceleration
L. M. 9
130 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
10. Two particles of masses m, m' are attached to the ends of a thread
passing over a pulley, and are held on two inclined planes each of angle a
placed back to back with their highest points beneath the centre of the pulley.
Prove that, if each portion of the thread makes an angle /3 with the corresponding
plane, the particle of greater mass m will at once pull the other oft' the plane if
m'/m < 2 tan a tan j8 — 1.
11. A straight smooth groove is cut in a horizontal table, and a straight
slit is cut in the bottom of the groove. A thread of length I, attached at one
end to a shot of mass m resting in the groove, passes through the slit and
supports a particle of mass urn. The suspended particle is held displaced in
the vertical plane containing the slit with the thread straight, and is let go.
Prove that its path is part of an ellipse of semi-axes I, and l/(l + K), the major
axis being vertical.
12. Two particles of masses M, m are connected by a cord passing over a
small smooth pulley ; the smaller (m) hangs vertically and the other ( J/)
moves in a smooth circular groove on a fixed plane of inclination a to the
vertical, the highest point of the groove being the foot of the normal from the
pulley to the plane. M starts from a point close to the highest point of the
groove without initial velocity. Prove that, if it makes complete revolutions,
the radius of the groove must not exceed
hmMcos a/(m2 — M2 cos2 a),
where h is the distance of the pulley from the plane.
13. A particle of weight W moves in a smooth elliptic groove on a
horizontal table, and is attached to two threads which pass through holes at
the foci, and each thread supports a body of weight W. One of the bodies is
pulled downwards with velocity Ve when the particle is at an end of the minor
axis. Prove that, if F2 < abzgj{e (3a2 - 262)}, the threads do not become slack,
and that in this case the horizontal pressures, R and R, on the groove when
the particle is at the ends of the axes are connected by the equation
fib3 ~ Ra (3a2 - 262) = 6 Wa*be2,
where 2a and 2b are the principal axes, and e is the eccentricity of the ellipse.
14. A smooth parabolic wire, on which is a bead of weight w, is fixed in
a horizontal plane. To the bead is attached a thread, which passes through
a smooth ring fixed at the focus of the parabola and carries, at its other end,
a weight wj(e - 1). Prove that the tension T of the thread at any stage of the
motion is given by an equation of the form
(eT-iv) (e?--a)2 = const.,
where r is the focal distance of the bead and 4a the latus rectum of the
parabola.
15. Two smooth straight horizontal non-intersecting wires are fixed at
right angles to each other at a distance d apart. Two small rings of equal
mass, connected by an inextensible thread of length I, slide on the wires, and
they are projected with velocities u and v from points at distances a and b
MISCELLANEOUS EXAMPLES 131
from the shortest distance between the wires. Prove that after the thread
becomes tight the motion is oscillatory and of period 2ir (I2 - d2)/(av ~ bit).
16. One end of a thread of length I is attached to the highest point of a
fixed horizontal circular cylinder of radius a. A particle attached to the other
end is dropped from a position in which the thread is straight and horizontal
and at right angles to the axis of the cylinder. Prove that, if I <£ 27ra, the
thread will become slack before the particle comes to rest, and that it will
then have turned through an angle whose circular measure is
17. Two particles, masses m, in', on a smooth horizontal table are con-
nected by a thread passing through a small smooth ring fixed in the table.
Initially the thread is just extended and in two straight pieces meeting at the
ring, the lengths of the pieces being a and a'. The particles are projected at
right angles to the thread with velocities v and v'. Prove that, if T is the
tension at any time and r, / the distances from the ring, then
a2v2 a'V2
r3 r'3
Prove also that the other apsidal distances will be equal if
18. A particle slides down a rough cycloid whose axis is vertical and
vertex downwards. Prove that the time of reaching a certain point on the
cycloid is independent of the starting point.
Prove also that, if X is the angle of friction, and if the tangent at the
starting point makes with the horizontal an angle greater than a, where a is
the least positive angle which satisfies the equation
the particle will oscillate.
19. A ring moves on a rough cycloidal wire with its axis vertical and
vertex downwards. Prove that, if it starts from the lowest point with velocity
w0, its velocity u when its direction of motion is inclined at an angle $ to the
horizontal is given by
u? = (u(? + 4ag sin2 e) e~2<t> tan e - lag sin2 (<£ + e),
where a is the radius of the generating circle and e is the angle of friction.
Prove also that, if it starts from a cusp with velocity w0, its velocity v
during its descent is given by
»« = (V + 4ag cos2 e) e-<"-W tan . _ 4ag s[^ ^ _ ^
20. A particle is projected from a point on the lowest generator of a rough
horizontal cylinder of radius a with velocity V at right angles to the generator,
and moves under no forces except the pressure and friction of the surface.
Prove that it returns to the point of projection after a time a (e2Fir — !)/(/* V),
where p is the coefficient of friction.
9—2
132 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
21. A point P moves along a plane curve which rotates in its plane about
a point 0 with uniform angular velocity a. Prove that the curvature of its
path is
V(<r V+ 2<a) ( lr+ rot sin ^) + ra ( Va> sin \//- —f cos >//• + raF)
( F2 + r2o>2 + 2 Vrta sin ^)T
where r is the length OP, <r is the curvature of the curve at P, \//- the angle
between OP and the tangent, Fthe velocity of P relative to the curve, and/
the rate of increase of V.
22. A bead is initially at rest on a smooth circular wire of radius a in a
horizontal plane ; the wire is made to rotate with uniform angular velocity o>
about an axis perpendicular to its plane and passing through a point on the
diameter through the bead at a distance c from the centre. When the bead
has moved a distance ad on the wire, the wire is suddenly stopped. Prove
that the bead will subsequently move with velocity
to {v/(a2 + c2 + 2ac cos 6}-(a + c cos 6}}.
23. Two small beads of masses mt, m.2 slide along two smooth straight
rods which intersect at an angle a, and the beads are connected by an elastic
thread of natural length c and modulus X. The rods are made to revolve
uniformly in their plane, about their point of intersection, with angular
velocity <a. Prove that throughout the motion
mi (*i2 — ^i2*"2) + »«2 (^ 22 — r22«>2) + ^e'2c = const.,
where ( is the extension of the thread, and r1} r.2 are the distances of the beads
from the intersection of the wires at any time.
24. A smooth elliptic tube rotates about a vertical axis through its
centre perpendicular to its plane with uniform angular velocity o>. Prove that
a particle can remain at an extremity of the axis major, and, if slightly
disturbed, will oscillate in a period 2?r v/(l —e?)/eu>, where e is the eccentricity.
25. A body is describing an ellipse of semi-axes «, 6 about a centre of
gravitation, and when it is at a distance r from this centre it comes under the
influence of a small disturbing force directed to the same point and varying
inversely as the cube of the distance. Prove that the effect is the same as if
the body described under the original force an orbit which at the same time
rotated (with the body) round the centre of force with angular velocity n times
the angular velocity of the body, where n is a small constant such that the
semi-axes of this new free orbit are equal to those of the original one reduced
by fractious 2ra&2,V2 and n (1 -»-&2/r2) of themselves.
26. A particle moves on a helical wire whose axis is vertical. Prove that
the velocity v after describing an arc « is given by the equations
o . , , ds . sec2 a cosh d>
t>2=aasecasmh<i, -7T=4«z r--, ,
d(f) • tan a-fjL cosh </>'
where a is the radius of the cylinder on which the helix lies, a the inclination
of the helix to the horizon, and p. the coefficient of friction.
MISCELLANEOUS EXAMPLES 133
27. A small smooth groove is cut on the surface of a right circular cone
whose axis is vertical and vertex upwards in such a manner that the tangent
is always inclined to the vertical at the same angle /3. A particle slides down
the groove from rest at the vertex ; show that the time of descending through
a vertical height h is equal to the time of falling freely through a height
h sec2/3. Show also that the pressure is constant and makes with the principal
normal to the path a constant angle
tan"1 {^sin a/>/(cos2o-cos2^)}, •*
where 2a is the angle of the cone.
28. A smooth helical tube of pitch a has its axis inclined at an angle
/3 (> a) to the vertical, and a particle rests in the tube. The tube is made to
turn about its axis with uniform angular velocity <a. Prove that the particle
makes at least one- complete revolution round the axis if
|«o>2/<7 >[(ir + 2y) sin y + 2 cos y] sin /3 cot a cosec3 a,
where sin y=tan a cot )3, and a is the radius of the helix.
29. A small ring can slide on a smooth plane curved wire which rotates
with angular velocity o> about a vertical axis in its plane. Find the form of
the curve in order that the ring may be in relative equilibrium at any point.
Prove that, if the angular velocity is increased to <a', the ring will still
be in relative equilibrium if the wire is rough and the coefficient of friction
between it and the ring is not less than i (a>'ja> — a>/a>').
30. A particle moves in a smooth circular tube of radius a which rotates
about a fixed vertical diameter with angular velocity u>. Prove that, if 0 is
the angular distance of the particle from the lowest point, and if initially it is
at rest relative to the tube with the value a for 6 where a> cos £a =
then at any subsequent time t
cot $d = cot £o cosh (a>t sin £a).
31. A particle moves under gravity on a right circular cone with a
vertical axis. Show that, if the equations of motion can be integrated without
elliptic functions, the particle must be below the vertex, and that its distance
r from the vertex at time t is given by an equation of the form
(rr)2 =2gcosa(r- r0) (r + 2r0)2,
where 2a is the vertical angle of the cone.
32. A right circular cone of vertical angle 2a is placed with one
generator vertical and vertex upwards. From a point on the generator o
least slope a particle is projected horizontally and at right angles to the
generator with velocity v. Prove that it will just skim the surface of the cone
without pressure if the distance of the point of projection from the vertex is
•£y2 cosec2 ajg.
33. A particle is projected horizontally from a fixed point on the interior
surface of a smooth paraboloid of revolution whose axis is vertical and vertex
downwards. Prove that when it is again moving horizontally its velocity is
independent of the velocity of projection.
134 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
34. Prove that, if the path of a particle moving on a right circular cone
cuts the generators at an angle x> the acceleration in the tangent plane to the
surface and normal to the path is
v2(dx/ds + r~l sin x),
where v is the velocity, and r the distance from the vertex.
If the axis of the cone is vertical, and the vertex upwards, and if the
velocity is that due. to falling from the vertex, prove that, when the particle
leaves the cone (supposed smooth),
2 sin2 x = tan2 a,
2a being the vertical angle of the cone. What happens when tan2 a > 2 ?
35. A particle moves on a surface of revolution. The velocity is v at a
point where the normal terminated by the axis of revolution is of length j/,
and this normal makes an angle 6 with the axis ; prove that, if ds is the
element of arc of the path, and x the angle at which it cuts the meridian, the
acceleration in the tangent plane to the surface and normal to the path is
2 fd\ sin x cot $\
V \ds + ~ ) •
36. A particle is placed at rest on the smooth inner surface of a vertical
circular cylinder, which rotates with uniform angular velocity CD about the
generator which is initially furthest from the particle. Prove that the
pressure vanishes when the particle has descended a distance
- 4
37. A particle is attached by a thread of length a to a point of a rough
fixed plane inclined to the horizon at an angle equal to the angle of friction
between the particle and the plane. The particle is projected down the plane
at right angles to the thread, which is initially straight and horizontal. Prove
that it comes to rest at the lowest point of its path if the square of the initial
velocity is (IT - 2) /^«/x/(l +/x2), where p. is the coefficient of friction.
38. A particle is projected horizontally with velocity V along the interior
surface of a rough vertical circular cylinder. Prove that, at a point where
the path cuts the generator at an angle 0, the velocity v is given by the
equation
ag/v2 = sin2 $ (ag\ F2 + 2M log (cot $ + cosec $)},
and the azimuthal angle and the vertical descent are respectively
MISCELLANEOUS EXAMPLES 135
39. A particle falls from rest under gravity through a distance x in a
medium whose resistance varies as the square of the velocity ; v is the velocity
acquired by the particle, V the terminal velocity, and v0 the velocity that
would be acquired by falling through a distance x in vacuo ; prove that
40. A particle is projected vertically upwards from the surface of the
Earth with velocity u, and when its velocity is v and its height above the
surface is z the resistance is KV^/(a+z), where a is the Earth's radius. Prove
that, if z is always small compared with a, the velocity V with which it
returns to the point of projection is given approximately by the equation
variations of gravity with height being taken into account.
41. Prove that, if the resistance is proportional to the square of the
velocity, the angle 6 between the asymptotes of the complete trajectory of a
projectile is given by the equation
U2/iv2 = cot 6 cosec 6 + sinh - 1 cot 6,
where U is the terminal velocity and w the velocity when the projectile moves
horizontally.
42. A particle moves under gravity in a medium whose resistance is
proportional to the velocity. Prove that the range on a horizontal plane is
a maximum, for given velocity of projection, when the angle of elevation at
first and the angle of descent at last are complementary.
43. A particle is projected up a plane of inclination a under gravity and
a resistance proportional to the velocity. The direction of projection makes
an angle /3 with the vertical, the range R is a maximum and t is the time of
flight. Prove that, if U is the terminal velocity and V the velocity of
projection, then
(i)
(ii)
(iii) U V'2 sin /3/( F+ U cos £) -gR cos a.
44. A pendulum oscillates in a medium of which the resistance per unit
of mass is K (velocity)2. Prove that, when powers of the arc above the first
are neglected, the period is the same as in the absence of resistance, but the
time of descent exceeds that of ascent by f «a -J(l3/g}, where a is the angular
amplitude of the descent, and I is the length of the pendulum.
45. A particle of mass m moves in a field of force having a potential V
in a medium in which the resistance is k times the velocity. Prove that, if
D is the quantity of energy dissipated in time t,
dD Zk
-=-4 — • (D - m V) = const.
at m
136 MOTION UNDER CONSTRAINTS AND RESISTANCES [CH. V
If the resistance is k (velocity)-, and if ds is the element of arc of the
path of the particle, then
dD afr , .
-j- H (/> — m l )=const.
a* m
46. A smooth straight tube rotates in one plane with uniform angular
velocity <a about a fixed end, and a particle moves within it under a resistance
equal to < times the square of the relative velocity. Prove that, if the particle
is projected so as to come to rest at the fixed end, the relative velocity at
a distance r from that end is
47. A particle moves on a smooth cycloid whose axis is vertical and
vertex upwards in a medium whose resistance is (2c)-1 (velocity)2 per unit
of mass, and the distance of the starting point from the vertex measured
along the curve is c ; prove that the time to the cusp is v/{8« (4a - c)/gc],
2a being the length of the axis.
CHAPTER VI
140. Direct impact of spheres. Let the centres of two
spheres move in the same line. This line must be that joining the
centres. The spheres will come into contact if their centres are
moving in opposite senses, or if one of them is at rest, and the
other is moving towards it, or if they are moving in the same
sense, and one overtakes the other. Let m, m' be the masses of
the spheres, determined by weighing them in a common balance.
Let U be the velocity of the centre of the sphere m before impact,
in the sense from m towards m, U' the velocity of the centre of
m! before impact, in the same sense; and let u and u be the
velocities of m and m' in the same sense after impact. When
proper arrangements are made for measuring the velocities, it is
found that
m(u— IT) = m ( U' — u).
141. Ballistic balance. An instrument by which experiments of
the kind just considered may be made is called a
"ballistic balance." In principle it comes to this*: —
The two spheres are suspended from two fixed points
at the same level by cords, and, when the cords
are vertical, the spheres are in contact and the line
of centres is horizontal (see Fig. 45). The distance
between the fixed points is equal to the sum of the
radii. One sphere is then raised, the cord attached
to it being kept taut, until its centre is at a known
height H above the equilibrium position. It is then
let fall. At the instant of impact its velocity is
-J(ZgH}. The velocities of the spheres immediately
after the impact are measured by observing the
heights to which the centres rise. Fig. 45.
* The actual construction and method of using the instrument are described by
W. M. Hicks, Elementary Dynamics of Particles and Solids, London, 1890. Experi-
mental investigations of the kind referred to in tbe text were made by Newton. See
Principia, Lib. i. "Axiomata sive leges motus."
138 THE LAW OF REACTION [CH. VI
142. Statement of the Law of Reaction. The result
stated in Art. 140 may be written
m'u' — m'U' = — (uiu — m U).
The left-hand member is the measure of the " change of momen-
tum" of the sphere m' ; the right-hand member, with its sign
changed, is the measure of the change of momentum of the sphere
m. These changes of momentum are produced, during the very
short time of the impact, by forces which the spheres exert one on
the other. The result can be stated in the form : — The impulses of
these forces are equal and opposite. This result leads us to conclude
that the forces also are equal and opposite. The result is generalized
in the statement : — In any action between bodies, by which the
motion of either is set up, altered or stopped, each body exerts
force on the other, and these forces are equal and opposite. The
statement may be made more precise when the bodies are replaced
by particles, and then it takes the form : —
The magnitude of the force exerted by one particle on another is
equal to the magnitude of the force exerted by the second particle on
the first, the lines of action of both the forces coincide with the line
joining the particles, and the forces have opposite senses.
This abstract statement may be regarded as an induction from
experience. The proof of its truth is found in the agreement of
results deduced from it with results of experiment.
The statement is frequently called the " Law of Reaction "
because it was briefly expressed by Newton in the phrase " action
and reaction are equal and opposite."
143. Mass-ratio. The result of Art. 140 may be expressed in
the form
-(u-U) = mf
u - U' m'
and this result may be generalized, and made precise, in the
statement : —
In any action between particles the changes of velocity are
inversely proportional to the masses.
This result enables us to assign for any two particles, or for
any two bodies treated as particles, a perfectly definite ratio,
which may be called the " mass-ratio." If the force between the
142-145] THE NOTION OF MASS 139
particles produces in them accelerations f arid/' respectively, the
mass-ratio is/' :/.
The mass-ratio of any two particles is the inverse ratio of the
accelerations which, by their mutual action, either produces in the
other.
144. Mass. Whenever two bodies can be treated as particles,
the mass-ratio of the particles is the ratio of the masses of the
bodies.
This statement enables us to assign masses to bodies without
weighing them in a common balance.
Whenever the bodies can be so weighed, the ratio of the
masses that is determined by the mutual action is, as a matter of
fact, the same as the ratio that is determined by the operation of
weighing.
It is clear that the definition of mass by means of mutual action is more
general and more fundamental than that by means of weighing. We shall
show in Chapter X that the determination of masses by weighing is a
particular case of the determination by means of mutual action.
Since we are accustomed to estimate the quantity of matter in a body by
weighing the body, it is customary to state that the quantity of matter in
a body is equal to the mass of the body.
To produce any alteration in the velocity of a moving body, to set the
body in motion, or to bring it to rest, applications of force are required.
This result leads us to recognize a tendency in bodies to maintain an estab-
lished state of motion when there are no forces which produce changes
of motion. This tendency is called "inertia." The impulse of the force
required to produce any assigned change of motion in a body is proportional
to the mass of the body. Thus the mass of the body provides a measure
of its inertia.
\
145. Density. The fraction
number of units of mass in the mass of a body
number of units of volume in the volume of the body
is the " mean density " of the body. In the same way we may
define the mean density of any portion of a body.
When the mean density of all parts of the body is the same,
the body is said to be " homogeneous," or " uniform," otherwise it
is " heterogeneous."
140 THE LAW OF REACTION [CH. VI
In the case of a heterogeneous body, we may define the density
at a point as the limit to which the mean density of a volume con-
taining the point tends when the volume is diminished indefinitely.
The densities of sensibly homogeneous substances in assigned
circumstances are physical constants. For example, the density
of pure water (at a temperature of 4° Centigrade and a barometric
pressure represented by 76 centimetres of mercury) is unity, the
centimetre and the gramme being the units of length and mass.
Density is a physical quantity of dimensions 1 in mass and
— 3 in length.
146. Gravitation. The periodic time of a particle describing
an elliptic orbit about a focus is 27ra*//,~^, where 2a is the major
axis of the orbit, and //. is the intensity of the field of force at unit
distance from the focus (Art. 48, Ex. 5). The result that the squares
of the periodic times of the Planets, describing orbits about the
Sun, are proportional to the cubes of the major axes of the orbits,
was noted by Kepler*. If the intensity of the field of the Sun's
gravitation is denoted by /t/(distance)2, the quantity p is the same
for all the Planets.
Let E be the mass of the Earth, P that of any Planet, r, r
the distances from the Sun to the Earth and the Planet respect-
ively. The forces of the Sun's gravitation, acting on the Earth
and the Planet respectively, are ^Ejrz and fiPjr*. These therefore
are the magnitudes of the forces which the two bodies exert on
the Sun, and they are proportional to the masses of the bodies.
Thus the force of the Earth's gravitation, and the force of the
Planet's gravitation, are proportional to the masses of the Earth
and the Planet respectively. We should accordingly expect the
force of the Sun's gravitation to be proportional to the mass of the
Sun, that is to say, we are led to take for /* the form yS, where $
denotes the mass of the Sun and 7 is a constant independent of
the masses. The force exerted by the Sun on the Earth, or by the
Earth on the Sun, is then expressed by the formula
yES
* Harmonice Mundi, 1619. The result is sometimes called Kepler's "third law
of planetary motion."
145-148] GRAVITATION 141
Such forces would arise if bodies were made up of small parts,
each of which may be treated as a particle, if these particles acted
upon each other with forces in the lines joining their positions,
and if the force between two particles of masses m and m' were an
attraction of amount
<ym>ri
^ '
The law of gravitation states that this formula expresses the
law of force between particles (taken to be small parts of bodies)
at all distances which can be measured by ordinary means (e.g. by
a divided scale), and at all greater distances.
The law can be verified by actual observation of the gravitational force
between bodies at the Earth's surface. By these observations also the value
of y can be determined. The best determination gives for y the value
(6'65) 10~8 in C.G.S. units*.
The quantity y is a physical constant ; it is called the " constant of
gravitation." It is of dimensions, 3 in length, — 1 in mass, - 2 in time.
Since the intensity of the field of the Sun's gravitation is yS/( distance)2,
a knowledge of the period of the Earth's revolution about the Sun (365^ days)
enables us to determine the mass of the Sun.
147. Theory of Attractions. When a body is regarded as made up
of particles, and the particles of a body, and those of other bodies, act upon
each other with forces according to the law of gravitation, the resultant force
acting on a particle of any one of the bodies may be calculated. The theory
by means of which the calculation is effected is the Theory of Attractions,
and accounts of it will be found in books on Statics. From our present
point of view, the most important result of the theory is that homogeneous
spheres, or spheres of which the material is arranged in concentric spherical
strata of constant density, attract an external particle as if their masses
were condensed at their centres t.
148. Mean density of the Earth. In consequence of the result ^
last stated, we are led to take the intensity of the field of the Earth's
gravitation, even at a moderate distance, to be yE/R2, where E is the mass
of the Earth, and R denotes distance from its centre. Now if we take R
to be the radius of the Earth, this quantity is the acceleration of a free
body at the surface. Apart from the correction on account of the rotation
of the Earth, it is the same as g. We denote it by g1. Then we find that
the mean density p of the Earth is given by the equation
~"4iry/r
* C. V. Boys, Proc. R. Soc. London, vol. 56 (1894).
t The result is due to Newton, Piincipia, Lib. i. Sect. xn.
142 THE LAW OF REACTION [CH. VI
If we ignore the distinction between g1 and g, or if we determine g' (cf.
Chapter X), this equation gives us p when y is known. Thus the law of
gravitation avails for the determination of the mass and the mean density
of the Earth. The mean density (in grammes per cubic centimetre) has
been determined* to be 5'527, or about 5| times the density of water.
149. Attraction within gravitating sphere. It is a known
result in the Theory of Attractions that a homogeneous shell bounded by
concentric spherical surfaces exerts no attraction at any point within its
inner surface.
It follows that the attraction at a point within a homogeneous gravitating
sphere is that of the concentric sphere which passes through the point.
If the Earth were a homogeneous sphere of radius a, the attraction of
the Earth upon an internal particle at a distance r from its centre would be
<i i- </, where g' is the attraction at the surface.
150. Examples.
1. Consider the motion of a particle under the action of a uniform fixed
gravitating sphere, of density p and radius a, and suppose the particle to
.start from rest at a distance b ( > a) from the centre. It will move directly
towards the centre with an acceleration ^n-ypa3/j72 at a distance x from the
centre, so long as x>a, and when #«=«, it will have a velocity given by
Now suppose a fine tunnel to be bored through the centre of the sphere
in the direction of motion of the particle. When the particle passes into
the tunnel its acceleration becomes &irypx at a distance x from the centre,
and it moves with a simple harmonic motion. The velocity at a distance
x from the centre is given by the equation
^x2 + $Trypx- = const.,
and the constant is determined from the expression given above for the velocity
at the instant of entering the tube.
Prove that the velocity at the centre is
and, taking b = a, find the time of passing through the tunnel.
2. Prove that, on taking a pendulum down a mine, the time of vibration
is increased or diminished according as the mean density of the surface rock
is greater or less than two-thirds of the Earth's mean density. [Neglect
the distinction between g' and g.]
* C. V. Boys, loc. cit.
148-153] SYSTEMS OF PARTICLES 143
THEORY OF A SYSTEM OF PARTICLES
151. The Sun and the Planets with their Satellites afford an
example of a system of bodies, which can be treated as particles
moving under their mutual attractions. The law of gravitation
avails for the determination of the masses of the system as well
as for .the determination of the motions. Much of theoretical
Mechanics has been developed from the theory of the motion of
such a system of particles. In general we shall suppose that each
particle of the system has an assigned mass, and moves under
forces, some of which are taken to arise from the mutual actions
of particles within the system, and others from the actions exerted
upon particles within the system by particles outside the system.
152. Centre of mass. Let x, y, z be the coordinates at time
t of a particle of the system, m the mass of the particle; and let a
point (z, y, z) be determined by the equations
"2, (mx) 2 (my) 2 (mz)
rp x /?/ N ** ' t? ^ J /
•c v > y ~~ ~~v > ^ — v ~ >
2w 2m 2m
where the summations extend to all the particles. This point is
denned to be the " centre of mass " of the system of particles.
The centre of mass coincides with the "centre of gravity"
defined in books on Statics. On account of the relation between
mass and inertia (Art. 144) it is sometimes called the "centre of
inertia." We shall denote it by the letter G.
153. Resultant momentum. The momentum of a particle
of mass m, which is at the point (x, y, z) at time t, has been
defined to be a vector, localized in a line through the point, of
which the resolved parts in the directions of the axes are mx, my,
mz. The momenta of the particles of a system are a system of
vectors localized in lines.
The general theory of the reduction of a system of localized
vectors (see Appendix to this Chapter) shows that the momenta of
the particles of a system are equivalent to a "resultant momentum,"
localized in a line through any chosen point, together with a vector
couple, which is a " moment of momentum." The resolved parts in
the directions of the axes of the resultant momentum are
2 (mx), 2 (my), 2 (mz),
where the summations extend to all the particles.
144 THE LAW OF REACTION [CH. VI
Now we have
Ix5.m = 2 (mx), y'Zm = 2 (my), ~z?,m — 1.(mz).
The left-hand members of these equations are the resolved parts
parallel to the axes of the momentum of a fictitious particle, of
mass equal to the sum of the masses of the particles, and moving
so as to be always at the centre of mass of the system of particles.
We call this fictitious particle the "particle G." Then we have
the result that the resultant momentum of the system of particles
is equal to the momentum of the particle G.
154. Resultant kinetic reaction. The kinetic reaction of a
particle of mass m, which is at the point (x, y, z) at time t, has
been defined as a vector, localized in a line through the point, of
which the resolved parts in the directions of the axes are mx,
my, mz.
The kinetic reactions of a system of particles are equivalent to
a "resultant kinetic reaction," localized in a line through any
chosen point, and a vector couple, which is a " moment of kinetic
reaction."
The components parallel to the axes of the resultant kinetic
reaction of a system of particles are
2 (mx), 2 (my), 2 (mz).
Now by differentiating the equations such as xS, (m) = 2 (mx), we
find such equations as S2 (m) = 2 (mx).
Hence the resultant kinetic reaction is the same as the kinetic
reaction of the particle G (i.e. of a particle of mass equal to the
mass of the system, placed at the centre of mass of the system,
and moving with it).
155. Relative coordinates. The resultant momentum and
resultant kinetic reaction are independent of the chosen point
which is used in reducing the system of momenta, or kinetic
reactions, to a resultant and a vector couple ; but the vector couples
depend upon the position of the point. For most purposes it is
simplest to take the point either at the origin of coordinates,
whichuian arbitrary fixed point, or at the centre of mass. We shall
take x, y, z to be the coordinates of the centre of mass, and put
] 53-156] MOMENTUM AND KINETIC REACTION 145
Then x', y' , z' are the coordinates of a point relative to the centre
of mass.
From the definition of x, y, ~z we have
2 (mac') = 0, 2 (my') = 0, 2 (mz) = 0,
and it follows that
156. Moment of Momentum. The sum of the moments of
the momenta of the particles of the system about any axis is the
moment of momentum of the system about the axis.
The moment of momentum of the system about the axis x is
2 [m (yz - zy)\
See Appendix to this Chapter. This expression is equal to
2 [m {(y + y') (z + z) -(z + z) (y + if')}],
and this reduces to
(yz - zy) 2 (m) + 2 [m (y'z' - z'y')].
The first term of this expression is the moment about the axis x
of the momentum of the particle G, and the second term is the
moment about an axis drawn through G parallel to the axis x of
the system of momenta mx, my', mz. These are the momenta
relative to parallel axes through G, or the momenta in the " motion
relative to G." We may therefore state our result in the words: —
The moment of momentum of a system about any axis is equal to
the moment of momentum of the particle G, together with the
moment of momentum in the motion relative to G about a parallel
axis through G.
When the momenta of a system of particles are reduced to a
resultant momentum at the centre of mass and a vector couple,
the couple is the moment of momentum in the motion relative to
the centre of mass. It may be called the " resultant moment of
momentum at the centre of mass " and its axis " the axis of re-
sultant moment of momentum." Its components are
^[m (y'z' -z'y')],....
" Moment of momentum " is often called " angular momentum."
L. M. 10
146 THE LAW OF REACTION [CH. VI
157. Moment of kinetic reaction. The sum of the moments
of the kinetic reactions about the axis x is
2 [m (yz - zy)], or ^ 2 [m (yz - zy}\
and this can be expressed in the form
(yz - zy) 2 (m) + 2 [m (y'z' - z'y')],
or ^ [(yz - zy) 2 (m)] + ^ 2 [m (y'z - zy )].
Hence the sum of the moments of the kinetic reactions about any
fixed axis is equal to the rate of increase (per unit of time) of the
moment of momentum about the same axis, and this is equal to
the moment of the kinetic reaction of the particle G about the
axis together with the moment of kinetic reaction in the motion
relative to G about a parallel axis through G.
When the kinetic reactions of a system of particles are reduced
to a resultant kinetic reaction at the centre of mass and a vector
couple, the couple is the rate of increase (per unit of time) of the
resultant moment of momentum at the centre of mass.
The sum of the moments of the kinetic reactions about a moving axis is
not in general equal to the rate of increase (per unit of time) of the moment
of momentum about the same axis. For example, let the axis be parallel to
the axis of z and be specified as passing through all the points whose x and y
are equal to £ and 17, some functions of t. Then the sum of the moments of
the kinetic reactions about the axis is
2 [m {(*-£) .?-(#- ,,)£}],
and the rate of increase of the moment of momentum about this axis is
jt[*rn{(x-Z)y-(y-Ti)x}l
which differs from the above by the addition of the terms
r)^(mx}-^(my}.
If the moving axis always passes through the centre of mass, the two
expressions are equal, as we saw before.
158. Kinetic energy. The kinetic energy of a particle is
half the product of its mass and the square of its velocity.
For a particle of mass m at (x, y, z) it is
157-160] MOMENTUM AND KINETIC REACTION 147
The kinetic energy of a system of particles is the sum of the
kinetic energies of the particles. It is the quantity
£2[>(^ + </2 + *2)].
This expression is equal to
| (a? + ~y* + z2) 2m + £2 [m (x"> + y* + *'*)].
We may state this result in words : — The kinetic energy of a
system of particles is the kinetic energy of the particle G together
with the kinetic energy in the motion relative to G.
159. Examples.
1. Two particles of masses m, m' move in any manner. F is the velocity
of the centre of mass, and •» the velocity of one particle relative to the other.
The kinetic energy is
1 mm'
,.
2 m+m
2, In the same case, if p is the perpendicular from the position of one
particle to the line drawn through the other in the direction of the relative
velocity, the resultant moment of momentum at the centre of mass is
mm
— / Pvi
m + m r. .
and the axis of resultant moment of momentum is at right angles to the
plane containing the particles and the line of the relative velocity.
160. Equations of motion of a system of particles. Let
•W&! be the mass of one particle of the system, xl,yl, zt its coordinates
at time t, X1} F1( Zt the sums of the resolved parts parallel to the
axes of the forces exerted on this particle by particles not forming
part of the system, X-^, F/, Z± the sums of the resolved parts
parallel to the axes of the forces exerted on the same particle by
the remaining particles of the system.
The equations of motion of this particle are
m^ = X, + X,', m^! = Yl + F/, m^ = Z1 + Zj1.
Similarly the equations of motion of a second particle of mass
m-j at (#2, 2/2) ^2) may be written
m&z = X2 + X2', m.2y2 = F2 + F2', m£z = Z.2 + Z2'.
We shall write as the type of such equations
mx = X + X', my=Y+ F', mz = Z+Z'.
Then (X, V, Z) is the type of the external forces, and
(X1, F', Z'} is the type of the internal forces.
10—2
148 THE LAW OF REACTION [CH. VI
161. Law of internal action. The sum of the resolved parts
parallel to any axis, and the sum of the moments about any axis, of
all the internal forces between the particles of a system are identically
zero.
The mutual action between any two particles of the system
consists of two equal and opposite forces acting upon the two
particles in the line joining their positions. The sum of the
resolved parts of these two forces parallel to any axis vanishes.
The moment of a force about an axis is the same at whatever
point in its line of action the force may be applied. Hence the sum
of the moments about any axis of two equal and opposite forces
acting in the same line vanishes.
In the notation of Art. 160 the result may be written
yX')= 0.
162. Simplified forms of the equations of motion. Adding
the left-hand members of all the ^-equations of motion, and remem-
bering that 2,X' = 0, we obtain the equation 2 (m'x) = 2-5T.
In like manner we have
2 (my) = 2 Y, and 2 (m'z) = %Z.
Again multiplying the ^-equations by the y's and the ^/-equa-
tions by the z's, and remembering that 2 (yZr — z Y') = 0, we form
the equation
In like manner we have
2 [m (zx - xz)~\ = 2(zX- xZ\ and 2 [m (ay - yx}} = 2 (xY - yX).
Our equations may be stated in words : —
(1) The sum of the resolved parts in any direction of the kinetic
reactions of a system of particles is equal to the sum of the resolved
parts of the external forces in the same direction.
(2) The sum of the moments about any axis of the kinetic
reactions of a system of particles is equal to the sum of the moments
of the external forces about the same axis.
The result may also be briefly stated in the form : — When the
external forces are regarded as localized in their lines of action, the
161-164] D'ALEMBERT'S PRINCIPLE 149
kinetic reactions and the external forces are two equivalent systems
of localized vectors.
This result, in a slightly different form, was first stated by
D'Alembert in his Traiti de Dynamique, 1743. It is known as
D'Alembert's Principle.
By integrating both members of the equations such as
2 (ma?) = 2X
with respect to the time, between limits which correspond to the
initial and final instants of any interval, we find such results as
Ctl
2 (mx)t=ti - 2 (mx)t=t<> = 2 Xdt,
J to
or, in words : — The change of momentum of the system in any
direction is equal to the sum of the impulses of the external forces
resolved in that direction.
163. Motion of the centre of mass. Since the resultant
kinetic reaction of a system is the kinetic reaction of a particle of
mass equal to the mass of the system placed at the centre of mass
and moving with it, we see that
xZm = 2X, |/2m = 2 F, £2ra = ZZ,
so that the centre of mass moves like a particle, of mass equal to
the mass of the system, under the action of the vector resultant of
all the external forces applied to the system.
164. Motion relative to the centre of mass. In the equations
such as 2 [m (y'z — z'y)\ = X (yZ — zY) put x = x + x , ____ The left-
hand member of the equation just written becomes
[(yz - ~zy} 2m] + 2 {m (y'z - z'y)},
and the right-hand member becomes
The terms in square brackets in the two members are equal, and
we thus have such equations as
H{m(y'z'-zy}} = Z(yZ-z'Y).
These can be stated in words : — The rate of increase (per unit
of time) of the moment of momentum in the motion relative to G,
about any line through G, is equal to the sum of the moments of
the external forces about the same line.
150 THE LAW OF REACTION [CH. VI
165. Independence of translation and rotation. From the
results of the last two Articles we see that the motion of the centre
of mass is determined by the external forces independently of any
motion relative to the centre of mass, and the motion relative to the
centre of mass is determined independently of the motion of the
centre of mass.
166. Conservation of Momentum. When the resultant ex-
ternal force on a system has no resolved part parallel to a particular
line, the sum of the resolved parts of the kinetic reactions of the
particles parallel to that line is zero. Hence the rate of increase
(per unit of time) of the resolved part of the resultant momentum
of the system parallel to that line is zero, or the resolved part of
the resultant momentum parallel to the line is constant.
In such a case the resolved part, parallel to the line, of the
velocity of the centre of mass is constant.
167. Conservation of moment of momentum. When the
sum of the moments of the external forces about any fixed axis
vanishes, the sum of the moments of the kinetic reactions about
that axis vanishes, and the moment of momentum of the system
about the axis is constant.
When the sum of the moments of the external forces about an
axis, drawn in a fixed direction through the centre of mass, vanishes,
the moment of momentum about that axis in the motion relative
to the centre of mass is constant.
168. Sudden changes of motion. As in Art. 160, let X + X'
be the sum of the resolved parts parallel to the axis x of all the
forces, external and internal, that act on a particle m ; and, as in
Art. 82, suppose that X and X' do not remain finite at time t, but
that the impulses of X and X' are finite, or that X and X', defined
by the equations
rt+fr rt+^r
LtT=0 Xdt=X, iX-o I X'dt = X',
Jt-k-r Jt-\r
are finite. Let x and £ be the resolved parts parallel to the axis x of
the velocity of m just after the instant t and just before this instant
respectively. Then we have the equation
mx- = X+ X'.
165-169] MOTION OF A SYSTEM OF PARTICLES 151
In like manner the impulsive changes of velocity parallel to the
axes y and z will be determined by equations which may be written
Now it follows from the law of internal action (Art. 1(51) that
T', . . . and 2 (yZ' — zY'), . . . vanish. Hence we have the equations
These equations can be expressed in words in the statements : —
(1) The change of momentum of the particle G in any direction
is equal to the sum of the resolved parts of the external impulses
in that direction. .
(2) The change of the moment of momentum of the system about
any axis is equal to the sum of the moments of the external impulses
about that axis.
169. Work done by the force between two particles. Let
#!, 2/1, zl and %2, y%, z2 denote the coordinates of the two particles at
time t, and r the distance between them, so that
r2 = (x, - xtf + (y, - ytf + (z, - z2)2.
Also let F denote the magnitude of the force between them, and,
for definiteness, take this force to be repulsive. The components
parallel to the axes of the forces exerted on the particles 1 and 2
respectively are
px\ #2 rr j/l ~ y* -riz\ Zti
r r
L7#2 — #1 jn Hi
and r - — » f
r r r
The rate (per unit of time) at which the first force does work is
,
z,,
r r
and the rate at which the second force does work is
Hence the sum of the rates at which the two forces do work is
V
- O (#! - 4) + (y, - y2) (JA - y2) + (^ - z.2} (^ - z.2}},
or Fr.
152 THE LAW OF REACTION [CH. VI
The work done in any displacement is the value of the integral
[Fr dt or iFdr,
taken between limits which correspond, to the positions of the
particles before and after the displacement.
If the distance between the particles remains unaltered through-
out the motion, no work is done by the force between them ; but
if the distance varies, the internal force does work.
170. Work function. We form as in Art. 86 the work done
by all the forces acting on any particle of a system as the particles
move from their positions at time t0 to their positions at time t. The
expression for the sum of the works of all the forces acting on all
the particles may be written
S f ' {(X + X')x + (Y+ Y')y + (Z+ Z') z\ dt,
Jto
where the summation extends to all the particles.
When this expression has the same value for all paths joining
the initial and final positions of the particles, it is a function of the
coordinates of the final positions, the initial positions being pre-
scribed. This function is the " work function."
We refer to the prescribed initial positions as constituting the
" standard position."
It is important to observe that the work done by the internal
forces may not in general be omitted from the sum.
When a work function exists the system is said to be "con-
servative."
171. Potential Energy. The work function in any position
A with its sign changed is the work that would be done by the
forces if the system passed from the position A to the standard
position. It is defined to be the Potential Energy of the system in
the position A.
For the sake of precision we present our previous statements
in the following form : — A system in which the work done by all
the forces on all the particles, as they pass from one set of positions
to another, is independent of the paths of the particles, is said to be
a conservative system; and the work done by the forces of such a
169-173] ENERGY OF A SYSTEM OF PARTICLES 153
system, as its particles pass from any set of positions to a prescribed
standard set of positions, is called the potential energy of the system
in the former set of positions.
172. Potential energy of gravitating system. When the
force between two particles of masses m, m' is an attraction ymm'/r2, the
work done in a displacement by which the distance r between them changes
from rn to r^ is
fri mm' .
and this is
ymm' ( ) .
Vi ro/
Hence in a gravitating system the work done in any displacement is
fmm! mm'\
where the summation extends to all the pairs of particles.
If we choose the standard position to be that in which all the distances
are infinite, the value of the work function in any other position is
mm'
y* — ,
and the potential energy in this position is
The negative sign indicates that there is less potential energy in any other
state than there is in the state of infinite diffusion.
173. Energy equation. From the equations of the type
mx = X + X'
we form the equation
of which the left-hand member may be written
~U-2|m(*2 + 2/2 + *2)]}.
We deduce the result that the rate of increase (per unit of time)
of the kinetic energy of the system is equal to the rate at which
work is done by all the forces internal and external ; and conse-
quently we deduce the result that the increment of kinetic energy
in any displacement is equal to the sum of the works done by all
the forces.
154 THE LAW OF REACTION [CH.'VI
When a work function exists this result gives us an integral of
the equations of motion, and this integral can be written in the
form
kinetic energy + potential energy = const.
This integral of the equations of motion is called the "energy
equation."
The work done by the internal forces may not, in general, be omitted.
Examples are furnished by a system of gravitating particles, or Toy the
tension in an elastic string.
In some cases the motion of some part of a system is assigned. For
example the system may contain a rigid body which is made to rotate
uniformly about an axis. In such cases force must usually be applied to the
system in order to maintain the assigned motion, and the required force
usually does work. It follows that in such cases there is not, in general, an
energy equation.
174. Kinetic Energy produced by Impulses. As in Art. 1 (JJS
let x, if, z be the resolved parts parallel to the axes of the velocity
of the particle of mass m just after an impulse, £, 77, £ the similar
resolved parts of the velocity just before the impulse, X, Y, Z the
sums of the resolved parts parallel to the axes of the external im-
pulses applied to m, X', Y', Z the sums of the similar resolved
parts of the internal impulses, T and T0 the kinetic energies of the
system just after and just before the impulses.
We have such equations as
m(x-£) = X + X'.
Also T - T, = -| 2, [m (x- + y- + z*)] - £2 [m (f + if- + £)]
= ^ 2 [m (x — £) ( x + £) + two similar terms]
= 2 [( X + X ') -£ (x + |) + two similar terms].
Thus, the change of kinetic energy produced by impulses is the
sum of the products of all the impulses and the arithmetic means of
the velocities, in their directions, of the particles to which they are
applied, just before and just after the impulsive actions.
It is very important to notice that the internal impulses may
not be omitted from the equation here obtained, just as the internal
forces may not be omitted from the energy equation of Art. 173.
THE PROBLEM OF THE SOLAR SYSTEM
175. The problem of n bodies. As we have already explained,
the bodies of the Solar system can be treated as a system of particles
173-176] THE PROBLEM OF TWO BODIES L55
moving under their mutual gravitation. The mathematical problem
of integrating the equations of motion of such a system of particles,
supposed to be n in number, is known as the " problem of n bodies."
The particular cases of two and three bodies are known as the
" problem of two bodies " and the " problem of three bodies." The
only one of these problems which has been solved completely is the
problem of two bodies.
176. The Problem of Two Bodies*. Two particles which
attract each other according to the law of gravitation are projected
in any manner. It is required to show that the relative motion is
parallel to a fixed plane, and that the relative orbits are conies, and
to determine the periodic time when the orbits are elliptic.
The principle of the conservation of momentum shows that the
centre of mass of the two particles moves uniformly in a straight
line. The accelerations of the particles, and the velocity of either
relative to the other, are unaltered, if we refer them to a frame
whose axes are parallel to those of the original frame of reference,
and whose origin is at the centre of mass. We shall suppose this
to be done.
Fig. 46.
Then the acceleration of each particle is in the line joining it
to the origin, and the velocities of the particles are localized in lines
which lie in a plane containing the origin ; the motion of each particle
therefore takes place in this plane.
Now let G be the centre of mass, ml} m.2 the masses of the
particles, ?'j, r.2 their distances from G at time t, 6 the angle which
* The Problem of Two Bodies was solved by Newton, Principia, Lib i, Sect, xi,
Props. 57—63.
156 THE LAW OF REACTION [CH. VI
the line joining them makes with any fixed line in the plane of
motion, also let r, = r, + r2, be the distance between the particles
at time t. The force between them is 71^™
Then the equations of motion of m^ are
Since ?-j = m^r Kir^ + ra^, these equations become
r — r& = — 7 (?/i!
and it is clear that the equations of motion of m.2 would lead us to
the same two equations.
The equations last written show that the acceleration of ?HJ
relative to mz, or of m? relative to m1, is 7(111^ + m2)/?-2, and that
there is no transverse acceleration. Thus either particle describes
a central orbit about the other with acceleration varying inversely
as the square of the distance, and, by Art. 51, this orbit is a conic
described about a focus.
Further, when the orbit is an ellipse, its major axis, 2«, is the
sum of the greatest and least distances between the particles, and
the periodic time is, by Ex. 5 of Art. 48, equal to
(m, + m2){ '
177. Examples.
1. If the particles are projected with velocities v, v' in directions con-
taining an angle a from points whose distance apart is R, prove that the
relative orbit is an ellipse, parabola, or hyperbola according as
V* + v'2 - 2vv' cos a < = or >
2. S, P, and E denote the masses of the Sun, a Planet, and the Earth ;
the major axis of the Planet's orbit is k times that of the Earth's orbit, and
its periodic time is n years ; prove, neglecting the mutual attractions of the
Planets, that
[Kepler's Third Law of Planetary motion quoted in Art. 146 states that
n-=b3 approximately. Kepler's law is approximately correct because S is
great compared with P or E.]
176-178] THE PROBLEM OF TWO BODIES 157
3. Two gravitating spheres of masses m, m', and radii a, a', are allowed
to fall together from a position in which their centres are at a distance c,
it is required to find the time until they are in contact.
We may suppose the centre of mass to be at rest, and take x for the
distance between the centres of the spheres a.t time t. Then their velocities
are equal in magnitude to
m'x , mx
-, and
m-\-m
Hence the kinetic energy of the system is
, / m'x \2 , / mx \2
km( — , +W — >) = i
\m + mj \m + mj
The potential energy, measured from the position in which the distance
was c as standard position, is (see Art. 172)
r
Hence the energy equation is
#2 = 2-y (m + m'} ( --- ) ,
\# CJ
and the time required is
J_ ('
(m + m')} J a + a,
If then we find an angle 6 such that a + a' = ccos?d, we shall have for the
required time
c% (# + sin 6 cos 6)
4. Two gravitating spheres, masses wi, m', moving freely with relative
velocity V when at a great distance apart, would, in the absence of gravitation,
pass each other at a minimum distance d. Prove that the relative orbits
are hyperbolic, and that the direction of the relative velocity will ultimately
be turned through an angle
5. Prove that, if two bodies of masses E and J/ move under their mutual
gravitation and that of a fixed body of mass S, so that the three are always
in a fixed plane, then
where h is the rate at which M describes area about E, and If is the rate at
which the centre of mass of E and M describes area about S.
Prove that, if all three bodies are free, the equation becomes
S ( E+ M )2 ff+(S+E+M) EMh = const.
178. General problem of Planetary motion. In the general
case of a system of particles moving under their mutual gravitation we know
seven first integrals of the equations of motion. The principle of the
Conservation of Momentum gives us three integrals representing the result
158 THE LAW OF REACTION [CH. VI
that the velocity of the centre of mass in any direction is constant. The
principle of the Conservation of Moment of Momentum gives us three
integrals representing the result that the moment of momentum of the
system about any axis drawn in a fixed direction through the centre of
mass is constant. The energy equation also is an integral of the equations
of motion.
Even in the case of three particles these integrals do not suffice for a
complete description of the motion. Except in particular circumstances of
projection, no other first integral has, so far, been obtained.
Thus we cannot deduce from the law of gravitation an exact account of
the motions of the bodies forming the Solar system. But there are a number
of circumstances which conduce to the possibility of deducing from this
law such an approximate account of the motions in question as shall be
sufficiently exact to agree with observation over a long period of time.
Among these we may mention (1) that the mass of the Sun is great com-
pared with that of the other bodies, even the mass of Jupiter being less than
_jLffth part of that of the Sun, (2) that all the orbits are nearly circular, and
all but those of a few Satellites lie nearly in one plane.
It would be outside the scope of this book to explain how these special
circumstances can be utilized for the purpose of integrating approximately
the equations of motion of the bodies of the Solar system. For this we must
refer to books on gravitational Astronomy. A comprehensive treatise i.s
F. Tisserand's Traite de Mecanique celeste, tt. 1 — 4, Paris, 1889-1896.
BODIES OF FINITE SIZE
179. Theory of the motion of a body. We deal with the
motion of a body in the same way as with the motion of a system
of particles. If the body is divided in imagination into a very large
number of very small compartments, and a particle is supposed to
be placed in each compartment, the motion of the body is deter-
mined when the motions of all the particles are known.
We suppose that the particles move under the actions of forces
obeying the law of reaction.
We adjust the masses of the particles so that the sum of the
masses of those particles which are in any part of the body shall
be equal to the mass of that part of the body. This comes to the
same thing as taking the mass of a particle, in any compartment,
to be equal to the product of the volume of the compartment and
the density of the body in the neighbourhood.
In general we do not attempt to determine the forces between
the particles, but we assume that they are adjusted so as to secure
the satisfaction of certain conditions. For example, when the body
178, 179] MOTION OF A BODY IN GENERAL 159
is regarded as rigid, we assume that they are adjusted so that the
distance between any two particles is invariable. When the body
is a string or chain, we assume that the forces between particles
situated on the two sides of a plane, drawn at right angles to the
line of the chain, are equivalent to a single force directed along
this line. This force is the tension of the chain. A more general
discussion will be given in Chapter XL
The centre of mass of a body is found by a limiting process from
the formulae of Art. 152. It coincides with the centre of gravity of
the body, as determined in books on Statics.
The momentum of a body is equivalent to a certain resultant
momentum and a certain moment of momentum. The resultant
momentum is that of a particle, of mass equal to the mass of
the body, placed at the centre of mass and moving with it. The
moment of momentum about any axis through the centre of mass
is the sum of the moments about that axis of the momenta of the
particles relative to the centre of mass.
Like statements hold for the kinetic reaction.
The kinetic energy of the body is equal to the kinetic energy of
a particle, of mass equal to the mass of the body, placed at the centre
of mass and moving with it, together with the kinetic energy of the
motion relative to the centre of mass.
The equations of motion of the body express the statements that
the resolved part of the resultant kinetic reaction in any direction
is equal to the sum of the resolved parts of the external forces in
the same direction, and the moment of kinetic reaction about any
axis is equal to the sum of the moments of the external forces about
the same axis.
The equations of motion of any part of the body are formed in
the same way. The forces exerted upon this part of the body across
the surface which separates it from the rest of the body are now
" external " forces acting on the part in question. The gravitational
attractions between particles within the surface and particles outside
it are also " external " forces acting on the part within the surface.
The rate (per unit of time) at which the kinetic energy of a body
increases is equal to the sum of the rates at which work is done by
all the forces external and internal. If the work done can be specified
160 THE LAW OF REACTION [CH. VI
by a " work function " there is an energy equation, which is an integral
of the equations of motion.
180. Motion of a rigid body. Solid bodies often move in
such a way that no apparent change of size or shape takes place in
any part of them. To represent the motions of such bodies by those
of systems of particles we subject the internal forces between the
hypothetical particles to the condition that the distance between
any two of the particles is to be maintained invariable.
The system of particles subjected to this condition is said to
represent a " rigid body."
The motion of a rigid body is determined when the motion
of three of its particles is determined. For the three particles
determine a frame of reference relatively to which all the particles
of the body have invariable positions.
To determine the positions of all the particles of a rigid body
relative to a frame is therefore the same thing as determining the
position of one frame, F, relative to another. This requires the
determination of the positions of the origin of the frame F, of one
of its lines of reference, and of a plane through that line. The position
of a point depends on three quantities, the coordinates of the point.
The position of a line through a point depends on two quantities,
since the line may make any angle with one of the axes, and the
plane through it parallel to that axis may make any angle with a
coordinate plane, but these two angles determine the line. The
position of a plane through a line depends on one quantity, which
may be taken to be the angle it makes with the plane passing through
the line and parallel to one of the axes of reference. Thus the
positions of all the particles of a rigid body relative to a frame are
determined when six quantities such as those specified are given.
When a rigid body moves without rotation, the motion of the
body is determined by that of a fictitious particle, of mass equal to
the mass of the body, placed at the centre of mass and moving with
it. The equations of motion of this particle are the same as if all
the external forces acting on the body were applied at the centre
of mass, their magnitudes, directions and senses being unaltered.
181. Transmissibility of force. The motion of every part of a
rigid body is known when the motion of any part of it is known.
Now the equations of motion of the body involve the external forces
179-183] MOTION OF A BODY IN GENERAL 161
by containing the sums of the resolved parts of these forces in assigned
directions and the sums of the moments of these forces about assigned axes.
The forces do not enter into the equations in any other way.
The resolved parts and moments in question depend upon the lines of
action of the forces, but not upon their points of application.
Hence the forces may be supposed to act at any points in their lines of
action without altering the motion of the body, or of any part of the body.
In the cases of a deformable body and a system of isolated particles, it is
manifest that the internal relative motion of the parts of the body or system
would be altered by transferring the point of application of a force from one
particle to another in the line of action of the force.
We conclude that a force acting on a rigid body may be regarded as a
vector localized in a line instead of a vector localized at a point. This result
is sometimes called the Principle of the transmissibility of force.
182. Forces between rigid bodies in contact. The surfaces
of two rigid bodies may be regarded as touching at a single point,
and the action between the two bodies (apart from their mutual
gravitation) may be regarded as consisting of a pair of equal and
opposite forces applied at the point of contact.
The force which one of the bodies A exerts upon the other B
at the point of contact can be resolved into components along and
perpendicular to the common normal. The normal component is
the " pressure " of A on B, and the tangential component is the
" friction " of A on B. The resultant of the pressure and friction is
often called the " total reaction."
In the system of two bodies in contact the pressure does no
work ; for, so long as the bodies remain in contact, the parts in
contact have the same velocity in the direction of the normal, and
the pressures acting upon the two bodies are equal and opposite.
In general, the pressure does (positive or negative) work on both
bodies, and the sum of the rates (per unit of time) at which it does
work on the two is zero.
183. Friction. Let P be the point of contact of two bodies
A, B, and let R denote the pressure and F the friction.
Each of the bodies is regarded as having a particle at P.
The particle of A at P will have a certain velocity, and similarly
for the particle of B at P. The velocity of the particle of A at P,
relative to axes parallel to the axes of reference drawn through the
L. M. 11
162 THE LAW OF REACTION [CH. VI
particle of B at P, is the velocity of the point of contact, considered
as a point of A, relative to B. In like manner there is an equal and
opposite velocity of the point of contact, considered as a point of
B, relative to A.
The condition of continued contact is that the relative velocity
just described is localized in a line in the tangent plane at P, or
that the resolved part of this velocity in the direction of the com-
mon normal vanishes.
The first law of Friction is that the friction acting upon
at P is opposite in sense to the velocity of the point of contact,
considered as a point of \ D [ , relative to \ . > .
(B) (A)
The second law of Friction is that the friction F and the pressure
R are connected by a relation of inequality F ^ pR, where JJL is a
constant depending only on the materials of which the bodies are
composed. The constant p is called the coefficient of friction.
When the relative velocity above described is zero, the motion
is described as rolling. In order that rolling may take place it is
generally necessary that the coefficient of friction should exceed a
certain number depending on the circumstances of the case. A
motion of two bodies in contact which is not one of pure rolling
is described as a motion of sliding or slipping. The rule for the
direction of friction may be stated in the form : — Friction tends to
prevent slipping. When slipping takes place .F* = /x.R. When the
bodies are sufficiently rough to prevent slipping throughout the
motion they are sometimes said to be perfectly rough.
When the motion is one of rolling, the friction does no work on
the system of two bodies, but it may do (positive or negative) work
on each of the bodies ; and then the sum of the rates (per unit of
time) at which it does work on the two is zero.
When the motion is one of sliding, the friction does work on the
system, and this work is always negative.
184. Potential energy of a body. For a body under the
gravitational attractions of other bodies, and regarded as made up
of particles, the external forces X, Y, Z of Art. 160 do work in any
displacement; and this work is specified by means of a work function.
1 83-186] ENERGY OF A BODY 163
Further the work done by those components of the internal forces,
which represent the mutual gravitation of the parts of the body, is
also specified by means of a work function. The other internal forces
may also do work, and this work may also be specified by a work
function. When this is the case the portion of the potential energy,
corresponding to this work function, represents what may be called
" internal potential energy."
In such a case the potential energy is divisible into three parts :
potential energy of the body in the field of the external attraction,
potential Energy of the mutual gravitation of the parts of the bod}-,
and internal potential energy.
The potential energy of a body in the field of the Earth's gravity is
represented by the expression
where m denotes the mass of any of the hypothetical particles, and z is the
height of that particle above a fixed level. This expression is equal to
Mgz,
where M is the mass of the body, and z is the height of its centre of mass
above the fixed level.
185. Energy of a rigid body. It follows from the result ot
Art. 169 that the internal forces between the particles of a rigid body
never do any work.
The potential energy of the mutual gravitation of the parts of
a rigid body and the internal potential energy of the body can both
be taken to be zero by choosing the actual state of aggregation of
the body as the " standard " state.
The kinetic energy of the body and the potential energy of the
body in the field of external force are variable quantities.
The equations of motion of a rigid body do not always possess
an integral in the form of an energy equation. For the body may
be in contact with other rigid bodies, or with deformable bodies such
as elastic strings, or with resisting media such as the air ; and the
forces exerted upon the rigid body by bodies with which it is in
contact may do work which is not specified by a work function.
186. Potential energy of a stretched string. Consider a
portion of the string of natural length 10, and let its extension be
e, so that its length is 10 (1 + e). Its tension is \e, where X is the
modulus of elasticity. For the purpose of calculating the potential
11—2
164 THE LAW OF REACTION [CH. VI
energy we may regard this portion as having one end fixed, and the
other attached to a body, which exerts upon it a tension Xe, and
we may also regard the portion as free from the action of any other
external forces. Now let the string be extended further. The rate
at which the terminal tension does work (per unit of time) is
Xe . I0e, for I0e is the velocity of the moving end. Hence the work
done in the extension of the string from its natural length to the
length /0(1 + e) is
IXe.
The integral is taken between limits which correspond to the
values 0 and e of the extension, and its value is ^X^e2.
We may regard the string as being extended so slowly that
no sensible kinetic energy is imparted to it. Then the work done
by the internal forces together with that done by the external
forces vanishes. It follows that the work done by the internal
forces is — ^XJ0e2.
Since this amount depends only on the initial and final states
we can regard it, with changed sign, as an amount of internal
potential energy (Art. 184). Hence the potential energy of a
portion of a stretched string, which is of natural length 10, is
£\/0e2, when its extension is e.
A similar result holds for a spring, whether extended or con-
tracted (cf. Art. 101).
When the string is not stretched uniformly, let s0 be the natural length of
any portion measured from one end, SO + ASO that of a slightly longer portion,
and let s, s+As be what these lengths become when the string is stretched.
Then we define the extension at the point corresponding to s0 to be
If this is denoted by *, the potential energy of any portion between s0= a and
«0=& is
187. Localization of Potential Energy. The potential energy of
a gravitating system and the potential energy of a stretched string are two
examples of the potential energy that arises from internal forces between the
parts of a system.
But the two cases present a marked difference. In the case of the string
we are able to assign a certain amount of the potential energy to each piece
of the string, in such a way that the amount so assigned corresponds to the
186-189] LOCALIZATION OF POTENTIAL ENERGY 165
state of that piece. We may therefore say that the energy is located in the
string, so much being located in each piece. The amount located in any piece
can be expressed as JAe2 per unit of length (in the natural state), e denoting
the extension at any point of the piece. We can think of this energy as
possessed by the piece of string, in the same way as kinetic energy is possessed
by a moving body.
In the case of the gravitating system we are not able to assign a certain
amount of the potential energy to any part of the system in such a way that
changes of the energy so assigned correspond to changes in the state of that
part, independently of changes in the position of the part relative to other
parts. We cannot, in any way that shall be completely satisfactory, locate
some portion of the energy in one part of the system, another portion in
another part of the system, and so on. For instance, in the case of a heavy
body near the Earth's surface we cannot locate the energy in the body, or in
the Earth, or in any definite proportion some of it in the body and some in the
Earth. We have to think of it as possessed by the system, not by the bodies
composing the system.
188. Power. When work is done by the action of a system S upon a
system S' the forces exerted by the particles of S upon the particles of S' do
work in the displacements of the particles of S'. In cases where the energy
can be localized, the energy of the system AS" is increased, and that of S
diminished, by a quantity equal to the amount of work so done. The number
of units of work done in any interval bears a definite ratio to the number of
units of time in the interval; and, when the interval is indefinitely short,
this ratio has a limit, which is the rate at which work is being done per unit
of time.
The power of a system acting on another system is the rate per unit of
time at which the first system does work upon the second.
Corresponding to each force between particles of the two systems there is
a certain power measured by the product of the magnitude of the force and
the resolved part, in its direction, of the velocity of the particle on which it
acts, or by the product of the magnitude of the velocity of the particle and the
resolved part, in its direction, of the force exerted upon it; either of these
products measures the rate at which the force does work. The sum of all
these powers is the power of the first system acting on the second.
The power can be measured equally by the rate at which work is done
upon the second system or by the rate at which the first system does work.
Thus, in any machine performing mechanical work, a certain amount of
energy is expended, and an equal amount of work done, per unit of time ; and
the machine is said to be "working up to a power" measured by the rate at
which the work is done. In general much of the work is done against friction.
189. Motion of a string or chain. In general we neglect
the thickness of the chain, but suppose that the mass of any finite
length of it is finite. When the mass of any portion is proportional
166 THE LAW OF REACTION [CH. VI
to the length of the portion, the chain is uniform. When the
chain is not uniform, the limit of the ratio of the number of units
of mass in the mass of any portion to the number of units of
length in the length of the portion, when the length is diminished
indefinitely, is the mass per unit of length, or the line-density.
If a (geometrical) plane cuts the line of the chain at right
angles at any point, the two parts of the chain which are separated
by this plane act one on the other with a force directed along the
line of the chain at the point. This force is the tension of the
chain.
Let the chain be divided in imagination into a very large
number of very short lengths. In each length let a particle be
supposed to be placed, and let the mass of the particle be the
mass of that length of the chain. Let each of the hypothetical
particles act upon its next neighbours with a force adjusted in
accordance with the law of reaction. The force between two
neighbouring particles is taken to be equal to the tension of the
chain at the corresponding point. The motion of the chain is
determined by forming the equations of motion of any particle,
and then passing to a limit by increasing the number of particles,
and diminishing the lengths of the small portions of the chain,
indefinitely.
If any of the short lengths is As, and if m is the line-density
of the chain in the neighbourhood, wAs is the mass of the cor-
responding particle.
k
The tensions in the two directions from the particle to its two
next neighbours are in general different, but the difference tends
to zero with As.
The other forces acting on the hypothetical particles are the
forces of the field, when the chain is in a field of force, and the
pressure and friction of any curve or surface with which the chain
is in contact.
190. String or chain of negligible mass in contact with
a smooth surface. The chain lies in a curve drawn on the surface.
We resolve the acceleration of any hypothetical particle of the
chain in the direction of the tangent to this curve at the point
occupied by the particle. We denote the resolved part of the
189, 190] MOTION OF A CHAIN 167
acceleration by/! We resolve the force of the field in the same
direction, and denote by F the force of the field per unit of mass
in that direction. The pressure of the surface on the hypothetical
particle is directed at right angles to the tangent to the curve at
the point.
Let T be the tension of the chain at the point ; and let I\ and
T2 be the forces acting between the hypothetical particle and its
two next neighbours, </>i and <£2 the angles which their lines of
action make with the tangent to the curve. In the limit
l\ = T2=T and <£2 = 0, 0, = TT.
Resolve along the tangent to the curve for the motion of the
hypothetical particle. Denoting the mass per unit of length by m,
we have
./= mAs .F+T.1 cos <f>.2 + T^ cos fa,
,»/= mF + ' + T, A' •
As Ax
The limiting form of this equation is
dT
mf=mF+ -5-.
as
dT
If m is very small this equation is nearly the same as -^- = 0.
as
Hence we conclude that, if the mass of the chain may be neglected,
the tension is constant.
The result is proved for any portion of the chain which is in
contact with a smooth surface. The form of the argument shows
that it holds also for any portion which is free.
MISCELLANEOUS EXAMPLES
1. A thin spherical shell of small radius, moving without rotation, describes
a circle of radius R with velocity V about a gravitating centre of force 0 ; and,
when its centre is at a point A, bursts with an explosion which generates
velocity v in each fragment directly outwards from the centre of the shell.
Prove that the fragments all pass through the line AO within a length
and that, if v is small, the stream of fragments will form a complete ring after
a time approximately equal to \-rrRjv.
2. Two equal particles are under the action of forces tending to a fixed
point and varying as the distance from that point, the force being the same at
the same distance in each case ; the particles also attract each other with a
168 THE LAW OF REACTION [CH. VI
different force varying as the distance between them ; prove that the orbit
of either particle relative to the other is an ellipse and the periodic time is
2irlJ(p + 2p.'\ /* and // denoting the forces on unit mass respectively at unit
distance.
3. A body, of mass km, describes an ellipse of eccentricity e and axis major
2a under the action of a fixed gravitating body of mass m. Prove that, if m is
let go when the distance between the bodies is R, the eccentricity e of the
subsequent relative orbit is given by the equation
4. Two gravitating particles of masses m, in' are describing relatively to
each other elliptic orbits of eccentricity e and axis major 2a, their centre of
mass being at rest. Prove that, if m is suddenly fixed when the particles are
at a distance /?, the eccentricity e' of the orbit subsequently described by m'
is given by the equation
,./2 m + m'l-ef*\ /2 IN
) I -a -- - ^ - « I = m [-£ -- .
J\R am 1-e2/ \R aj
5. A body of mass M is moving in a straight line with velocity £', and is
followed, at a distance r, by a smaller body of mass m moving in the same line
with velocity u. The bodies attract each other according to the law of gravi-
tation. Prove that the smaller body will overtake the other after a time
IT — ,/(! — w2) — cos"1 w
r-u
where l-w = —7^7 - -. .
y(M+m)
6. Two bodies, masses m, m', are describing relatively to each other circular
orbits under their mutual gravitation, a and a' being their distances from the
centre of mass. If V is the relative velocity, and m receives an impulse m V
towards TO', prove that the two bodies proceed to describe, relatively to the
centre of mass, parabolas whose latera recta are 2a and 2a'.
7. In a system of two gravitating bodies, M and m, initially M is at rest,
and m is projected with velocity ^{y (M+m)/d} at right angles to the line
joining the bodies, d being the distance between the bodies. Prove that the
path of M is a succession of cycloids and that M comes to rest at successive
cusps after equal intervals of time.
8. In a system of two gravitating bodies of masses M and m the relative
orbit is an ellipse of semi-axes a and b. Prove that, if the mass of the second
body could be suddenly doubled, the eccentricity of the new orbit would be
where v is the relative velocity at the instant of the change.
MISCELLANEOUS EXAMPLES 169
9. Two gravitating particles, whose distance is r, are describing circles
uniformly about their common centre of gravity with angular velocity o>, and
a small general disturbance in the plane of motion is communicated to the
system, so that after any time t the distance is r + u, and the line joining the
particles is in advance of the position it would have occupied if the steady
motion had not been disturbed by the angle <£ ; obtain the equation
2« - ra><$> = 3a>t (rfy + 2&>w) +const.,
squares of u and 0 being neglected.
10. Two equal particles P, Q are projected from points equidistant on
opposite sides of a third particle S, with a velocity due to their distance under
the attraction of S only. All three particles are gravitating, and the directions
of projection are at right angles to PQ. If b is the conjugate axis of the orbit
described by either P or Q, e its eccentricity, and b', e' those of the relative
orbit of P and S (in the absence of Q\ P being projected in the same manner
as before, then 6 = 26', and
11. If three bodies of masses m^ m2, ms, subject only to their mutual
attractions P,a, P3l, P12, remain at constant distances from one another, those
distances are in the ratios
12. Three equal particles A, J3, C, attracting each other with a force
proportional to the distance and equal to p, per unit mass at unit distance,
are placed at the corners of an equilateral triangle of side 2a. The particle A
is projected towards the centre of the triangle with velocity c^p, the other
particles being set free at the instant of projection. Prove that the three
particles will first be in a straight line after a time
1 a
13. Two particles, each of unit mass, attracting each other with a force
fj. (distance), are placed in two rough straight intersecting tubes at right angles
to each other and the friction is equal to the pressure on each tube : prove
that, if they are initially at unequal distances from the point of intersection,
one moves for a time %-rr/Jp, before the other starts, and that, while they are
approaching the point of intersection of the tubes, they move in the same
manner as the projections of the two extremities of a diameter of a circle upon
a straight line on which the circle rolls uniformly.
14. Two particles move in a medium, the resistance of which is pro-
portional to the mass and the velocity, under the action of their mutua
attraction, which is any function of their distance. Prove that their centre
of mass either remains at rest or moves in a straight line with a velocity
which diminishes in geometric progression as the time increases in arithmetic
progression.
170 THE LAW OF REACTION [CH. VI
15. A particle placed at an end of the major axis of a normal section of
a uniform gravitating elliptic cylinder of infinite length is slightly disturbed
in the plane of the section. Prove that it can move round in contact with the
cylinder, and that its velocity v when at a distance y from the major axis of
the section is given by the equation
v*=4:irypy*a(a-b)l{b (a + &)},
where p is the density of the cylinder, and 2a, 26 are the principal axes < >f a
normal section.
16. A particle is projected along a circular section of the surface of a
smooth uniform oblate spheroid given by the equation (#a+y2)/a'- + «2/c-=l.
Prove that, if it describes the circle with uniform angular velocity o> under
the attraction of the spheroid, then
where Ax, Ay, Cz are the components of attraction of the spheroid at a point
(*t .% *)•
17. A ring moves on a rough elliptic wire, of semi-axes a, 6, under the
attraction of a thin uniform gravitating rod of mass M in the line of foci.
Prove that, if it is projected from an end of the minor axis and comes to rest
at the end of the major axis through which it first passes, the velocity ,• of
projection is given by the equation
2_
~
where /* is the coefficient of friction, and a = (a-b)l(a
APPENDIX TO CHAPTER VI
REDUCTION OF A SYSTEM OF LOCALIZED VECTORS
(a) Vector COUple. Two equal vectors, localized in parallel lines,
and having opposite senses, are said to form a "vector couple," or, briefly, a
"couple."
Draw any line L at right angles to the plane of the couple, and choose a
sense for this line. The sum of the moments (with their proper signs) of the
two vectors about this line L is always the same, both in magnitude and in
sign, whatever line L we take, so long as the chosen sense of the line L remains
the same. This sum of moments is the moment of the couple. Its magnitude
is the product of the measure of either vector of the couple and the measure
of the perpendicular distance between the lines in which the vectors are
localized. Its sign is determined when the sense of the line L is chosen.
The rule of signs is the rule of the right-handed screw, and may be stated as
follows : — If the line L meets one of the vectors, and the sense of the line L
and that of the other vector are related like the senses of translation and
rotation of a right-handed screw, the sign is 4- ; otherwise, it is — .
When the sense of the line L is such that the moment is positive, a vector
(unlocalized), of which the magnitude is the magnitude of the moment of the
couple, and the direction and sense are those of the line //, is called the axis
of the couple.
We shall obtain the result that a couple can be represented in all respects
by this unlocalized vector.
(b} Equivalence of couples in the same plane. We shall
prove that two couples in the same
plane, of equal moments, in opposite
senses, are equivalent to zero.
The lines in which the vectors
are localized, being two pairs of
parallel lines, form a parallelogram.
Let this be ABCD (Fig. 47).
Let the vectors of one couple
be of magnitude P, and be localized
in the lines AB, CD ; and let the
vectors of the other couple be of
magnitude Q, and be localized in Fig. 47.
the lines A D, CB.
Let the unit of length be so chosen that AB represents P in magnitude,
Then the area of the parallelogram is of magnitude equal to the moment
of the couple.
172 THE LAW OF REACTION [CH. VI
Hence A D represents Q in magnitude.
Now the vectors P and Q localized in the lines AB, AD, and proportional
to those lines, are equivalent to a vector localized in the line AC, and propor-
tional to that line. The sense of this vector is AC.
Also the vectors P and Q localized in the lines CD, CB, and proportional
to those lines, are equivalent to a vector localized in the line CA, and propor-
tional to that line. The sense of this vector is CA.
It follows that the set of four vectors P, P and Q, Q are equivalent to zero.
This theorem shows that a couple may be replaced by any other couple in
the same plane having the same moment and sense.
(c) Parallel vectors. Let P, Q be the magnitudes of two vectors
localized in parallel lines, A, B any points on these lines, d the distance
between the lines.
When P and Q are in like senses, let two vectors each of magnitude Q be
introduced in the line of the vector P and in opposite senses. Then the
vectors P and Q are equivalent to a vector of magnitude P+Q, localized in
the line of P, and having the sense of P, and a couple of moment Qd. See
Fig. 48. Replace the couple of moment Qd by two vectors, each of magnitude
P+ Q, localized in parallel lines, one of which is the line of P, and let the
sense of the vector in this line be opposite to that of P. The line of the other
vector is at a distance from the line of P which is equal to Qd:(P+Q), it lies
between the lines of P and Q, and the sense of the vector P+Q in it is that
of P or Q. See Fig. 49. The two vectors P and Q are equivalent to a single
vector P+ Q in this line.
Fig. 48. Fig. 49.
When P and Q are in unlike senses, let Q be the greater. Introduce two
vectors each of magnitude Q into the line of action of P. Then the vectors P
and Q are equivalent to a vector of magnitude Q - P localized in the line of P,
and having the opposite sense to P, and a couple of moment Qd. See Fig. 50.
Replace the couple of moment Qd by two vectors each of magnitude Q — P
localized in parallel lines, one of which is the line of P, and let the sense of
the vector in this line be the same as that of P. The line of the other vector
is at a distance from the line of P which is equal to Qd/(Q-P), it lies on
APP.]
SYSTEM OF LOCALIZED VECTORS
173
the side of the line of Q which is remote from the line of P, and the sense of
the vector Q— P in it is that of Q. See Fig. 51. The two vectors P and Q are
equivalent to a single vector Q- P in this line.
B.
Fig. 50.
R=Q-P
Fig. 51.
Hence two vectors localized in parallel lines, when they are not equal and
opposite, are equivalent to a resultant vector localized in a parallel line, and
the moment of the resultant about any axis is equal to the sum of the
moments of the components about the same axis.
(cT) Equivalence of couples in parallel planes. We shall prove
that two couples in parallel planes having equal moments and opposite senses
are equivalent to zero.
Let the vectors of one couple be of magnitude P, and be localized in the
lines AS, CD; and let the vectors of the other couple be of magnitude Q, and
be localized in the lines A'D', C'B'.
Through A'D' and B'C' let there pass a pair of parallel planes meeting the
lines of the couple P in the points A, D, B, C.
Through AB and CD let there pass a pair of parallel planes meeting the
lines of the couple Q in the points A', B', C', D '.
174
[CH. VI
These two pairs of planes with the planes of the two couples form a
parallelepiped.
Replace the couple Q in its plane by an equivalent couple consisting of
vectors localized in the lines S A' and UC'. These vectors are both of magni-
tude P, and have the senses indicated by the order of the letters.
Now parallel vectors P localized in lines AB, DC', and having the senses
indicated, are equivalent to a vector of magnitude 2P localized in the line
MM' joining the middle points of AD1 and BC'. The sense of this vector
is MM1.
Also parallel vectors P localized in lines CD, B'A' are equivalent to a
vector of magnitude 2P localized in the same line MM'. The sense of this
vector is M'M.
It follows that the set of four vectors P, P and Q, Q are equivalent to
zero.
This theorem shows that a couple may be replaced by any couple of the
same moment in any parallel plane.
(e] Composition of couples.
Fig. 53.
Let the planes of two couples meet in the line AB.
Replace the couple in one plane by any couple having one of its vectors
localized in AB in the sense AB.
Let the two vectors be of magnitude P, and let the other be localized in
the line CD.
Replace the couple in the other plane by a couple having one of its vectors
localized in BA in the sense BA.
We can take these vectors also to be of magnitude P, and then the other
will be localized in a certain line FE in the plane of the second couple.
Let AB represent P in magnitude, and through the points AB let there
pass planes at right angles to AB cutting the lines CD and EF in the points
named C, D, E, F.
Then the two couples are seen to be equivalent to a single couple, whose
vectors are of magnitude P, and are localized in the lines CD, FE.
The figures A BCD, ABEF, CDFE&re rectangles, and their areas are pro-
AFP.] SYSTEM OF LOCALIZED VECTORS 175
portional to the momenta of the couples. These areas are in the ratios of the
lengths of BC, BE, CE.
Hence if we turn the triangle BCE through a right angle in its plane its
sides will be parallel and proportional to the axes of the couples. Let B'C'E'
be the new triangle. See Fig. 53. It is clear that, if E'B' represents the axis
of the second couple in sense, the sense of the first is B'C', and the sense of
the resultant is E'C'.
Thus the axis of a couple which has the magnitude, direction, and sense of
a line E'C' is the axis of the resultant of two component couples, the axes of
the components having the magnitudes, directions, and senses of two lines E'B'
and B'C'. This is the vector law.
It follows from the preceding theorems that a couple can be regarded as
an unlocalized vector represented by its axis.
(/) Systems of localized vectors in a plane. Let a vector of
any magnitude P be localized in a line AB, and let 0 be any point not in the
line AB. Through 0 draw a line parallel to AB, and let
there be two vectors each of magnitude P and of opposite
senses localized in this line. Then the system of vectors is
equivalent to a vector localized in the line through 0 parallel
to AB, of magnitude P, and having the sense of the original
vector in A B, together with a couple of moment Pp, where
p is the distance of AB from 0. This couple has a definite
sense, and its axis is perpendicular to the plane AOB.
Any given system of vectors in a plane can in this way
be replaced by a resultant vector localized in a line passing
through a chosen point 0 in the plane together with a
couple. The resultant vector is the resultant of vectors
localized in lines through 0, equal and parallel to the given .
vectors, and having the same senses as those vectors. The
axis of the couple is perpendicular to the plane and its moment is 2 (±Pp),
where P is the magnitude of any one of the original vectors, p the perpen-
dicular on its line from 0, and the sign of each term is determinate.
Let R be the resultant of the vectors at 0, and G the moment of the couple.
If R is not zero, replace G by two localized vectors, each of magnitude R, one
localized in the line of R through 0 and in the sense opposite to R, and the
other in a parallel line at a distance 0/R from 0. The whole system is then
equivalent to this last vector. See Fig. 55.
If R is zero the whole system is equivalent to the couple G.
If R and G are both zero the system is equivalent to zero.
Thus any system of vectors localized in lines lying in a plane is equivalent
to a single vector localized in a line lying in the plane, or- to a couple whose
axis is perpendicular to the plane, or to zero.
The single vector or the couple, in the cases where the system is equiva-
lent to a single vector or a couple, are determinate and unique.
176
THE LAW OF REACTION
[CH. VI
G/R
The conditions of equivalence of two systems of vectors localized in lines
|R lying in a plane are these: (1) When one system is
equivalent to a single vector, the other is equivalent
to a single vector, of the same magnitude and sense,
localized in the same line. (2) When one system is
equivalent to a couple, the other is equivalent to a
couple, of the same magnitude and sense. (3) When
one system is equivalent to zero, the other is equiva-
lent to zero.
(g) Reduction of a system of vectors
localized in lines. Take any origin 0, and any
rectangular axes of a:, y, z. Let X, F, Z be the re-
solved parts parallel to the axes of one of the vectors,
and x, y, z the coordinates of a point on the line in
which it is localized. Introduce a pair of equal and
opposite vectors localized in a line through 0 parallel
to the line of this vector, and resolve them into com-
ponents localized in the axes. The magnitudes of these components are X, Y,
Z. The original vector is thus replaced by vectors X, I7, Z localized in the
axes, and by three couples about the axes, whose moments are
yZ-zY, zX-xZ, xY-yX
respectively. Cf. Art. 84.
Fig. 56.
Hence any system of vectors localized in lines can be replaced by a resultant
vector localized in a line through the origin, whose resolved parts parallel to
the axes of coordinates are 2 Jf, 2 Y, 2Z, together with a couple equivalent to
component couples about the axes, whose moments are 2 (yZ- zY\^ (zX - xZ\
2 (xY-yX\ where X, Y, Z are the resolved parts of any one of the original
vectors parallel to the axes, and x, y, z are the coordinates of any point on the
line in which that vector is localized. The resultant vector, of which the
components are 2 X, ... , is independent of the position of the origin ; but the
vector couple, of which the components are 2 (yZ-zY\ ..., takes different
values for different origins.
CHAPTER VII
191. WE propose in this Chapter to bring together a number
of methods and theories relating to general classes of problems
which can be solved by the principles laid down in previous
Chapters. One of the great difficulties of our subject is the
integration of the differential equations of motion of a system of
bodies, but there are a number of cases in which all the desired
information can be obtained without any integration. Such cases
include sudden changes of motion, and initial motions, or the
motions which ensue upon release from constraint. There are other
cases in which the method of integration is known. Such cases
include small oscillations, and problems in which the principles of
energy and momentum supply all the first integrals of the equations
of motion.
SUDDEN CHANGES OF MOTION
192. Nature of the action between impinging bodies.
When two bodies collide, at first their surfaces come into contact
at a point of each, but a little observation shows that, before
separation, they must be in contact over a finite area ; for example,
if one body is smeared over with soot, the other, after separation,
will show a sooty patch. It is clear therefore that during the
impact the bodies undergo deformation. There are numberless
cases in which the deformation is permanent, there are others in
which the recovery of form is practically complete. Now it is
clear that, if the bodies are rigid, no deformation can take place,
and accordingly we shall be unable to give an account of the
circumstances if we treat the bodies as rigid. On the other hand,
the problem of calculating the deformation from the elastic proper-
ties of the bodies is generally beyond our power. Further, we shall
find that one inevitable result of every impulsive action between
parts of a system is a loss of kinetic energy in the system, and this
apparent loss of energy can frequently be calculated. Nor have we
far to seek for the form of energy that is developed in compensation
L. M. 12
178 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
for the apparent loss. It is a fact of observation that, when one
body strikes against another, the temperature of both is raised,
and it has been abundantly proved that the production of thermal
effects of this kind is of the nature of a transformation of energy.
We must therefore expect that in impulsive changes of motion
some mechanical energy will be transformed into heat. In order to
formulate in a simple and general manner the mechanical effects
produced in two bodies by collision it is necessary to have recourse
to special experiments and subsidiary hypotheses.
193. Newton's experimental Investigation. Newton made
an elaborate series of experiments* on the impact of spheres which
come into contact when their centres are moving in the line joining
them. He found that the relative velocity of the two spheres after
impact was oppositely directed to that before impact, and that the
magnitude of the velocity of separation bears to the velocity of
approach a ratio which is less than unity. He found that this ratio
depends upon the materials of which the spheres are made.
To express this result, let U and V be the velocities of the
two spheres in the line of centres, and in the same sense, before
impact, u and u' their velocities in the same line and in the same
sense after impact, then
u-u> = -e(U-U'\
where e is a positive number less than unity.
194. Coefficient of restitution. The number e is called
the " coefficient of restitution." For very hard elastic solids, such
as glass and ivory, e is little different from unity ; for very soft
materials, such as wool or putty, it approaches zero. The con-
nexion between e and the elasticity of the impinging bodies has
led to its being sometimes called the "coefficient of elasticity," but
we avoid this phrase because it is sometimes used (in a different
meaning) in the Theory of Elasticity. For a like reason we avoid
the phrase "coefficient of resilience" which has also been sometimes
used. Materials for which e is zero or unity may be regarded as
ideal limits to which some bodies approach. We shall speak of
such materials as being "without restitution" and "of perfect
restitution " respectively, ordinary materials we shall speak of as
* Loc. cit, ante, p. 137.
192-195 IMPACT OF ELASTIC SPHERES 179
having " imperfect restitution." It is, of course, to be understood
that any such phrase refers to an action between two bodies of the
same or different materials. The coefficient e depends on both the
materials, just as the coefficient of friction between two bodies
depends on the materials and degree of polish of both.
195. Direct impact of elastic spheres. Let the masses
of the spheres be m, m' ; let the velocities of their centres just
before impact be U, U', and just after impact, u, u', these velocities
being parallel to the line of centres. We suppose all the velocities
to be estimated in the same sense, which is that from the centre
of the sphere m to the centre of the sphere m.
For the determination of u, u' we have the equation given by
Newton's experimental result, viz.
and the equation of constancy of momentum of the system, viz.
mu 4- m'u' = mU+m'U '.
Hence we find
(m-m'e)U+m'(l+e)U'
u —
m + m'
, (m' - me) U'
u =
m + m
Let R be the impulsive pressure between the spheres. R is
regarded as the impulse of a force acting on the sphere m in the
direction opposite to that of U. Then we have
' 7(U-U').
The kinetic energy lost in the impact is
(%mU* + %m'U''2) - (\mu* 4
or \m(U- u)(U + u)+ | w' (U' - u')(U' 4- u),
or ^E[(U+u)-(U' + u')-].
This expression accords with the result of Art. 174.
In virtue of the equation
u-u' = -e(U-U')t
12—2
180 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
the expression for the kinetic energy lost becomes
and, when we substitute for R, we find that this is equal to
. mm' . 2W/T jj'Y
m + m'
196. Generalized Newton's rule. For the purpose of appli-
cations to problems of collision in which the circumstances are less
simple than in the case of direct impact of spheres we state the
following generalization of Newton's experimental result : —
The relative velocities, after and before impact, of the points of
two impinging bodies that come into contact, resolved along the
common normal to their surfaces at these points, are in the ratio
— e:l, where e is the coefficient of restitution.
197. Oblique impact of smooth elastic spheres. Let two
smooth uniform spheres, of masses m, m', impinge.
Let U, V be the resolved veloci-
ties of m in the line of centres
and at right angles thereto before
impact, U', V corresponding ve-
locities of m', and let u, v and u', v'
be corresponding velocities for m
and m' after impact.
The spheres being smooth, there
is no friction between them, and
Fig. 57.
the pressure between them is directed along the line of centres.
Hence the resolved part of the momentum of either sphere at right
angles to the line of centres is unaltered by the impact. We have
therefore the equations
v = V, v'=V.
The generalized Newton's rule gives the equation
u-u' = -e(U-U'\
and the equation of constancy of momentum parallel to the line of
centres is
mu + m'u' = mlf + m U'.
195-199] IMPACT OF ELASTIC SPHERES 181
Solving these equations we find
_(m-m'e)U + m'(l + e)U'
i /
m + m
,_(m'-me)U'
It1 —
m + m
Hence the velocity of each sphere after impact is determined.
The impulsive pressure between the spheres is found in the
same way as in Art. 195 to be
(i+e)^L/(U -to,
m + m
and the kinetic energy lost is found in the same way as in that
Article to be
-
m + m
198. Deduction of Newton's rule from a particular as-
sumption. In the motion before impact, let u, .v denote the components
of velocity of the centre of mass of the two spheres parallel to the line of
centres arid at right angles to this line, W, 17 the components of the velocity
of m relative to m' parallel to the same directions. Then u, v, 17 are unaltered
by the impact. Let W be changed into w by the impact. The quantities W
and w are the " relative velocity of approach " and the " relative velocity of
separation." The kinetic energy before impact is equal to
Cf. Art. 159, Ex. 1. The kinetic energy after impact can be expressed in a
similar form. Hence the kinetic energy lost in the impact is
If we assume that the kinetic energy lost is proportional to the square
of the relative velocity of approach, we have the result that w has a constant
ratio to W, and this is Newton's rule.
199. Elastic systems. The method followed in applying the above
rule is to treat the impact as instantaneous, and the impinging bodies as rigid
both before and after it. This method is adequate for the discussion of many
questions. It cannot however give an exact account of the effects of impact in
elastic systems. In such systems no internal forces are developed except after
gome deformation has taken place, so that at the beginning of a motion which is
suddenly produced some part of the system yields at once, and starts to move
with a finite velocity ; after a finite time a finite deformation is produced,
and is opposed by finite elastic forces, which continue to act as long as there
182 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
is any deformation. This statement may conveniently be summed up in the
proposition: — An elastic system cannot support an impulse. It is now clear
that the method founded on Newton's result is of the nature of a compromise,
the time of the action in which the elasticity of the bodies is concerned being
treated as negligible. An example of the statement that an elastic system
cannot support an impulse will be found in the action of elastic strings
attached to rigid bodies whose motion is altered suddenly. There is no
impulsive tension in such a string, and the motion of the body immediately
after the impulse is exactly the same as if the string were not attached to
it (cf. Art. 213). On the other hand, an inextensible string is regarded as
capable of supporting an impulsive tension.
200. General theory of sudden changes of motion. So
far we have been confining our attention to the impulsive action
between impinging bodies, but there are many other changes of
motion which take place so rapidly that it is convenient to regard
them as suddenly produced. The general method of treating such
changes of motion depends simply on repeated applications of the
principle that for every particle in a connected system, and for
each rigid body in such a system, the changes of momentum are a
system of vectors equivalent to the impulses that produce them.
We shall illustrate the application of this principle by means of
some problems.
201. Illustrative problems.
I. Two equal smooth balls, whose centres are A and B, lie nearly in contact
on a smooth table, and a third ball of equal size and mass impinges directly
on A, so that the line joining its centre C to A makes ivith the line AB an angle
CAB,=ir-0. Prove that, if sin 6 > (1 - e)/(l + e), the ball A will start off in
a direction making with AB an angle tan'1 {2(1 -e}~1 tan 6}, e being the
coefficient of restitution for either pair of balls.
Let Pbe the velocity of C before striking A ; since the impact is direct,
Pis localized in CA. Let w be its velocity after striking A ; the direction of
w is that of V. Let u' be the velocity of A immediately after C strikes it,
u its velocity just after A strikes B, v the velocity of B after A strikes it, then
the direction of u' makes an angle 6 with AB. Suppose the direction of u to
make an angle <£ with AB. The direction of v is AB.
We have the equations of momentum
V=u' + w, u' cos d = u cos <f> + v, u'sin0=usin(f),
and the equations given by Newton's rule
u'-w=eV, ucoa<j)-v = -eu'cosd;
whence 2w=V(l-e), 2w'=K(l+e), 2«cos<£ = (l -e)«'cos<9,
tan d> =
\-e
199-201] SUDDEN CHANGES OF MOTION 183
Thus A moves oft' as stated, provided that there is no second impact
between A and C. The condition for this is u cos (<£ — 6) > w,
1 — e
or i (1 - e} u' cos2 d + u' sin2 6 > n ?t'
1+e
which leads to sin d > (I - e)/(l +e).
Fig. 58.
II. A particle is projected with velocity V from the foot of a smooth fixed
plane of inclination 6 in a direction making an angle a with the horizon (a > 6}.
Find the condition that it may strike the plane n times, striking it at right angles
at the nth impact, e being the coefficient of restitution between the plane and the
particle.
Since the velocity parallel to the plane is unaltered by impact, the motion
of the particle parallel to the plane is determined by the same equation as if
there were no impacts, thus at the end of any interval t from the beginning
of the motion the velocity parallel to the plane is Fcos (a -ff)—gt sin 6.
Let ti, t^, ... tn be the times of flight before the first impact, between the
first and second, and so on. Then t± is given by
\\ sin (a - ff) - \gt? cos (9 = 0,
and tlms £1 = 2Fsin (a — ff)jg cos 6. The velocity perpendicular to the plane
at time ^ is Fsin (a — 6)—gti cos 6 or - Fsin(a-#). Immediately after the
impact the velocity at right angles to the plane becomes eFsin(a-0) away
from the plane. We thus find that t2 = eti, <s = e^> —
1 — en 2 V sin (a — 6)..,., , r
Hence ti + t., + ...+tn. <= — — •*-*-, 1S the interval from the be-
1 - e
ginning of the motion till the «th impact. By supposition, at the end of
this interval the velocity parallel to the plane vanishes, or this interval is
Fcos (a - 6)/g sin 6. The required condition is therefore
cot 6 = 2 tan (a - (9) (1 - e'l)/(l - e).
184 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
III. A smooth sphere of mass m is tied to a fixed point by an inextensibl<-
thread^ and another sphere of mass m' impinges directly on it with velocity v in
a direction making an acute angle a with the thread. Find the velocity with
which m begins to move.
The impulse Itetween the spheres acts in the line of centres so that the
direction of motion of m' is unaltered.
Let its velocity after impact be v'.
There is an impulsive tension in the
thread and the sphere m is constrained
to describe a circle about the fixed end.
It therefore starts to move at right
angles to the thread. Let u be its
velocity.
Resolving for the system at right
angles to the thread we have the equa-
Fig. 59. t,jon of momentum
mu + m'v' sin a = m'v sin a.
By the generalized Newton's rule we have
v' — u sin a = — ev.
m'sin a (1 +e)
— __ _*__ ' M
Whence
,. ,
m+m sin2 a
IV. Two particles A, B of equal mass are connected by a rigid rod of
negligible mass, and a third equal particle C is tied to a point P of the rod at
distances a, b from the two ends, C is projected with velocity u perpendicular
to AB. Find the velocity of C immediately after the string becomes tight.
Let v be the velocity of C immediately after the string becomes tight.
Fig. 60.
Since the impulse on C is along the string its direction of motion is unaltered.
The velocity with which P starts to move is v along the string.
Let o> be the angular velocity with which the rod begins to turn. The
velocity of A is compounded of the velocity of P and the velocity of A
201, 202] SUDDEN CHANGES OF MOTION 185
relative to P. Thus A starts with velocity v + aa>. So B starts with'velocity
v - bco.
The equation of momentum parallel to the string is
mv + m (v + aw) + in (v — ba>} = imi,
m being the mass of either particle.
The equation of moment of momentum about P is
ma
giving
Eliminating w we find
202. Examples.
[In these examples e is the coefficient of restitution between two bodies.]
1. The sides of a rectangular billiard table are of lengths a and b. If a
ball is projected from a point on one of the sides of length b to strike all
four sides in succession and continually retrace its path, show that the angle
of projection 6 with the side is given by aecot d=c + ec', where c and c' are
the parts into which the side is divided at the point of projection.
2. Prove that, in order to produce the greatest deviation in the direction
of a smooth billiard ball of diameter a by impact on another equal ball at rest,
the former must be projected in a direction making an angle
™~i C" \/fe
with the line (of length c) joining the two centres.
3. A particle is projected from a point at the foot of one of two smooth
parallel vertical walls so that after three reflexions it may return to the point
of projection ; and the last impact is direct. Prove that e3 + e2 + e = 1, and that
the vertical heights of the three points of impact are in the ratios
e2 : 1-e2 : 1.
4. A particle is projected from the foot of an inclined plane and returns
to the point of projection after several rebounds, one of which is at right
angles to the plane ; prove that, if it takes r more leaps in coming down than
in going up, the inclination 6 of the plane and the angle of projection a are
connected by the equation
cot 6 cot (a - ff) = 2 { v/( 1 - O - (1 - er)}/{er (1 - e)}.
5. A particle is projected from the foot of a plane of inclination y in a
direction making an angle /3 with the normal to the plane, in a plane through
this normal making an angle a with the line of greatest slope on the inclined
186 MISCELLANEOUS METHODS AND APPLICATIONS [CH. Vfl
plane. Prove that, for the particle to te on the horizontal through the point
of projection when it meets the plane for the nth time, the angles a, ft y must
satisfy the equation
(1 - e") tan y=(l -e) cos a tan /3.
6. Three equal spheres are projected simultaneously from the corners of
an equilateral triangle with equal velocities towards the centre of the triangle,
and meet near the centre. Prove that they return to the corners with velocities
diminished in the ratio e : 1.
7. A smooth uniform hemisphere of mass M is sliding with velocity V
on a plane with which its base is in contact ; a sphere of smaller mass m is
dropped vertically, and strikes the hemisphere on the side towards which it
is moving, so that the line joining their centres makes an angle TT '4 with the
vertical. Show that, if the coefficient of restitution between the plane and
the hemisphere is zero, and that between the sphere and the hemisphere is e,
the height through which the sphere must have fallen if the hemisphere is
.stopped dead is
8. A particle of mass M is moving on a smooth horizontal table with
uniform speed in a circle, being attached to the centre by an inextensible
thre<od, and strikes another particle of mass m at rest. Show that, if the two
particles adhere, the tension of the thread is diminished in the ratio
If there is restitution between the particles and the second one is describ-
ing the same circle as the first, prove that the tensions T and t in the two
threads after impact are connected with their values T0 and t0 before impact
by the equation
T+ 1 = T0 + t0 - (1 - e2) {J(mTJ - J(Mt0)}2j(M+ m).
9. A bucket and a counterpoise, of equal mass M, connected by a chain
of negligible mass passing over a smooth pulley, just balance each other, and
a ball, of mass ?«, is dropped into the centre of the bucket from a height k
above it ; find the time that elapses before the ball ceases to rebound, arid show
that the whole distance descended by the bucket during this interval is
4meh/{C2M+m) (1 - e)2}.
10. Three equal particles are attached to the ends and middle point of a
rod of negligible mass, and one of the end ones is struck by a blow so that it
starts to move at right angles to the rod. Prove that the magnitudes of the
velocities of the particles at starting are in the ratios 5:2:1.
11. An impulsive attraction acts between the centres of two sphere*
which are approaching each other so as to generate kinetic energy E. If v is
their relative velocity before the impulse, and 0, & the angles which the
202-204] INITIAL MOTIONS 187
directions of the relative velocity, before and after, make with the line of
centres, then
where M is the harmonic mean of the masses.
12. Two small bodies of equal mass are attached to the ends of a rod
of negligible mass ; the rod is supported at its centre and is turning
uniformly, so that each of the bodies is describing a horizontal circle, when
one of the bodies is struck by a vertical blow equal in magnitude to twice
its momentum. Prove that the direction of motion of each of the bodies is
instantaneously deflected through half a right angle.
INITIAL MOTIONS
203. Nature of the problems. We suppose that a system
is held in some definite position in a field of force, and that at a
particular instant some one of the constraints ceases to be applied ;
then the system begins to move, each particle of it with a certain
acceleration. Our first object in such a case is to determine the
accelerations with which the parts of the system begin to move.
When the accelerations have been found -there is generally no
difficulty in determining the initial values of the reactions of
supports, or internal actions between different bodies of the system ;
and the determination of the unknown reactions is our second
object.
The senses of the accelerations with which a conservative
system moves away from a position of instantaneous rest can
sometimes be determined by help of the observation that the
motion must be one by which the potential energy is diminished.
This is evident since the kinetic energy must be increased above
the value (zero) which it has in the position of rest.
The problem of determining the curvature of the path of a
particle whose velocity is not zero offers no difficulty when the
velocity and acceleration are known, since the resolved acceleration
along the normal to the path is the product of the square of the
resultant velocity and the curvature. This remark enables us
easily to determine the initial curvature of the path of a particle
when its motion is changed suddenly.
204. Method for initial accelerations. It is always possible
to determine expressions for the accelerations of all the points of
188 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
a connected system in terms of a small number of independent
accelerations, and there is always the same number of equations of
motion free from unknown reactions, so that all the accelerations
can be found. The expression of the initial accelerations in the
proposed manner is facilitated by observing (1) that the velocity
of every particle initially vanishes, (2) that every composition and
resolution may be effected by taking the position of the system to
be that from which it starts. The method will be better under-
stood after the study of an example. We purposely choose one of
a somewhat complicated character in order to illustrate the various
details of the method.
205. Illustrative Problem. Four equal rings A, B,C, Dare at equal
distances on a smooth fixed horizontal rod, and three other equal and similar
rings P, Q, R are attached by pairs of equal inextensible threads to the pairs of
rings (A, B), (B, C}, (C, Z>). The system is held so that all the threads initially
make the same angle a with the horizontal, and is let go. It is required to find
the acceleration of each ring.
From the symmetry of the system the accelerations of A, D are equal and
Fig. 61.
opposite, so are those of S, C, and those of P, R. Also the acceleration of Q
is vertical.
Let/,/' be the accelerations of A, B along the smooth horizontal rod.
Now relatively to A, P describes a circle, and thus the acceleration of P
relative to A is made up of a tangential acceleration /j at right angles to AP,
and a normal acceleration proportional to the square of the angular velocity
of A P. Since the initial angular velocity vanishes, we have, as the relative
Acceleration,/! at right angles to AP. Again, since the threads AP, BP are
equal, the particle P is always vertically under the middle point of AB and
thus its horizontal acceleration is | (/+/')•
Hence *(/+/>/-/, sin a,
givmg /, sin a
204-206] INITIAL MOTIONS 189
Again, the horizontal acceleration of Q vanishes, and we have therefore the
acceleration j\ of Q relative to B given by the equation
/2sina=/'.
Thus the accelerations of the particles are expressed in terms of / and /' ;
in particular the vertical accelerations of P and Q are ^ (/—/') cot a and
/' cot a downwards.
Now let m be the mass of each particle and T\, 7^, Ts the tensions in the
threads as shown in the figure. Then resolving horizontally for A, P, and B
we have
mf=Tlcosa, ^m(f+f') = (T^-T1)cosa, mf' = (T3- T2)cos a...(l) ;
and resolving vertically for P and Q we have
\m (/—/') cot a= - (7Ti + 7T2)sin a + mg, mf cota= - 273 sin a + m<?...(2).
From the set of equations (1) we have
T! cos a=mf, T2 cos a=m (f/+iA Z'scos a = m f (/+/') ;
and from (2), on substituting for 7\, T2, T3, we have
(/-/') cota + (5/+/') tan a=2#, /' cot a + 3 (/+/') tan a=g ;
f f q sin 2a
wnpncp ___ ~~ _ — _ — ~^~~ _ — _
4--cos2a cos 2a 12 — 11 COS 2a + cos2 2a '
206. Initial curvature. As an example of initial curvatures when
the motion does not start from rest we take the following problem :
Two particles of masses m, m' connected by an inextensible thread of length I
are placed on a smooth table with the thread straight, and are projected at right
angles to the thread in opposite senses. It is required to find the initial
curvatures of their paths.
Let «, v be the initial velocities of the particles, and &> the initial angular
velocity of the thread, then
Let G be the centre of mass
of the two particles. Then G
moves uniformly on the table
with velocity
(mu — m'v)/(m + m').
The acceleration of G vanishes, p^ g2.
and the acceleration of m relative
to G is that of a particle describing a circle of radius m'l/(m + m') with angular
velocity « ; thus the acceleration of m along the thread is m'l<M>*j(m-\-m'), and
this is the acceleration of m along the normal to its path. Hence, if p is the
initial radius of curvature of the path of m,
u2 m'l /t< + v\2
p m + m' \ I / '
giving 1/p = m' (u + v)z/{(m + m,') ht?}.
190 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
In like manner the initial curvature of the path of m' is
m (u + v)'*j {(m + m') Iv*}.
207. Examples.
1. Two bodies A and B of equal weight are suspended from the chains
of an Atwood's machine ; A is rigid, while B consists of a vessel full of
water in which is a cork attached to the bottom by a string. Supposing
the string to be destroyed in any manner, determine the sense in which A
begins to move.
2. A particle is supported by equal threads inclined at the same angle a
to the horizontal. One thread being cut, prove that the tension in the
remaining thread is suddenly changed in the ratio 2 sin2 a : 1.
3. Particles of equal mass are attached to the points of trisection C, D
of a thread A CDB of length 3£, and the system is suspended by its ends from
points A, B distant £(1 + 2 sin a) apart in a horizontal line, so that CD is
horizontal and equal to /. Prove that, if the portion DB of the thread is cut,
the tension of A C is instantly changed in the ratio 2 cos2 a : l+cos2a, and
that the initial direction of motion of D is inclined to the vertical at an
angle $ such that
cot 0 = tan a + 2 cot a.
4. Three small equal rings rest on a smooth vertical circular wire at the
corners of an equilateral triangle with one side vertical, the uppermost being
connected with the other two by inextensible threads. Prove that, if the
vertical thread is cut through, the tension in the other thread is instantly
diminished in the ratio 3 : 4.
5. A set of 2n equal particles are attached at equal intervals to a thread,
and the ends of the thread are attached to equal small smooth rings which
can slide on a horizontal rod. The rings are initially held in such a position
that the lowest part of the thread is horizontal and the highest parts make
equal angles y with the horizontal, and the rings are let go. Prove that in
the initial motion (i) the acceleration of each particle is vertical, (ii) the
tension in the lowest part of the thread is to what it was in equilibrium in
the ratio m' : mn cot2 y + m' , where m is the mass of a particle and m' the
mass of a ring.
6. Three particles A, B, C of equal masses are attached at the ends and
middle point of a thread so that AB = BC=a, and the particles are moving
at right angles to the thread, which is straight, with the same velocities,
when B impinges directly on an obstacle. Prgve that, if there is perfect
restitution, the radii of curvature of the paths which A and C begin to
describe are equal to Ja.
7. Two particles, of masses M and nM, are attached respectively to a
point of a thread distant a from one end and to that end, and the other end
is fixed to a point on a smooth table on which the particles rest, the thread
being in two straight pieces containing an obtuse angle TT - a. Prove that, if
206-208] APPLICATIONS OF THE ENERGY EQUATION 191
the particle nM is projected on the table at right angles to the thread, the
initial radius of curvature of its path is a (1 +n sin2 a).
8. Two particles P, Q, of equal mass, are connected by a thread of length I
which passes through a small hole in a smooth table. P being at a distance c
from the hole and Q hanging vertically, P is projected on the table at right
angles to the thread with velocity v ; prove that the initial radius of curvature
of P's path is 2cv*l(v'2 + cg). Prove also that, if Q is projected horizontally
with velocity v, while P is not moved, the initial radius of curvature of Q's
path is
APPLICATIONS OF THE ENERGY EQUATION
208. Equilibrium. The possible positions of equilibrium of
a system are distinguished from other positions by the condition
that, if the system is at rest in an equilibrium position, so that all
the velocities vanish there, the accelerations also vanish there.
Now let the equations of motion be taken in the forms
mx = X + X'
of Art. 160; and let the system pass through a position of equi-
librium with any velocities denoted typically by x, y , z'. The
equation which expresses the result that the rate of change of
kinetic energy (per unit of time) is equal to the rate at which
work is done (Art. 173) is
2 \m (xx1 + yy' + zz}} = 2[(X + X')x' + (Y+ Y'} y' + (Z + Z') z'}.
Since, by hypothesis, the position is one of equilibrium, the left-
hand member of this equation vanishes. Hence the right-hand
member also vanishes, or we have the result : —
The rate at which work is done when a system passes through a
position of equilibrium with any velocity vanishes.
This result is usually stated in a form involving infinitesimals,
and is called the " Principle of Virtual Work " or of " Virtual
Velocities."
In forming the expression for the rate at which work is done,
or the expression for the virtual work, the velocities must be such
as are compatible with the connexions of the system. Further, if
there are any resistances which depend upon velocities, and vanish
with the velocities, the rate at which these resistances would do
work is to be omitted, for manifestly such resistances do not affect
the positions of equilibrium.
192 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
When there is a work function W, the rate at which work is
done is — - . If W is a function of any quantities which define the
dt
position of the system, say 6, <$>, ..., then
dW
If the position is one of equilibrium, this vanishes for all values of
0, <j>, . . . . Hence we have the equations
and the values of 6, <j>, ... which satisfy these equations determine
the positions of equilibrium.
If we sought the positions in which W is a maximum or mini-
mum, we should have to begin by solving the equations
8Tf-0
W
and then proceed to determine which among the various sets of
solutions make W a true maximum or a true minimum. In the
positions in which
we should say that W is stationary, whether it is a true maximum
or minimum or not. Since the potential energy of the system in
any position is — W, we have the result :
The equilibrium positions of a conservative system are those
positions in which the potential energy is stationary.
209. Machines. In all the so-called " simple machines " or
"mechanical powers" the positions of all the parts can be expressed
in terms of a single variable, and consequently the potential energy
is determined in terms of a single variable. The condition that
the potential energy is stationary in the position of equilibrium
becomes a relation between the masses of two moving parts : the
" power " and the " weight." This result is worked out in books on
Statics.
In any conservative system in which the positions of all the
parts can be expressed in terms of a single variable, the equation
of energy determines the whole motion. We had an example in
Atwood's machine [Ex. 1 of Art. 74].
208-211] EQUILIBRIUM AND SMALL OSCILLATIONS 193
210. Examples.
1. Two bodies are supported in equilibrium on a wheel and axle, and
a body whose mass is equal to that of the greater body is suddenly attached
to that body. Prove that the acceleration with which it moves is bg/(2b + a),
a and b being the radii of the wheel and the axle respectively, and the inertia
of the machine being neglected.
2. In any machine without friction and inertia a body of weight P
supports a body of weight W, both hanging by vertical cords. These bodies
are replaced by bodies of weights P and W't which, in the subsequent
motion, move vertically. Prove that the centre of mass of P' and W will
descend with acceleration
g ( WP'- W'P)*I( W*P' + FT2) ( W' + P').
211. Small oscillations. We have to consider the small
motion of a system which is slightly displaced from a position of
equilibrium. We confine our attention to cases where any position
of the system is determined by assigning the value of a single
geometrical quantity 6, as in the case of the simple circular pen-
dulum (Article 119). We can always choose 6 to vanish in the
position of equilibrium ; for, if it has been chosen in any other way
so that its value in the position of equilibrium is 00, then 9 — 0Q
can be used instead of 6.
Now the velocity of each particle of the system can be expressed
in terms of 6 and 0, and the kinetic energy T is thus of the form
^A02, where A may depend upon 0, but does not vanish with 0.
Also the potential energy V vanishes with 0, if the standard
position is the position of equilibrium. Thus Fis a function of 0
which may be expanded in powers of 0 and the series contains
no term independent of 0. Again, the principle of virtual work
dV
shows that -j^- vanishes with 0, or that the term of the first order
du
is missing from the series for V. Thus V can be expressed as a
series beginning with the term in $2, and more generally we may
say that, when 0 is sufficiently small, F= %C0'2, where G is a function
of 0 which is finite when 0 — 0.
The equation of energy accordingly is
^40a + £C0a = const.,
and on differentiating we have
L. M. 13
194 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
Omitting small quantities of an order higher than the first we
have
where A and C have their values for 6 = 0. Thus, if these two
quantities have the same sign, the motion in 6 is simple harmonic
with period 2?r ^(A/C).
Now A must be positive since otherwise the expression \AQ-
could not represent an amount of kinetic energy. Hence there are
oscillations in a real period if C is positive.
72 \T
The value of C for 6 = 0 is the value of -^ for 6 = 0, and thus
the conditions for a real period of oscillation are the same as the
conditions that V may have a minimum value in the position of
equilibrium.
If the period is real, the motion can be small enough for the
approximation to be valid ; otherwise it soon becomes so large that
we cannot simplify the equation of motion by neglecting 6'2. In
the former case the equilibrium is stable and in the latter unstable.
We learn that in a position of stable equilibrium the potential
energy is a minimum*.
The process which has been adopted shows that we might have
reduced the expression for T by substituting zero for 0 in A, and
the expression for V might have been taken to be simply the
term of the series which contains 62. These simplifications might
have been made before differentiating the energy equation. If we
express the kinetic energy correctly to the second order of small
quantities in the form %A62, and the potential energy also correctly
to the second order of small quantities in the form ^C62, the
period of the small oscillations is 27r*J(A/C). In the case of a
simple pendulum of mass m and length I, A is ml2 and C is mgl,
so that
A/C^lfg.
In any other case we may compare the motion with that of a
simple pendulum, and then the quantity gA/C is the length of a
simple pendulum which oscillates in the same time as the system.
It is called the length of the equivalent simple pendulum for the
small oscillations of the system.
* This result, here proved for a special class of cases, is true for all conservative
systems.
211-213] SMALL OSCILLATIONS 195
212. Examples.
1. Two rings of masses m, m' connected by a rigid rod of negligible
mass are free to slide on a smooth vertical circular wire of radius a, the rod
subtending an angle a at the centre. Prove that the length of the equivalent
simple pendulum for the small oscillations of the system is
(m + m1) a/*/(m2 + m'2 + 2mm' cos a).
2. One end of an inextensible thread is attached to a fixed point A, and
the thread passes over a small pulley B fixed at the same height as A and at
a distance 2a from it and supports a body of mass P. A ring of mass M can
slide on the thread and the system is in equilibrium with M between A and B.
Prove that the time of a small oscillation is
3. A particle is suspended from two fixed points at the same level by
equal elastic threads of natural length a, and hangs in equilibrium at a depth
h with each thread of length I. Prove that, if it is slightly displaced parallel
to the line joining the fixed ends of the threads, the length of the equivalent
simple pendulum for the small oscillations is
4. Prove that, if the fixed points in Ex. 3 are at a distance 2c apart,
and the particle is displaced vertically, the length of the equivalent simple
pendulum is
5. A pulley of negligible mass is hung from a fixed point by an elastic
cord of modulus X and natural length a, and an inextensible cord passing
over the pulley carries at its ends bodies of masses M and m. Prove that the
time of a small oscillation in which the pulley moves vertically is
213. Principles of Energy and Momentum. We have re-
marked that there are numerous cases in which the principles of energy and
momentum supply all the first integrals of the equations of motion of a system,
and thus suffice to determine the velocities of the parts of the system in any
position.
To illustrate these principles further we take the following problem :
Two particles A and B, placed on a smooth horizontal table, are connected
by an elastic string of negligible mass. When the string is straight, and of its
natural length, one of the particles is struck by a blow in the line of the string
and away from the other particle ; determine the subsequent motion.
Let m be the mass of the particle struck, m' that of the other, F the
velocity with which m begins to move. There is no tension in the string
until it is extended, and thus at first m' has no velocity.
The centre of mass moves on the table with uniform velocity u,
= mVl(m + m'), in the line of the string. Let x be the increase in the
13—2
196 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
length of the string at time t, then the velocities of the particles are
m'x
— , , - • / •
m+m m + m
1 1 mm' .„
Hence the kinetic energy is - (m + m ) K^- m,m, x •
The potential energy is - - x2 so long as x is positive, a being the natural
— ' '
length of the string, and X the modulus of elasticity.
Thus the energy equation is
1 m? T,9 1 mm' 1 X 0 1 JT,
-- --- ,
2 m + m' 2 TO + OT' 2 a 2
showing that the motion in x is simple harmonic motion of period
2ir J {mm? a I (m + m') X},
so long as # remains positive. Whenever the string is unstretched we have
x= + V. When x vanishes the string has its greatest length
a + V*J{mm'a/(m + m) X}.
We can thus describe the whole motion : — m moves off with velocity F
which gradually diminishes, and m' moves in the same direction from rest
with gradually increasing velocity ; the string begins to extend, and continues
to do so until it attains its greatest length ; this happens at the end of a
quarter of the period of the simple harmonic motion, and at this instant the
particles have equal velocities u. The velocity of m continues to diminish
until it is reduced tc V (m — m')/(m + wi'), and the velocity of m' continues to
increase until it has the value 2»i F/(TW + TO'), these values being attained at the
same instant ; in the meantime the string contracts to its natural length cr,
which it attains at the instant in question, and this happens at the end of
half a period from the beginning of the motion. The particles then move
with the velocities they have attained until m' overtakes m, when a collision
takes place. The subsequent motion depends on the coefficient of restitution.
If this is unity, the relative motion is reversed. In any case the description
of the subsequent motion involves nothing new.
214. Examples.
1. A shot of mass m is fired from a gun of mass M placed on a smooth
horizontal plane and elevated at an angle a. Prove that, if the muzzle velocity
of the shot is F, the range is
2F2
g
2. A smooth wedge of mass M whose base angles are a and )3 is placed
on a smooth table, and two particles of masses m and m' move on the faces,
being connected by an inextensible thread which passes over a smooth pulley
at the summit. Prove that the wedge moves with acceleration
(TO sin a ~ m' sin /3) (m cos a + m' cos /3)
3 (in + m') (M+m+m') - (m cos a + m' cos /3)2 '
213, 214] ENERGY AND MOMENTUM 197
3. Two bodies of masses »i1} m2 are connected by a spring of such strength
that when ml is held fixed mt makes n complete vibrations per second. Prove
that, if m2 is held, m1 will make n ^(m^/m^, and that, if both are free, they
will make » /v/{(wi + TO2)/wh} vibrations per second, the vibrations in all cases
being in the line of the spring.
4. Three equal particles are attached at equal intervals to an inextensible
thread, and, when the thread is straight, the two end ones are projected with
equal velocities in the same sense at right angles to the thread. Prove that,
if there are no external forces, the velocity of each of the end particles
(at right angles to the part of the thread which is attached to it) at the
instant when they impinge is J ^/3 of their initial velocity.
5. A particle is attached by an elastic thread of natural length a to a
point of a smooth plank which is free to slide on a horizontal table, and the
thread is stretched to a length a + c, in a horizontal line passing over the
centre of mass of the plank, and the system is let go from rest. Prove that,
if the plank and particle have equal masses, and the modulus of elasticity of
the thread is equal to the weight of the particle, the velocity of the particle
relative to the plank when the thread has its natural length is that due to
falling through a height c2/a.
6. A spherical shell of radius a and mass m contains a particle of the
same mass, which is attached to the highest point by an elastic thread of
natural length a, stretched to length a + c, and is also attached to the lowest
point by an inextensible thread ; and the shell rests on a horizontal plane.
Suddenly the lower thread breaks, the particle jumps up to the highest point
of the shell and adheres there, and it is observed that the shell jumps up
through a height h. Prove that the modulus of elasticity of the upper
thread is
Zmga (a + c + 4h)/c2.
What external forces produce momentum in the system as a whole ?
7. Three equal particles are connected by an iuextensible thread of length
a + 6, so that the middle one is at distances a and b from the other two.
The middle one is held fixed and the other two describe circles about it with
the same uniform angular velocity so that the two portions of the thread are
always in a straight line. Prove that, if the middle particle is set free, the
tensions in the two parts of the thread are altered in the ratios 2a + b : 3a
and 2b + a : 36, there being no external forces.
8. Two equal particles are connected by an inextensible thread of length
I ; one of them A is on a smooth table and the other is just over the edge,
the thread being straight and at right angles to the edge. Find the velocities
of the particles immediately after they have become free of the table, and
prove (i) that in the subsequent motion the tension of the thread is always
half the weight of either particle, and (ii) that the initial radius of curvature
of the path of A immediately after it leaves the table is j^\/5 • I-
198 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
MISCELLANEOUS EXAMPLES
1. Two equal balls lie in contact on a table. A third equal ball impinges
on them, its centre moving along a line nearly coinciding with a horizontal
common tangent. Assuming that the periods of the impacts do not overlap,
prove that the ratio of the velocities which either ball will receive according
as it is struck first or second is 4 : 3 — e, where e is the coefficient of resti-
tution.
2. Two unequal particles are attached to a thread which passes over a
smooth pulley. Initially the smaller is in contact with a fixed horizontal
plane, and the other at a height h above the plane. Prove that, if the co-
efficient of restitution for each impact is e, and if e is a root of any equation
of the form en — 2e + l=0 with n integral, the system will come to rest after
a time 2h(l + e)/v(l — e), where v is the velocity of the particle of greater
mass immediately before its first impact on the plane.
3. Two balls of masses J/, m and of equal radii, connected by an
inextensible thread, lie on a smooth table with the thread straight, and a
ball of the same radiu.s, and of mass m', moving parallel to the thread with
velocity v, strikes the ball m so that the line of centres (m', m) makes an
acute angle a with the line of centres (J/, m). Prove that, if e is the co-
efficient of restitution between m and ?«', M starts with velocity
vmmf (l+e) cos2 al{Mm' sin2 a + m ( M + m + m')} .
4. Two balls are attached by inextensible threads to fixed points, and
one of them, of mass m, describing a circle with velocity M, impinges on the
other, of mass m\ at rest, so that the line of centres makes an angle a with
the thread attached to m, and the threads cross each other at right angles.
Prove that m' will start to describe a circle with velocity
mu sin a cos a ( 1 + e)/(m cos2 a + m! sin2 a),
where e is the coefficient of restitution between the balls.
5. A shell of mass M is moving with velocity V. An internal explosion
generates an amount E of energy, and thereby breaks the shell into two
fragments whose masses are in the ratio m\ : m%. The fragments continue
to move in the original line of motion of the shell. Prove that their
velocities are
6. Three particles A, B, C of equal mass are placed on a smooth plane
inclined at an angle a to the horizontal, and B, C are connected with A by
threads of length h sec a which make equal angles a with the line of greatest
slope through A on opposite sides of it, the line BC being below the level of
4. If A is struck by a blow along the line of greatest slope, so as to start to
move down this line with velocity F, find when the threads become tight, and
prove that the velocity of A immediately afterwards is
T"/(3 - 2 sin2 a) + fyh sin a/ F.
MISCELLANEOUS EXAMPLES 199
7. Three particles of equal mass are attached at equal intervals to a
rigid rod of negligible mass, and, the system being at rest, one of the extreme
particles is struck by a blow at right angles to the rod. Prove that the
kinetic energy imparted to the system, when the other extreme particle is
fixed, and the rod turns about it, is less than it would be if the system were
free in the ratio 24 : 25.
8. Two equal rigid rods AB, EC of negligible masses carry four equal
particles, attached at A, C and at the middle points of the rods. The rods
being freely hinged at B, and laid out straight, the end A is struck with an
impulse at right angles to the rods. Prove that the magnitudes of the
velocities of the particles are in the ratios 9:2:2:1.
9. Four particles of equal masses are tied at equal intervals to a thread,
and the system is placed on a smooth table so as to form part of a regular
polygon whose angles are each TT — a. Prove that, if an impulse is applied
to one of the end particles in the direction of the thread attached to it, the
kinetic energy generated is greater than it would be if the particles were
constrained to move in a circular groove, and the impulse were applied
tangentially, in the ratio cos2 o + 4 sin2 a : cos2 a + 2 sin2 a.
10. Four small smooth rings of equal mass are attached at equal
intervals to a thread and rest on a circular wire in a vertical plane. The
radius of the wire is one-third of the length of the thread, and the rings are
at the four upper corners of a regular hexagon inscribed in the circle, the two
lower rings being at the ends of the horizontal diameter. Prove that, if the
thread is cut between one of the extreme particles and one of the middle
ones, the tension in the horizontal part is suddenly diminished in the
ratio 5 : 9.
11. Particles of masses m and m' are fastened to the ends of a thread,
which rests in a vertical plane on the surface of a smooth horizontal circular
cylinder of mass M. The cylinder can slide on a horizontal plane. The
system is initially held at rest so that the radii of the circular section, which
pass through the particles, make angles a and /3 with the vertical. Prove
that, when the system is released, the tension of the thread immediately
becomes
, M (sin a + sin /3) + (m sin a + m' sin /3) {1 — cos (a + /3)}
(m + m!) (M+ m sin2 a + m' sin2 /3) + mm' (cos a - cos £)2 '
12. A particle P, of mass M, rests in equilibrium on a smooth horizontal
table, being attached to three particles of masses m, m', m" by cords which
pass over smooth pulleys at points J, B, (7 at the edge of the table. Prove
that, if the cord supporting m" is cut, M will begin to move in a direction
making with CP an angle
_ , /i (m ~ m'} {(m + m')2 - m"2}
4Mmm'm"2 + (m + m') fj?
where ^ = 2m'2m"2 + 2wi"2m2 + 2»i2»i'2 - ?n4 - m'4 - »i"4.
200 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
13. A circular wire of mass Mis held at rest in a vertical plane, so as to
touch at its lowest point a smooth table ; and a particle of mass m rests
against it, being supported by an inextensible thread, which passes over the
wire, and is secured to a fixed point in the plane of the wire at the same level
as the highest point of the wire. Prove that, if the wire is set free, the
pressure of the particle upon it is immediately diminished by an amount
w2<7sin2a/(J/+4wisin'2|a), where a is the angular distance of the particle
from the highest point of the wire.
14. Four particles A, B, C, D of equal mass, connected by equal threads,
aro placed on a smooth plane of inclination a(< JTT) to the horizontal, so
that AC is a line of greatest slope, and AB, AD make angles a with AC on
opposite sides of it. If the uppermost particle A is held, and the particles
B and D are released, prove that the tension in each of the lower threads is
instantly diminished in the ratio
(l-2sin2a)/(l+2sin2a).
15. A particle of mass m on a smooth table is joined to a particle of
mass m' hanging just over the edge by a thread of length a at right angles to
the edge. Prove that, if the system starts from rest, the radius of curvature
of the path of m immediately after it leaves the table is
2m'a
16. Two particles A, B are connected by a fine string ; A rests on a rough
horizontal table (coefficient of friction = n] and B hangs vertically at a dis-
tance I below the edge of the table. If A is on the point of motion, and B is
projected horizontally with velocity «, show that A will begin to move with
an acceleration /i«2/ {(/* + !) I}, and that the initial radius of curvature of B's
path will be
17. A particle of mass m is attached to one end of a thread which passes
through a bead of mass M and the other end is secured to a point on a
smooth horizontal table on Avhich the whole rests. Initially the two portions
of the thread are straight and contain an obtuse angle a, the portion between
m and M being of length a, and m is projected at right angles to this portion.
Prove that the initial radius of curvature of the path of /// is
a (1+ 4m M'1 cos2 $«).
18. A window is supported by two cords passing over pulleys in the
framework of the window (which it loosely fits), and is connected with
counterpoises each equal to half the weight of the window. One cord breaks
and the window descends with acceleration /. Show that the coefficient of
friction between the window and the framework is
where a is the height and b the breadth of the window.
MISCELLANEOUS EXAMPLES 201
19. A bucket of mass M is raised from the bottom of a shaft of depth
h by means of a cord which is wound on a wheel of mass in. The wheel
is driven by a constant force, which is applied tangentially to its rim for a
certain time and then ceases. Prove that, if the bucket just comes to rest at
the top of the shaft t seconds after the beginning of the motion, the greatest
rate of working is
2hM2g'2t/{Myt'2 — 2h (M+m)},
the mass of the wheel being regarded as condensed uniformly on its rim.
20. An engine is pulling a train, and works at a constant power doing
H units of work per second. If Mis the mass of the whole train and Ffhe
resistance (supposed constant), prove that the time of generating velocity v
from rest is
'MH, H Mv^
21. Two pulleys each of mass 8m hang at the ends of a chain of negligible
mass which passes over a fixed pulley ; a similar chain passes over each
of the two suspended pulleys and carries at its ends bodies of mass 2m. A
mass m is now removed from one of the bodies and attached to one of those
which hang over the other pulley ; prove that the acceleration of each pulley
is ^y. Prove also that the two descending bodies move with the same
velocity, and that the velocity of one of the ascending bodies is five times
that of the other.
22. A chain of negligible mass passes over a fixed pulley B, and supports
a body of mass m at one end and a pulley C of mass p at the other. A similar
chain is fastened to a point A below B, passes over (7, and supports a body of
mass /»'. Prove that the acceleration of the pulley is
g (2m' -
23. Two equal particles of mass P sin a are attached, at a distance 2a sin a
apart, to a thread, to the ends of which particles of mass P are attached.
The thread is hung over two pegs distant 2a apart in a horizontal line. Prove
that the period of the small oscillations about the position of equilibrium is
the same as that for a simple pendulum of length a tan a.
24. A particle of mass M is placed at the centre of a smooth circular
horizontal table of radius a ; cords are attached to the particle and pass over
n smooth pulleys placed symmetrically round the circumference, and each
cord supports a mass M. Show that the time of a small oscillation of the
system is 2?r {a
25. A triangle ABC is formed of equal smooth rods each of length 2a,
and small equal rings rest on the rods at the middle points of AB, AC, being
attached to A by equal elastic threads of natural length I, and connected
together by an inextensible thread passing through a fixed smooth ring at
202 MISCELLANEOUS METHODS AND APPLICATIONS [CH. VII
the middle point of BC. Prove that, if there are no external forces, and if
one of the rings is slightly displaced, the period of the small oscillations is
where m is the mass of each ring and E is the modulus of elasticity.
26. A circular hoop of negligible mass and of radius b carries a particle
rigidly attached to it at a point distant c from its centre, and its inner
surface is constrained to roll on the outer surface of a fixed circle of radius
a, (b > a), under the action of a repulsive force, directed from the centre of
the fixed circle and equal to p times the distance. Prove that the period of
small oscillations of the hoop will be
b + c fb-a
V CM
2»r
27. An equilateral wedge of mass M is placed on a smooth table, with
one of its lower edges in contact with a smooth vertical wall, and a smooth
ball of mass M' is placed in contact with the wall and with one face of the
wedge, so that motion ensues without rotation of the wedge. Prove that the
ball will descend with acceleration
28. Two particles A, B of masses 2m and m are attached to an inextensible
thread OAB, so that OA = A B, and lie on a smooth table with the thread
straight and the end 0 fixed. The particle B is projected on the table at
right angles to AB. Prove that, in the subsequent motion, when GAB is
again a straight line, the velocity of B is half that of A.
29. A gun is suspended freely at an inclination o to the horizontal by
two equal parallel vertical cords in a vertical plane containing the axis of the
gun, and a shot whose mass is l/n of that of the gun is fired from it. Prove
that the range on a horizontal plane through the muzzle is 4» (1 + n) h tan o,
where h is the height through which the gun rises in the recoil.
30. A railway carriage of mass J/, moving with velocity r, impinges on a
carriage of mass M' at rest. The force necessary to compress a buffer
through the full extent I is equal to the weight of a mass m. Assuming that
the compression is proportional to the force, prove that the buffers will not be
completely compressed if
Prove also that, if v exceeds this limit, and if the backing against which
the buffers are driven is inelastic, the ratio of the final velocities of the
carriages is
31. Two particles of masses m and m', joined by an elastic thread of
natural length I and modulus X, are placed on a smooth table with m at the
edge and m' at a distance I in a line perpendicular to the edge. The particle
MISCELLANEOUS EXAMPLES 203
m is then just pushed over the edge. Prove that, if the length of the thread
at any time is I + s, then
s2 = 2gs - Xs2 (m + m')/mm'l.
Also, if at time t, m has fallen through z and m' is at a distance x from
the edge, prove that
32. Two particles each of mass m are connected by a rod of negligible
mass and of length I, and lie on a rough horizontal plane (coefficient of
friction /*). One of the particles is projected vertically upwards with velocity
F, prove that the other particle will begin to move when the rod makes with
the plane an angle a, where a is the least angle which satisfies the equation
( F2 — 3gl sin a) (cos a+p. sin a)=p.gl,
provided that V2/gl is less than 3 sin a + cosec a. Find also the radius of
curvature of the path immediately afterwards.
33. Two particles, each of mass m, are connected by an inextensible
thread of length I, passing over a smooth pulley at the top of a smooth plane
of inclination a, on which one of the particles rests at a distance a from the
top (a < I). Prove that, in the motion which ensues after the system is free of
the plane, the tension of the thread is constant and equal to
\mgal~1 cos2 a (1 - sin a),
and that the radius of curvature of the path of the upper particle immediately
after it leaves the plane is
1 - sin a [cos2 a + J ( 1 — sin a)2]5
a
cos a l+%al~l cos2 a (1 -sin a)'
34. A spherical shell contains a particle of mass equal to \jk times that
of the shell, supported by springs of equal length and strength, which are
attached at opposite ends of a diameter ; and the system, all parts of which
are moving in the line of the springs with the same velocity, strikes directly
a fixed plane. Show that, if the coefficient of restitution between the shell
and the plane is unity, the shell will or will not strike the plane again
according as k < or > 1 + 2 cos a, where a is the least positive root of the
equation tan a=a + ir.
35. In a smooth table are two small holes A, B at a distance 2« apart ;
a particle of mass M rests on the table at the middle point of AB, being
connected with a particle of mass m hanging beneath the table by two
inextensible threads, each of length a(l+seca), passing through the holes.
A blow J is applied to M at right angles to AB. Prove that, if
J2 > 2Mmag tan a,
M will oscillate to and fro through a distance 2a tan a, but if
,/2 = ZMmag (tan a — tan /3),
where tan /3 is positive, the distance through which M oscillates will be
— sec /3) (sec a - sec /3 + 2)} .
CHAPTER VHIf
MOTION OF A RIGID BODY IN TWO DIMENSIONS
215. IN this Chapter we propose to discuss the motion of
a rigid body in cases where every particle of the body moves
parallel to a fixed plane, for example the plane (x, y} of a frame
of reference. In such a case the x and y of a particle of the body
vary with the time, but the z of each particle remains constant
throughout the motion. The motion is said to be " in two dimen-
sions," or "in one plane." Now we saw in Art. 180 that to
determine the position of a rigid body it is requisite and sufficient
to determine the positions of a particle of the body, of a line of
particles passing through that particle, and of a plane of particles
passing through that line. In the case now under discussion we
may take the line and plane in question to be parallel to the plane
(x, y). Then the position of the plane is invariable ; and the
position of the line is determined by the angle which it makes
with a fixed line in the plane, for instance the axis of x ; further,
the position of the chosen particle is determined by its coordinates
x and y. Thus the determination of the position of the rigid body
(moving in two dimensions) requires the determination of three
numbers, representing the coordinates of the position of one of the
particles, and the angle which a line of the body drawn through
that particle, and moving in the plane of its motion, makes with a
fixed line.
We can now see what is meant by the angular velocity of a
rigid body moving in one plane. Let one line of particles, fixed in
the body, and parallel to the plane, make an angle 6 at time t with
a line fixed in the plane. Then this angle is increasing at a rate
6. Let any other line of particles be drawn also parallel to the
plane, and let a. be the angle which it makes with the first line.
Then a is invariable, for if it were to change the body would be
deformed. Now the second line of particles makes an angle 6 + a
t Articles in this Chapter which are marked with an asterisk (*) may be omitted
in a first reading.
•215, 216] MOMENTS OF INERTIA 205
with the fixed line, and this angle also increases at a rate 6. We
thus see that every line of particles parallel to the plane turns with
the same angular velocity, and this is the angular velocity of the
rigid body.
216. Moment of Inertia. Consider a rigid body turning
about an axis with angular velocity w. Let m be the mass of
a particle of the body at a distance r from the axis. Then this
particle is describing a circle of radius r with velocity ra). Hence
its moment of momentum about the axis is mr2aj, and its kinetic
energy is -|?7ir2&>2.
It follows that the moment of momentum of the rigid body
about the axis is
and the kinetic energy is
the summations referring to all the particles.
These expressions become
p (x2 + t/2) dxdydz,
and £o>2 HI p (#2 + y2) dx dy dz,
for a body of density p at a point (x, y, z), the axis of rotation
being the axis of z.
The integrals are volume integrals taken through the volume
of the body ; that is to say we must divide the volume of the
body into a very large number of volumes, very small in all their
dimensions, multiply the value of p (x2 + t/!) at a point in one of
these volumes by this volume, sum the products for all the volumes,
and pass to a limit by diminishing the volumes indefinitely. The
process will be exemplified in Art. 218.
The multiplier of &> and ^o>2 in these expressions is called the
moment of inertia of the body about the axis. We shall see
presently that it enters into the expressions for the kinetic energy
and moment of momentum of a rotating body, whether the axis of
rotation is fixed or not.
The moment of inertia of a body about an axis depends only
on the shape of the body, its situation with reference to the axis,
and the distribution of density within it.
206 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
217. Theorems concerning Moments of Inertia. I. The
moment of inertia of a system about any axis is equal to the moment of
inertia about a parallel axis through the centre of mass together with the
moment of inertia about the original axis of the whole mass placed at the
centre of mass.
Let x, y, z be the coordinates of any particle of the system, m its mass,
x, y, * the coordinates of the centre of mass, af, /, z' those of the particle m
relative to the centre of mass.
Then x=x+x', y=y+y', z=z+z',\
Now
z+z',\
j' = O.J
So 2w/ = y2<Sm + 2m/2.
Hence 2»i (x2 +y2) = 2m (a/2 +/2) + (x2 +y2) 2?n,
which is the theorem stated.
II. The moment of inertia of a plane lamina, of any form, about any axis
perpendicular to its plane, is the sum of those about any two rectangular axes
in the plane which meet in any point on the first axis.
For, if the axes are taken to be those of z, x, y, the moments of inertia
about the axes of x and y are respectively 2my2 and 2m,?2, and the moment of
inertia about the axis of z is 2m (x2+y2).
III. To compare the moments of inertia of a lamina about different axes
in its plane.
For parallel axes we cau use Theorem I and it will therefore be sufficient
to consider axes in different directions through the origin. Let 6 be the angle
which any line makes with the axis x. The distance of any point (x, y) from
this line is -xsind+y cos 0, and thus the moment of inertia about the line
is 2m (y cos 6 — x sin 0)2 = sin2 02 (ma2) + cos2 02 (my2) — 2 sin 6 cos 02m.ry.
The expression for the moment of inertia about a perpendicular line
would be
cos2 02 (mx2) + sin2 02 (my2) + 2 sin 0 cos 02 (mxy).
The quantity 2 (mxy) is known as the product of inertia with respect to
the axes of x and y (in two dimensions). For new axes obtained by turning
through an angle 0 it has the value
(cos2 0 - sin2 0) 2 (mxy) -J-sin 0 cos 0 (2 (my2) - 2 (mx2)}.
We can always choose the axes of (x, y) so that this quantity 2 (mxy)
vanishes. When this is done the axes of x and y are called Principal axes
of the lamina. The directions of the principal axes vary with the point chosen
as origin.
Now let the axes of x and y be principal axes of the lamina at the origin.
Let A, =2 (my2), be the moment of inertia about the axis x, and B, =2 (m#2),
217, 218] MOMENTS OF INERTIA 207
be the moment of inertia about the axis y. Then the moment of inertia
about a line through the origin making an angle 6 with the axis x is
A co
If an ellipse whose equation is Ax2 + J3y2 = const, is drawn on the lamina,
then the moment of inertia about any diameter of it is inversely proportional
to the square of the length of that diameter. This ellipse is known as the
ellipse of inertia.
IV. If two plane systems in the same plane have the same mass, the
same centre of mass, the same principal axes at the centre of mass, and
the same moments of inertia about these principal axes, they have the same
moment of inertia about any axis in or perpendicular to the plane.
For, in the first place, the two systems have by Theorem III the same
moment of inertia about any axis lying in the plane and passing through the
common centre of mass, by Theorem I they have the same moment of
inertia about any other axis in the plane, and by Theorem II they have the
same moment of inertia about any axis perpendicular to the plane.
Such systems are described as momental equivalents.
It is clear that two plane systems are momental equivalents if they have
the same mass, and the same centre of mass, and if their moments of inertia
about any three assigned axes in the plane are equal.
218. Calculations of moments of inertia.
I. Uniform ring. Radius of gyration of a body. For a circular ring of
mass m and radius a, and of very small section, the moment of inertia about
the axis is ma2, since every element of the mass can be taken to be at the
same distance a from the axis.
In the case of a body of any shape, and of mass m, we can always express
the moment of inertia about any axis in the form mk2, where k represents the
length of a line ; and thus we see that k is the radius of a ring such that, if
the mass of the body were condensed uniformly upon the ring, the moment
of inertia of the ring about its axis would be the same as the moment of
inertia of the body about the axis in question. The quantity k for any body
and any axis is known as the radius of gyration of that body about that axis.
II. Uniform rod. Let m be the mass of the rod, and 2a its length, r the
distance of any section from the middle point. The mass of the part between
the sections r and r + 8r is ^- or. Therefore, if the thickness of the rod is
disregarded, the moment of inertia about an axis through the middle point at
right angles to the rod is
(* ^L id =ima>
J-«2^'
The radius of gyration of the rod is a/,/3.
III. Circular disk. The mass per unit of area of a uniform thin circular
disk of radius a and mass m is m/na2. The area of the narrow ring contained
208 MOTION OF A RIGID BODY IN TWO DIMENSIONS [cH. VIII
between two concentric circles of radii r and r + 8r is 2n (r + ^8r) 8r. All the
particles in such a ring are at distances from the centre which lie between r
and r+8r. Hence the moment of inertia of the disk about an axis drawn
through its centre at right angles to its plane is
f
i o
— 2 ra . 2irrdr,
o ira
which is fyna?. The radius of gyration of the disk about this axis is a/^-2.
IV. Uniform sphere. Let a be the radius of the sphere, p the (constant)
density of the material, and let the origin of coordinates be the centre of the
sphere. According to the general formula of Art. 216 we must integrate
(x2 +y2) through the volume of the sphere. Now it follows from the symmetry
of the sphere that
I I Lv2dxdydz= I I iy^dxdydz— I I \z*dxdydz,
J J J J J J J J J
where the integrations are taken through the volume of the sphere. Hence
each of these integrals is equal to
^ I j ((Xi+y2+z*}dxdydz or Jj J (r^dxdydz,
where the integration is taken through the volume of the sphere, and r
denotes the distance of the point (#, y, z) from the centre.
To evaluate this integral we have first to divide the sphere into a very
large number of very small volumes, next to multiply the value of r1 for a
point within one of the small volumes by this volume, then to sum the
products so formed, and finally to pass to a limit by diminishing all the small
volumes indefinitely.
Now the volume contained between two concentric spheres of radii r and
r + 8r is 4ir {rs+r8r+$ (8r)2}8r, and the distances from the centre of all the
points in this volume lie between r and r + 8r. Hence the required integral
f f [
The moment of inertia of the sphere about any diameter is therefore
„ 47ra5
lp -g- , or ?ma2,
where m, = $irpas, is the mass of the sphere.
219. Examples.
1. Prove that a momental equivalent of a thin rod of mass m consists of
three particles : one of mass %m at the middle point, and one of mass %m at
each of the ends.
2. Prove that the moments of inertia of a uniform rectangular lamina of
mass m and sides 2a, 26 about axes through its centre parallel to its edges are
and
218-220] MOMENTS OF INERTIA 209
3. Prove that the radius of gyration of a circular disk about a diameter
is half the radius. Hence evaluate the integral I \x*dxdy taken over the
area of a circle of radius a, the origin being at the centre of the circle.
(Of. II of Art. 217 and IV of Art. 218.)
4. To evaluate the integral I I x^dxdy taken over the area within an ellipse
which is given by the equation x2la?+yz/b2 = 1, change the variables by putting
#=«£> y — b*!- We have to find the value of a?b I l^d^dr) where the
integration extends over a range of values given by the inequality £2 + ?72 ^> 1.
This is the same thing as an integration over the area of a circle of unit
radius. Hence prove that the moments of inertia of a uniform thin elliptic
lamina of semi-axes a, b and mass m about its principal axes are %mbz
and jma2.
5. An ellipsoid is given by an equation of the form #2/a2+y2/&2+.s2/c2=l.
To find the value of I I I x^dxdydz taken through the volume of the ellipsoid,
change the variables by putting x = a%, y = brj, z = c£. We get
a?bc
where the integration extends over a range of values given by the inequality
£2 + »72 + f2;t> 1. This is the same thing as an integration through the volume
4
of a sphere of unit radius. According to IV of Art. 218 the result is — TT.
15
Hence prove that the moments of inertia of the ellipsoid (supposed to be of
uniform density p) about the axes of #, y, z are
where m, —^irpabc, is the mass of the ellipsoid.
6. Prove that a momental equivalent of a uniform triangular lamina
consists of three particles, each one-third of its mass, placed at the middle
points of its sides.
7. Prove that the moment of inertia of a uniform cube of mass m and
side 2a about an axis through its centre parallel to an edge or at right angles
to an edge is §«ia2.
[It can be shown that the same formula holds for any axis drawn through
the centre of the cube.]
220. Velocity and Momentum of rigid body.
Let G be the centre of mass of a rigid body moving in two
dimensions, and let u and v be resolved parts of the velocity of G
parallel to the axes x and y. Let P be any other particle of the
body, r its distance from G, and x' , y' its coordinates relative to G
L. M. 14
210 MOTION OF A KIGID BODY IN TWO DIMENSIONS [CH. VIII
at time t. Then the line GP is turning with the angular velocity
to of the rigid body, and the velocity of P relative to G is ro>
at right angles to GP ; the resolved parts of this relative velocity
parallel to the axes are — wy' and wx ', since the line GP makes with
the axis x an angle whose cosine is x'/r and whose sine is y'/r.
Hence the resolved velocities of P parallel to the axes are
u —(ay and v + <ax'.
Fig. 63.
Let ra be the mass of the particle at P. Then the resultant
momentum of the body parallel to the axis x is
2m (u — <ay'\
which is equal to Mu, where M, = Sm, is the mass of the body.
Similarly the momentum of the body parallel to the axis y is Mv.
Thus the resultant momentum of the body is the same as the
momentum of a particle of mass equal to the mass of the body
placed at the centre of mass and moving with it. (Art. 153.)
The moment of momentum of the body about an axis through
the centre of mass perpendicular to the plane of motion is
2ra [x (v + tax) — y (u - (ay')\,
220, 22l] VELOCITY OF RIGID BODY 211
which is equal to &>2w (x'z + y'*) or to Mk2w, where k is the radius
of gyration about the axis.
The moment of momentum about any parallel axis is the
moment about that axis of the momentum of the whole mass
placed at the centre of mass and moving with it together with
the moment Mk*a> (Art. 156). Thus the momentum of the rigid
body is specified by the resultant and couple of a system of vectors
localized in lines. The resultant is localized in a line through G,
and has resolved parts Mu, Mv in the two chosen directions ; and
the moment of the couple is Mk"*a).
Again, the kinetic energy of the body is
£2w {(u - cay'Y + (v + owe')8}
u? + v2 + &2o>2,
which is the kinetic energy of the whole mass, moving with the
centre of mass, together with the kinetic energy of the rotation
about the centre of mass (Art. 158).
The formulae for the velocity of a point show that at each
instant the point whose coordinates relative to G are — v/m and
u/w has zero velocity, so that the motion of the body at the
instant is a motion of rotation about an axis through this point
perpendicular to the plane of motion. The point is called the
instantaneous centre of no velocity, or frequently "the instantaneous
centre." The fact that the motion of a rigid plane figure in its
plane is equivalent to rotation about a point is of importance in
many geometrical investigations.
221. Kinetic Reaction of rigid body. With the notation of
the last Article, the point P moves relatively to G in a circle of
radius r with angular velocity equal to &> at time t\ its acceleration
relative to G may therefore be resolved into rw at right angles to
GP, and r&>2 along PG. Hence the resolved parts of the acceleration
of P parallel to the axes are
u — a>y' — &>V and v + wx' — ury .
The kinetic reactions may be reduced to a resultant kinetic
reaction localized in a line through the centre of mass and a
couple. The resultant in question has resolved parts parallel to
14—2
212 MOTION OF A. RIGID BODY TN TWO DIMENSIONS [cH. VIII
the axes which are
2m (u — coy' — wV) and 2w (v + cox' — co2y'),
and these are Mu and Mv.
The couple is the moment of the kinetic reactions about a line
through the centre of mass perpendicular to the plane of motion ;
this moment is
2m \x (i) + tax — co2y) — y'(u — coy' — o>2#')},
and this is Mk'2co.
Fig. 64.
The moment of the kinetic reactions about any axis perpen-
dicular to the plane of motion is the moment about that axis of
the kinetic reaction of a particle of mass equal to the mass of the
body, moving with the centre of mass, together with the moment
of the couple Mk^w. (Art. 157.)
The formulae for the acceleration of any point of the body show
that at each instant there is a point which has zero acceleration.
This point is called the instantaneous centre of no acceleration. It
is of much less importance than the instantaneous centre of no
velocity.
221-223] KINETIC REACTION OF BIGID BODY 213
222. Examples.
1. Prove that, at any instant, the normal to the path of every particle ^
passes through the instantaneous centre (of no velocity).
[It follows that this centre can be constructed if we know the directions
of motion of two particles.]
2. Calculation of the moment of the kinetic reactions about the instan-
taneous centre (of no velocity).
The coordinates of the instantaneous centre /being - v/o> and u/m referred
to axes through the centre of mass G parallel to the axes of reference, the
moment in question is
The velocity of G is rco at right angles to the line joining it to /, where
r=IG, or we have %2 + #2=r2co2.
Hence the above is — 7- (imr2co2) + mFo>,
« at
If we take an angle 6 such that 6=a>, and write K for the moment of
inertia about the instantaneous centre /, then K=m (&2+r2) by I of Art. 217,
and the result obtained may be written -^
When the point / is fixed in the body this can be replaced by K&. Other
cases in which this formula can be used are noted in Arts. 235 and 236 infra.
3. Prove that those particles which at any instant are at inflexions on
their paths lie on a circle.
[This circle is called the " circle of inflexions."]
4. Prove that the curvature of the path of any particle which is not on
the circle of inflexions is a>3p2/ V3, where p2 is the power with respect to the
circle of the position of the particle, co is the angular velocity of the body,
and V is the resultant velocity of the particle.
5. Prove that, in general, that particle which is at the instantaneous
centre (of no velocity) is at a cusp on its path.
223. Equations of motion of rigid body. The equations of
motion express the conditions that the kinetic reactions and the
external forces may be equivalent systems of vectors.
Let M be the mass of the body,/,^ the resolved accelerations
214 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
of the centre of mass in any two directions at right angles to each
other in the plane of motion, &> the angular velocity of the body.
Let the forces acting on the body be reduced to a resultant
force at its centre of mass and a couple. Let P, Q be the resolved
parts of the force in the directions in which the acceleration of the
centre of mass was resolved, and let N be the couple.
Then the system of vectors expressed by Mfl} Mfz, M&a> has
the same resolved part in any direction, and the same moment
about any axis, as the system P, Q, N.
In particular we have
Mf, = P, Mf2 = Q, Mteio = N,
and the equations of motion of the body can always be written in
this form.
In the formation of equations of motion diversity can arise
from the choice of directions in which to resolve, and of axes
about which to take moments. As in the case of Dynamics of a
Particle, the equations arrived at are differential equations, and no
rules can be given for solving them in general. If however the
circumstances are such that there is an equation of energy, or an
equation of conservation of momentum, such equations are first
integrals of the equations of motion.
224. Continuance of motion in two dimensions. The question
arises whether a body, which at some instant is moving in two dimensions
parallel to a certain plane, continues to move parallel to that plane or will
presently be found to be moving in a different manner. A general answer to
this question cannot be given here, but it is clear that there is a class of cases
in which the motion in two dimensions persists. This class includes all the
cases in which the body is symmetrical with respect to a plane and the forces
applied to it are directed along lines lying in that plane, or, more generally,
when the forces can be reduced to a single resultant in the plane of symmetry
and a couple about an axis perpendicular to that plane.
225. Rigid Pendulumf . A heavy body free to rotate about
a fixed horizontal axis is known as a "compound pendulum" to
t Ch. Huygens was the first to solve the problem of the motion of the pendulum,
and the principles which he invoked were among the considerations which ulti-
mately led to the establishment of the Theory of Energy. His work, De horologio
oscillatorio, was first published in 1673.
223-226] RIGID PENDULUM 215
\
distinguish it from the "simple pendulum" whose motion was
discussed in Arts. 95 and 119.
Let G be the centre of mass of the body,
GS the perpendicular from G to the axis, 6
the angle which GS makes with the vertical
at time t. Then the whole motion takes
place in the vertical plane which passes
through G and is at right angles to the axis;
and the position of the pendulum at any
time depends only on the angle 6.
Let GS — h. Let M be the mass of the body, k its radius of
gyration about an axis through G perpendicular to the plane of
motion.
The velocity of the centre of mass is hd, and the kinetic
energy is
|Jf (#+#)&
The potential energy of the body in the field of the earth's
gravity is
Mgh (1 - cos 6),
the standard position being the equilibrium posicion.
Hence the energy equation can be written /
%M (h2 + fc8) 6* = Mgh cos 6 + const."*
Comparing this equation with that obtained in Art. 119, we
see that the motion is the same as that of a simple pendulum of
length (k2 + h?)/li.
A point in the line SG at this distance from S is known as the
"centre of oscillation," S is called the "centre of suspension." The
distance between these centres is the "length of the equivalent
simple pendulum."
226. Examples.
1. A rigid pendulum, for which S and 0 are respectively a centre of
suspension and the corresponding centre of oscillation, is hung up so that it
can oscillate in the same vertical plane as before, but with 0 as centre of
suspension instead of S ; prove that S will be the centre of oscillation.
2. A uniform rod moves with its ends on a smooth circular wire fixed in
a vertical plane. Prove that, if it subtends an angle of 120° at the centre,
the length of the equivalent simple pendulum is equal to the radius of the
circle.
216 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
3. A compound pendulum consists of a rod, which can turn about a
fixed horizontal axis, and a spherical bob, which can slide on the rod. Prove
that the period of oscillation will be prolonged by sliding the bob up or down,
according as the length of the equivalent simple pendulum is > or < twice
the distance of the centre of gravity of the bob from the axis of rotation.
4. Two rigid pendulums of masses m and m' turn about the same hori-
zontal axis. The distances of the centres of mass and of oscillation from
the axis are h, h' and I, I' respectively. Prove that, if the pendulums are
fastened together in the position of equilibrium, the length of the equivalent
simple pendulum for the compound body will be (mhl + m'h'l')/(mh + m'h').
227. Illustrative Problems. We exemplify the application of the
principles that have been laid down by partially working out some problems.
The most important matters to be illustrated are actions between two rigid
bodies whether smooth or rough, and the expression of the effects of the
inertia of a rigid body by means of the moment of inertia. Other matters
•of subsidiary interest are the kinernatical expression of velocities and
accelerations in terms of a small number of independent geometrical quanti-
ties, the expression of kinematical conditions, and the calculation of resultant
stresses.
I. Inertia of machines. We shall consider At wood's machine. To avoid
having to take account of the motion of the pulley in our preliminary notice
of Atwood's machine (Art. 73) we assumed the pulley to be perfectly smooth
or that the rope slides over it without frictional
resistance and without setting it in motion. It will
now be most convenient, in order to get some idea
of the way in which the motion of the pulley affects
the result, to suppose the pulley to be so rough that
the particles of the rope and the pulley in contact
move with the same velocity along the tangents to
the pulley.
Now let M be the mass of the pulley, a its radius,
k its radius of gyration about its axis, 6 the angle
through which it has turned up to time t.
Let m and m' be the masses of the bodies at-
tached to the rope, and x the distance through
which m has fallen at time t. Then x=aQ.
D
Fig. 32 (bis)
The mass of the rope being neglected, the kinetic energy is
and the work done is (m — m'} gx,
so that the energy equation is
&
\M -g #2 -f \ (m + m') .r2 = (m — m') gx + const.
226, 227] ILLUSTRATIVE PROBLEMS
Thus the acceleration with which m descends is
217
It appears that the effect of the inertia of the pulley is equivalent to an
increase of each of the masses in the simple problem (where the pulley is
regarded as smooth and its mass is neglected) by |J/F/a2.
II. Wheel set in motion by couple. Let a wheel, the plane of which is
vertical, be in contact with rough horizontal ground; and let the wheel be
set in motion by a couple about its axis.
Let « be the radius of the wheel, k the radius of gyration about the axis,
m the mass, 0 the applied couple, F the friction and R the pressure at the
point of contact with the ground.
Let w be the angular velocity with which the wheel turns, v the velocity
with which its centre moves.
Fig. 66.
The left-hand figure is the diagram of the kinetic reactions, and the right-
hand figure is the diagram of the applied forces.
The equations of motion, obtained by resolving horizontally and vertically
and taking moments about the centre, are
mv =F, 0 = R — rug, mkzta = G — Fa.
We have drawn the figure, and written down the equations, on the
supposition that v does not exceed aa>. When v < a<a, the point of contact
slips on the plane in the sense opposite to that of v, and then the friction
acts in the sense shown.
If v = a<a, so that the wheel rolls, we may eliminate F from two of our
o(i nations, and obtain the equation
TO (k" + a2) cb = G.
The sense of <a is the same as that of G ; and therefore, if the motion
starts from rest, the sense of a> is the same as that of G. In the same case
F=Gaj(k2 + a*}, which is positive, so that the friction acts in the sense in
which the centre of the wheel moves (the sense shown in Fig. 66).
In order that this motion may take place it is necessary that F/R or
<•« [f£2 + a2) mg] should not exceed the coefficient of friction,
218 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
We conclude that, if the ground is sufficiently rough, the wheel will begin
to roll along the road, and that the friction at the point of contact is the
horizontal force which produces the horizontal momentum.
III. Wheel st't iii motion by force. Again let the wheel of No. II be set in
motion by a horizontal force P applied at its centre in its plane. With the
same notation as before, we have the equations of motion
mv=P+F, 0 = R- mg, mk?u> = - Fa.
If the wheel rolls, so that «? = «<«>, we have, on eliminating F,
Hence &> is positive, and F is negative, and equal to — Pk'* / (£2 + a2). The
friction in this case acts in the sense opposite to that in which P acts, or
the centre of the wheel moves (i.e. in the sense opposite to that shown in
Fig. 66). The motion will be one of rolling if Pk- {>»// (F + a2)} is less than
the coefficient of friction.
The problems of Nos. II and III illustrate the forces that affect the
motion of a railway train. The machinery is so contrived that a couple is
exerted on the driving wheel of the locomotive. If this couple is too great, or
the friction is too small, the wheel slips or "skids" on the rail ; but, if the
friction is great enough, the wheel starts to roll. The direction of the friction
at the point of contact is that of the motion of the train as in No. II.
The motion of a wheel of any coach or truck attached to the train is of
the character considered in No. III. The tension in the coupling is a hori/ontal
force setting the vehicle in motion, and the frictions at the points of contact
of the wheels with the rails act as resistances.
It appears that the "pull of the engine" (Art. 71) is really the friction of
the rails on the driving wheel. This is the "force" which sets the train in
motion, and keeps it in motion against the resistances. The condition for the
production of the motion is the existence of a source of internal energy,
which can be transformed into work done by the couple acting on the driving
wheel. The way in which a source of internal energy may result in the
production of motion, through the agency of external forces, has already been
illustrated in simple cases in Ex. 1 of Art. 207 and Ex. 6 of Art. 214. All
the characteristic motions of machines and of living creatures are examples of
the same principles, but the working out of the details is in general a matter
of difficulty. The external forces, such as the friction in this problem, are
necessary to the successful action of the animal or machine. (Cf. R. S. Ball,
Experimental Mechanics, 2nd Edition, London, 1888, pp. 83, 84.)
IV. Rolling and sliding. We take the problem presented by a uniform
cylinder of mass J/"and radius a which is set rolling and sliding on a rough
horizontal plane, the angular velocity being initially such that the points on
the lowest generator have the greatest velocity.
227]
ILLUSTRATIVE PROBLEMS
219
Let F be the velocity of the axis, and <B the angular velocity at time t, the
.senses being those shown in Fig. 67.
The system of kinetic reactions reduces to MV
horizontally through the centre of mass, in the sense
of V, and a couple Mk*<a in the sense of a>, where k
is the radius of gyration about the axis of the
cylinder.
Taking moments about the point of contact we
have
Now let F be the friction between the cylinder
and the plane. The particles on the lowest generator have velocity F+a« in
the sense of lr, and therefore F has the opposite sense.
Resolving horizontally we have
MV= -F,
where F is positive. Hence F is negative and &> is also negative.
The velocity F diminishes and the angular velocity a> also diminishes ac-
cording to the equation
where F0 and o>0 arc the values of F and o> in the beginning of the motion.
We shall proceed with the case where F0<o>0£2/a. Then there must come an
instant at which F vanishes, and at this instant <a has the value w0-aF0/F.
At this instant the lowest point has velocity ata0— F0«2/F in the same sense
as before, the friction is still finite and in the same sense as before, and a
velocity of the centre in the opposite sense begins to be generated.
At any later stage of the motion let U be the velocity in the sense opposite
to F0. See Fig. 68. Then so long as «o> > U the friction F acts in the same
sense, and we have
MU= F, \
whence U increases and «o diminishes according to
the equation
a U+ F» = £2a>o — aV0.
When U becomes equal to «o> the value of either is
and at this instant the cylinder is rolling on the
plane. Thereafter the cylinder rolls on the plane
uniformly.
It is to be noticed that, in this problem, so long
as the cylinder slips, the friction is constantly equal
to p.Mg, where /u is the coefficient of friction between the cylinder and the plane.
220 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
228. Examples.
1. In the problem just considered prove that the time from the beginning
of the motion until the motion becomes uniform is -.
a? + K* pcf
2. A homogeneous cylinder of mass M and radius a is free to turn about
its axis which is horizontal, and a particle of mass m is placed upon it close to
the highest generator. Prove that, when the particle begins to slip, the angle
ft which the radius through it makes with the vertical is given by the equation
/i {(M + 6m) cos d - 4m} = M sin 6,
where p is the coefficient of friction between the particle and the cylinder.
3. A uniform thin circular hoop of radius a spinning in a vertical plane
about its centre with angular velocity » is gently placed on a rough plane of
inclination a equal to the angle of friction between the hoop and the plane
so that the sense of rotation is that for which the slipping at the point of
contact is down a line of greatest slope. Prove that the hoop will remain
stationary for a time aoo/f/sina before descending with acceleration ^sina.
4. A locomotive engine of mass M has two pairs of wheels of radius a
such that the moment of inertia of either pair with its axle about its axis of
rotation is A. The engine exerts a couple G on the forward axle. Prove
that, if both pairs of wheels bite at once when the engine starts, the friction
between one of the forward wheels and the line capable of being called into
play must not be less than %G (A + Ma?)ja (2A + Nai}. Prove also that, if the
only action between an axle and its bearings is a frictional couple varying as
the angular velocity of the axle, the final friction called into play between
either forward wheel and the line is O/4a.
5. A uniform sphere rolls down a rough plane of inclination a to the
horizontal. Prove that the acceleration of its centre is f g sin a, and that the
ratio of the friction to the pressure is | tan a.
*229. Kinematic condition of rolling. Consider the following
problem : —
A cylinder of radius b rolls on a cylinder of radius «, which rolls on a
horizontal plane. It is required to determine the motion.
Let m and m' be the masses, A and B the centres, V the horizontal velocity
of m, Q the angular velocity of m, 6 the angle which AB makes with the
vertical, 00 the angular velocity of m', k and kf the radii of gyration of m and
m' about their axes.
The condition that m rolls on the plane is I'=«Q (1).
The velocity of B relative to A is (a + b)0 at right angles to AB, and the
velocity of B is therefore compounded of this velocity and V horixontally.
(Fig. 69.)
The velocity of P (considered as a point of m') relative to B is ba> at right
angles to A B, in the sense of (a + b) 6.
The velocity of P (considered as a point of m) relative to A is aO at right
angles to AB, but in the opposite sense.
228, 229]
ILLUSTRATIVE PROBLEMS
221
The condition of rolling is that the particles of m and m' that are at P
have the same velocity along the common tangent to the two circles.
We therefore have (a + b)6 + b(o — -aQ, (2).
Fig. 69.
In the diagram of accelerations (Fig. 70) we have introduced the value of
Ffrom equation (1).
Since B describes a circle relative to A with angular velocity 6, the accele-
ration of B relative to A is compounded of (« + 6) 6 at right angles to AB, and
(a+b)6* in BA. This gives us the diagram.
Fig. 70.
Now, to form the equations of motion, take moments about P for »i', and
about 0 for the system. We have
and
mk2Q + ma2fl -f m'aQ. [a + (a + 6) cos 6} + ?wT2a>
...(4).
222 MOTION OF A RIGID BODY IN TWO DIMENSIONS [cH. VIII
One of the quantities o> and Q can be eliminated by means of equation (2),
and there then remain two unknown quantities in terms of which the motion
can be completely expressed by solving the equations that are obtained by
substituting from (2) in (3) and (4).
Two first integrals of these equations can be obtained ; one of them is the
energy equation.
*230. Examples.
1. Prove that, in the problem just considered, there is an integral equation
of the form
maQ. ( 1 + £2/ «2) + in' (aQ. - (a + 6) 0 cos 0 - «£'2/6} = const. ,
and that 6 and 6 are connected by an equation of the form
\ (a + b) fr [(1+ VW) - m' (cos & - #2/62)2/{m (1 + £2/a2) + m! ( 1 + Itf 2/62)}] +g cos 6
= const.
2. A uniform rod of length I rests on a fixed horizontal cylinder of radius
« with its middle point at the top ; prove that, if it is displaced in a vertical
plane, so as to remain in contact with the cylinder, and if it rocks without
slipping, the angle 6 which it makes with the horizontal at time t is given
by the equation
$(^l2 + az82)d2+ga (costf + tfsin 0)=const.,
and the length of the equivalent simple pendulum for small oscillations is
3. A thread unwinds from a reel of radius a, the uppermost point of the
thread being held fixed, the unwound part of the thread being vertical, and
the axis of the reel being horizontal. Prove that the acceleration of the
centre of the reel is ga2/(a? + £2), where k is the radius of gyration of the reel
about its axis, and that the tension of the thread is &2/(£2 + a2) of the weight
of the reel.
4. A thread passes over a smooth peg and unwinds itself from two cylin-
drical reels freely suspended from it and having their axes horizontal. Prove
that each reel descends with uniform acceleration.
5. A ball is at rest in a cylindrical garden roller, when the roller is seized
and made to roll uniformly on a level walk ; to find
the motion of the ball, assuming that it does not
slip on the roller.
Let a be the radius of the ball, b of the roller,
6 the angle which the line of centres makes with
the vertical, V the velocity of the roller.
Prove (i) that the angular velocity of the roller
is F/6,
(ii) that the angular velocity o> of the ball is
V [a- (b- a) 6 la.
Fig. 71.
cxw
229, J>30] ILLUSTRATIVE PROBLEMS 223
Let k be the radius of gyration of the ball, supposed uniform, about an
a,xis through its centre, m the mass of the ball. Initially all the impulsive
forces acting on the ball pass through the point of contact, and therefore the
moment of momentum of the ball about any axis through this point is zero
initially. Hence obtain the equation
m&2w0 - ma {(b — a) 00 - I '} = 0
for the initial values co0 of co and #0 of 8. Prove also that o>0 vanishes, and find
the value of 00 •
Obtain the equations of motion
nik^t) — ma (b - a) 6 = mga sin $,
m(b-a)62 = R- mg cos 0,
where R is the pressure of the roller on the ball. Prove that the motion in 6
is the same as that of a simple pendulum of length %(b- a). Prove also that
the value of R in any position is
mg (V- cos 6 - \°-) + m V2/(b - a).
Deduce the condition that the ball may roll quite round the interior of the
roller.
6: A cube containing a spherical cavity slides without friction down a
plane of inclination a, and a homogeneous sphere rolls in the cavity. Prove
that the angle 0, between the normal to the plane and the common normal to
the sphere and the cavity, is connected with the angular velocity « of the
sphere by the equation (a — b)6 = ba>, where a is the radius of the cavity, and
b is the radius of the sphere.
Further, taking M and m for the masses of the cube and sphere, and x for
the distance described by the cube in time t, obtain the equations of motion by
resolving for the system down the plane and at right angles to it and taking
moments for the sphere about its point of contact with the cavity.
Finally obtain the equation
${l(M+m)- m cos2 6} ft - (M+ m) cos a cos 6 g/(a -b) = const.
7. Prove that, when the plane of Ex. 6 is rough, and e is the angle of friction
between it and the cube, the value of 6 at time t is given by the equation
I™ [(I (M + m)cose-m cos 6 cos (6 - 1)} 0-] - £ m6- sin e
.
+ (M+ m) cos a sin (6 - e) g/(a - b) = 0.
8. Motion of a circular disk rolling on a given curve under gravity.
Let c be the radius of
the disc, $ the angle which
the normal at the point of
contact makes with the
vertical, p the radius of
curvature of the curve at
this point. The centre of
the disk describes a curve
224 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
parallel to the given curve and at a distance c from it, and the instantaneous
centre of rotation of the disk is at the point of contact, so that if o> is the
angular velocity of the disk, we have
Velocity of centre = ceo = (p + c) </>.
Hence obtain the equation of energy
where k is the radius of gyration of the disk about its centre of mass, supposed
to coincide with its centre of figure. Investigate the corresponding equation
when the curve is concave to the disk.
Prove that the disk can roll inside a cycloid the radius of whose generating
circle is a and whose vertex is lowest so that the angular velocity 0 is uniform
and equal to
Prove that, when the disk is uniform and rolls outside a cycloid, the radius
of whose generating circle is jc and whose vertex is highest, the motion is
determined by the equation
3cc/>2 cos4 ^</> =g (3 + cos </>) sin2 5$,
and that the disk leaves the cycloid when cos <£ = § .
9. A uniform rod slides in a vertical plane between a smooth vertical wall
and a smooth horizontal plane. To determine the motion.
Let AB be the rod, 2a its length, m its mass, and let the end A move
vertically in contact with the
L.. wall and the end B horizon-
tally in contact with the plane.
The instantaneous centre / is
the intersection of the horizon-
tal through A and the vertical
through Z?, and the figure
OBIA is a rectangle, so that
the centre of mass &', which
is the middle point of A B, is
always at a distance a from 0.
The system of kinetic re-
B actions is therefore equivalent
to a resultant kinetic reaction
at G having components mad and maS2 perpendicular to OG and along GO.
and a couple mkzd in the sense of increase of the angle 6 which the rod BA
makes with the vertical BI.
The forces acting on the rod are its weight at G, the horizontal pressure
at A, and the vertical pressure at B. The lines of action of the two latter
forces meet in /. If then we take moments about / the pressures do not
enter into the equation.
230]
ILLUSTRATIVE PROBLEMS
225
Hence prove that the motion in 6 is the same as that of a simple pendulum
of length j^a.
By resolving horizontally and vertically find the pressures at A and B,
and show that the rod leaves the wall when cos$=-|cosa, a being the initial
value of 6.
10. When the plane and the wall of Ex. 9 are both rough, with the same
angle of friction 6, prove that the value of Q at time t is given by the equation
a (I + cos 2e) 6 - 06* sin 2e =g sin (6 - 2e).
11. A wheel, whose centre of gravity is at its centre, rolls down a rough
plane of inclination o, dragging a particle of mass m, which slides on the
plane, and is connected with the centre of the wheel by a thread ; the whole
motion takes place in a vertical plane, and the thread makes an angle j3 with
the line of greatest slope down which the particle slides. Prove that the
system descends with uniform acceleration
acos(/3-f)+mcos£sin(a — t) 2
""
where a is the radius of the wheel, M its mass, k its radius of gyration about
its axis, m the mass of the particle and e the angle of friction between it and '
the plane.
12. Two smooth spheres are in contact, and the lower slides on a horizontal
plane.
Let M, m be the masses, a and b the radii, 6 the angle which the line of
centres makes with the vertical
at time t. If the whole system
starts from rest, the centre of
mass G descends vertically, for
there is no resultant horizontal
force on the system. Further,
since all the forces acting on
either sphere pass through its
centre, neither acquires any
angular velocity. Let x be the
distance of the centre of the
lower sphere (J/) from the ver-
tical through the centre of mass
at time t, then the distance of FlS- 74m
(7 from the centre of M is m(a + b}/(M+m), and thus the horizontal velocity
of <7is
m
M +m
By equating this to zero we express x in terms of 6 and 6.
Hence prove that the equation of energy can be put in the form
\(^ ?L-^A\fr±-9
2
L. M.
.( 1 ??L_ cos2<A <92+ -2-j- cos 0 = const.
V M + m ) a + b
15
226 MOTION OF A RIGID BODY IN TWO DIMENSIONS [OH. VIII
Find the pressure between the spheres in any position, and prove that, if 6 = a
initially, the spheres separate when
m
cos 6 3 -
cos20 ) = 2 cos a.
M + m )
*231. Stress in a rod. As an example of the resultant force between
two parts of a body we consider the case
of a rigid uniform rod swinging as a
pendulum about one end.
If m is the mass of the rod, 2a its
length, 6 the angle which it makes with
the vertical at time t, we have, since
the radius of gyration about the centre
of mass is a/\/3,
%azd = -agsin 0,
and § ad2 = g (cos 6 - cos a),
where a is the amplitude of the oscil-
lations.
Now consider the action between
the two parts of the rod exerted across
a section distant 2.r from the free end.
Let P be the centroid of .this section. We may suppose the action of A P on
BP reduced to a force at P and a couple, and we may resolve the force into a
tension T in the rod, and a shearing force S at right angles to it. We call the
couple <?, and suppose the senses of T, S, and G to be those shown in the
figure. The action of BP on AP is then reducible to a force at P having com-
ponents T, S, and a couple G, in the opposite senses to those shown.
Now BP is a rigid uniform rod of mass m.vja, turning with angular velocity
0, while its centre describes a circle of radius 2a — x with the same angular
velocity. It moves in this way under the action of the forces T, S, the weight
mgje/a vertically downwards through its middle point, and the couple G. By
resolving along AB and at right angles to it, and by taking moments about /*,
we obtain the equations of motion of BP in the form
- (2o -
^
= T- mg - cos 6,
y a
••
m - (2a— x) 6 = S—
m- |*(2o— ar)tf+— 6\— — G-mg-xsin 6,
(t ^ o J a '
and by these equations 7*, S, and G are determined, 6 and 6'2 being known. In
particular the couple G resisting bending is
x* AP
-^(a-x\ or %mgsin0
230-232] SUDDEN CHANGES OF MOTION 227
232. Impulsive motion. We apply the theory of sudden
changes of motion of any system (Oh. VI, Art. 168) and the theory
of the momentum of a rigid body (Art. 220).
We have three equations of impulsive motion expressing that
the change of momentum of the body is equivalent to the impulses
exerted upon it.
The momentum of the body was shown to be equivalent to a
resultant momentum localized in a line through the centre of mass,
and equal to the momentum of the whole mass of the body moving
with the centre of mass, together with a couple, of amount equal
to the product of the angular velocity of the body and the moment
of inertia about an axis through the centre of mass perpendicular
to the plane of motion.
Let m be the mass of the body, U, V the resolved velocities of
the centre of mass in two directions (at right angles to each other)
in the plane of motion, and ft the angular velocity before impact ;
let u, v be the resolved velocities of the centre of mass in the
same two directions after impact, and o> the angular velocity ; also
let k be the radius of gyration of the body about an axis through
the centre of mass perpendicular to the plane of motion.
The change of momentum of the system can be expressed aa
a vector localized in a line through the centre of mass, whose
resolved parts in the two specified directions are m (u — U) and
r/i (w— V); together with a couple, in the plane of motion, of
moment mk'2 (o> — ft).
The impulses exerted on the body can be expressed as a single
impulse at any origin and an impulsive couple.
The equations of impulsive motion express the equivalence of
the two systems of vectors.
Thus if the impulses are reduced to an impulse at the centre
of mass, whose resolved parts in the specified directions are X
and F, together with a couple N, we can take the equations of
impulsive motion to be
m(u-U)=X, m(v-V)=Y, mk*(a)-ty = N.
More generally, the resolved part, in any direction, of the vector
whose resolved parts, in the specified directions, are m(u—lT) and
15—2
228 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
m (v — V} is equal to the resolved part, in the same direction, of
the vector whose resolved parts, in the specified directions, are
% and Y; and the moment about any axis of the vector system
determined by m(u— U), m(v — V), mk2((a — Q} is equal to the
moment about the same axis of the vector system determined by
x, Y, #
233. Kinetic energy produced by impulses. Let the
body move in one plane. Let m be the mass of the body, U, V
resolved velocities of its centre of mass parallel to the axes of
reference, and H its angular velocity, just before the impulses act,
u, v, <w corresponding quantities just after.
Let X, Y be the resolved parts parallel to the axes of the
impulse applied to the body at any point whose coordinates
relative to the centre of mass are x, y.
The equations of impulsive mdlion are
m(u- tf) =
mk* (&> - O) =
Multiply these equations in order by
i(u+Z7), MV+V), i(« + n),
and let T be the kinetic energy of the body after the impulses, T0
that before. Then we have i
The right-hand member of this equation is the sum of the
products of the external impulses and the arithmetic means of the
velocities of their points of application resolved in their directions
before and after.
Now the theorem of Art. 174 asserts that the change of kinetic
energy is -equal to the value of the like sum for all the impulses
internal and external. It follows that the internal impulses
between the parts of a rigid body, which undergoes a sudden
change of motion, contribute nothing to this sum.
234. Examples.
1. A uniform rod at rest is struck at one end by an impulse at right angles
to its length. Prove that, if the rod is free, it begins to turn about the point
of it which is distant one-third of its length from the other end, and that the
kinetic energy generated is greater than it would be if the other end were fixed
in the ratio 4 : 3.
232-236] IMPULSES, INITIAL MOTIONS, OSCILLATIONS 229
2. A free rigid body is rotating about an axis through its centre of mass,
for which the radius of gyration is k, when a parallel axis at a distance c
becomes fixed. Prove that the angular velocity of the body is suddenly
diminished in the ratio k* : c2 + k2.
3. An elliptic disk is rotating in its plane about one end P of a diameter
PP', when P' is suddenly fixed. Find the impulse at P' and the angular
velocity about P', and prove that, if the eccentricity exceeds ,/f , the diameter
PP' may be so chosen that the disk is reduced to rest.
4. A uniform rod of length 2a and mass m is constrained to move with
its ends on two smooth fixed straight wires which intersect at right angles,
and is set in motion by an impulse of magnitude m V. Prove that the kinetic
energy generated is |r/i72jo2/a2, where p is the perpendicular from the inter-
section of the fixed wires on a line parallel to the line of the impulse and such
that the centre of mass is midway between the two parallels.
235. Initial motions. No new method is required for the
solution of problems concerning rigid bodies of the same kind as
those which were considered in Arts. 203 — 206; but attention
must be paid to the proper expression of the kinetic reaction of
a rigid body. The kinetic reactions are equivalent as we saw in
Art. 221 to a resultant kinetic reaction and a couple; and the
resultant kinetic reaction is the same as that of a particle of mass
equal to the mass of the body placed at the centre of mass and
moving with the acceleration of the centre of mass.
Sometimes it is convenient to form an equation of motion by
taking moments about the instantaneous centre. It is then to be
remarked that, at an instant when the velocity of the centre of
mass vanishes, the moment of kinetic reaction is Ka>, where K is
the moment of inertia about an axis drawn through the instan-
taneous centre at right angles to the plane of motion, and &> is the
angular acceleration. Cf. Ex. 3 of Art. 222.
236. Small oscillations. When the method of Art. 211 is
applied, the most important matter to attend to is the expression
of the potential energy correctly to the second order of the small
quantity 6 by which the displacement from the equilibrium position
is specified.
As in the case of initial motions, so also in the case of small
oscillations, it is sometimes convenient to form an equation of
motion by taking moments about the instantaneous centre. If we
take moments about the instantaneous centre in the position of
230 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
equilibrium the equation is nugatory. This position is, of course,
occupied by the body at one instant during the period of oscillation,
and at any other instant during the period the instantaneous
centre is in a slightly different position. The method which is
now effective is to take moments about the instantaneous centre
in a displaced position. The moment of the kinetic reaction about
the instantaneous centre is expressed correctly to the first order in
the displacement by the formula Kw, where the letters have the
same meanings as in Art. 235. This approximation is sufficient
for the purpose of forming the equation of oscillatory motion.
237. Illustrative problem.
A uniform rod can slide with its ends on two smooth straight wires which are
equally inclined to the horizontal and fixed in a vertical plane. It is required
to find the oscillations about the horizontal position.
Let OA, OB be the two wires, a the angle which each of them makes with
the horizontal, AB the horizontal position of equilibrium of the rod. A'B' a
displaced position, 6 the angle between AB and A'B' . Then 0 is the angular
velocity, and 6 the angular acceleration of the rod. The instantaneous centre
in any position is the point of intersection of perpendiculars to OA, OB drawn
from the ends of the rod. We denote by /, /' the positions of the instantaneous
centre corresponding to AB and A'B', and by G, G' the corresponding positions
of the centre of mass.
The moment of the kinetic reaction about /' is m (tf + I'G'2) 6, where m is
236-238] INITIAL MOTIONS AND OSCILLATIONS 231
the mass of the rod and k its radius of gyration about its centre of mass.
With sufficient approximation we may put IO for I'G'.
The forces acting on the rod are its weight and the pressures at its ends,
and the lines of action of the pressures pass through /'. Now 01' is a
diameter of a circle of which A'B' is a chord subtending an angle 7r-2a at
the circumference, and thus 01' is of constant length and //' is therefore
ultimately at right angles to 01 and horizontal. Also GO' being ultimately
at right angles to IO is horizontal, and thus the moment of the weight about
/' is - mg (II1 — GO'}. Hence we have the equation of moments
m(t* + IG*) e=-mg (II'- GO'}.
Now let 2« be the length of the rod. We find
//' = BB ' sec a = IBB sec a = aB cosec a sec a,
and the equation becomes
ma? ( J + cot2 a) 6 = — mgaQ (sec a cosec a — cot a).
The right-hand member is - mgad tan a, and therefore the motion in Q is the
same as that of a simple pendulum of length
a cot a(^ + cot2a).
238. Examples.
1. A uniform rod of length 2a and mass m is supported in a horizontal
position by two equal inextensible cords each of length L The ends of the
cords are attached, one to either end of the rod, and the other to a fixed
point, so that the cords make equal angles a with the vertical. Prove that, if
one cord is cut, the tension in the other immediately becomes
mg cos a/(l+ 3 cos2 a),
and that the initial angular accelerations of the remaining cord and the rod
are in the ratio
a sin a : 31 cos2 a.
2. A uniform triangular lamina is supported in a horizontal position by
three equal vertical cords attached to its corners. Prove that, if one cord is
cut, the tension in each of the others is instantly halved.
3. Into the top of a smooth fixed sphere of radius a is fitted a smooth
vertical rod. A uniform rod of length 26 rests on the sphere with its upper
end constrained to remain on the vertical rod, the centre of mass being at a
distance c from the point of contact. Prove that, if the constraint is removed,
the pressure on the sphere is instantly diminished in the ratio
b(b-c]
4. A uniform rod of length 2« rests in a horizontal position in a smooth
bowl in the form of a surface of revolution whose axis is vertical ; the ends of
the rod are at points where the radius of curvature of the meridian curve is p
and the normal makes an angle a with the vertical. Prove that the length of
232 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
the equivalent simple pendulum for small oscillations in the vertical plane
through the equilibrium position of the rod is
^ap cos a (1 +2 cos2 a) /(a - p sin3 a),
provided that this expression is positive.
5. A uniform rod of length 2or passes through a .smooth ring, which is
fixed at a height b above the lowest point of a smooth bowl in the form of a
surface of revolution whose axis is vertical. The rod rests in a vertical
position. Prove that, if c denotes the radius of curvature of the meridian
curve at the lowest point, the length of the equivalent simple pendulum for
small oscillations is
provided that this expression is positive.
6. A uniform rod of length 2a is supported in the way explained in
Ex. 1, the distance between the fixed points of attachment of the cords being
2(a.+ £sina). Prove that the length of the equivalent simple pendulum for
small oscillations in the vertical plane through the cords is
i( al cos a (1 + 2 cos2 a)/(a + 1 sin3 a).
1. If any circle is drawn through the instantaneous centre of no accelera-
tion, prove that the accelerations of all other points on this circle are directed
to a common point.
2. A straight rod moves in any manner in its plane. Prove that, at any
instant, the directions of motion of all its particles are tangents to a parabola.
3. A rope passes round a rough pulley, which moves in any manner in
its plane, so that the rope remains tight. Prove that the directions of motion
of all the points of the rope, which are in contact with the pulley at any
instant, are tangenjts to a conic.
4. A uniform triangular lamina ABC is constrained to move in a vertical
plane with its corners on a fixed circle. Prove that the motion is the same as
that of a simple pendulum of length
R (1 - 2 cos A cos B cos C)/J(1 - 8 cos A cos B cos C),
where R is the radius of the circle circumscribing the triangle.
5. The pendulum of a clock consists of a rod with a moveable bob clamped
to it, the position of the centre of mass of the bob on the central line of the
rod being adjustable. Prove that, if xl, x2, x3 are the distances of the centre
of mass of the bob from the axis of suspension when the clock gains nl, n2, %
minutes a day respectively, the length of the equivalent simple pendulum
when the clock keeps correct time is
2 #M2 ( ~
MISCELLANEOUS EXAMPLES 233
where I, m, n are the numbers 1, 2, 3 in cyclical order, $1 = 1+«1/1440, ..., and
each of the sums contains three terms obtained by putting 1, 2, 3 successively
for^.
6. Two circular rings, each of radius a, are firmly joined together so that
their planes contain an angle 2a and are placed on a rough horizontal plane.
Prove that the length of the equivalent simple pendulum is
| a cos a cosec2 a (1 + 3 COS2 a).
7. A thin uniform rod, one end of which can turn about a smooth hinge,
is allowed to fall from a horizontal position. Prove that, when the horizontal
component of the pressure on the hinge is a maximum, the vertical component
is J^ of the weight of the rod.
8. A uniform rectangular block, of mass M, stands on a railway truck
with two faces perpendicular to the direction of motion, the lower edge of the
front face being hinged to the floor of the truck. If the truck is suddenly
stopped, find its previous velocity if the block just turns over. Prove that, in
this case, the horizontal and vertical pressures on the hinge vanish when the
angle which the plane through the hinge and the centre of mass of the block
makes with the horizontal has the values sin-1§ and sin"1 ?j respectively, and
that the total pressure is a minimum, and equal to \Mg -J^, when the angle
is sin"1 §§.
9. The door of a railway carriage, which has its hinges (supposed smooth)
towards the engine, stands open at right angles to the length of the train
when the train starts with an acceleration /. Prove that the door closes in
time A /f^ffi [*' ** with an angular velocity V{2«/7(a2+F)}, where
\ \ -Zaj /Jo v/(sm t>)
2a is the breadth of the door, and k the radius of gyration about a vertical axis
through the centre of mass.
10. A uniform sphere is placed on the highest generator of a rough cylinder,
which is fixed with its axis horizontal. Prove that, if slightly displaced, the
sphere will roll on the cylinder until the plane through the centre of the sphere
and the axis of the cylinder makes with the vertical an angle o satisfying the
equation
17/n cos a — 2 sin o= lOju,
where p is the coefficient of friction.
11. A uniform circular ring moves on a rough curve under no forces, the
curvature of the curve being everywhere less than that of the ring. The ring
is projected from a point A of the curve, and begins to roll at a point B. Prove
that the angle between the normals at A aud_J3 is /i~1log2, where p is the
coefficient of friction.
12. A uniform sphere of mass M rests on a rough plank of mass M', which
is on a rough horizontal plane ; 'the plank is suddenly set in motion along its
234 MOTION OF A RIGID BODY IN TWO DIMENSIONS [cH. VIII
length with velocity V. Prove that the sphere will first slide and then roll on
the plank, and that the whole system will come to rest after a time
from the beginning of the motion, where /u is the coefficient of friction at each
of the places of contact.
13. A reel, of mass M and radius a rests on a rough floor, /i being the
coefficient of friction. Fine thread is coiled on the reel so as to lie on a
cylinder of radius 6(<a) and coaxal with the reel. The free end of the
thread is carried in a vertical line over a smooth peg at a height /; above
the centre of the reel and supports a body of mass m. Prove that, if either
p < mb/(M- m} a, or if M< m [1 - V (1 +a//« - a2/6A)/(aa + ^)],
the thread will be unwound from the reel.
14. A garden roller, in which the mass of the handle may be neglected, is
pulled with a force P in a direction making an angle a with the horizontal
plane on which it rests. Show that it will not roll without slipping unless
P {sin a sin (f> + cos a cos <£ . &-/(a2 + &2)} < JFsin <£,
where a, k, W are the radius, the radius of gyration about the axis, and the
weight of the roller, and $ is the angle of friction between it and the ground.
15. Two rough cylinders of radii ;-x, r2 are put on a rough table, and on
them is placed a rough plank. Prove* that, under certain conditions, the
system can start from rest and move so that each cylinder rolls on the tal >le
with the constant acceleration
Mg sin 2a/{m! (1 +*,2/*1!2) +m.2 (1 +*2V22) + 4J/cos* a},
where sin a = (rt ~ r2)/o?, and d is the initial distance between the axes of the
cylinders.
16. On the top of a fixed smooth sphere rests a fine uniform ring with its
centre in the vertical diameter, and its diameter subtends an angle 2a at the
centre of the sphere. Prove that, if the ring is slightly displaced, it will first
begin to leave the sphere when its plane has turned through an angle 6 which
is given by the equation
sin (a - 6} sin a= 2 cos2 a (2 — 3 cos 6).
[Assume that the pressure between the sphere and the ring acts only at the
highest and lowest points of the ring.]
17. A uniform rod, lying at rest in a smooth sphere, is of such length that
it subtends a right angle at the centre. The rod is set in motion so that its
ends remain on the sphere and make complete revolutions in a vertical plane.
Prove that, if V is the initial velocity of the centre, and a the radius of the
sphere,
MISCELLANEOUS EXAMPLES 235
18. Two equal uniform rods, each of mass m and length 2«, are free to
turn about their middle points, which are fixed at a distance 2a apart in a
horizontal line. The rods being horizontal, a uniform sphere of mass M and
radius c is gently placed upon them at the point where their ends meet. Prove
that, if 9J/{a2 + c2}2 = 2»z{a2-c2}2, the sphere will, as it leaves the rods, have
half the velocity which it would have had after falling freely through the same
height.
1 9. An elastic thread of modulus X is wound round the smooth rim of a
homogeneous circular disk of mass m, one end being fastened to the rim, and
the other to the top of a smooth fixed plane of inclination a to the horizontal,
down which the disk moves in a vertical plane through a line of greatest
slope, which is the line of contact of the straight portion of the thread with
the plane. Initially the thread has its natural length I and is entirely wound
on the rim of the disk which is at rest at the top. Prove that at any time t
before the thread is entirely unwound the tension is
\m.g sin a sin2 {}>t J(3\/lm)}.
20. Two equal cylinders of mass m, bound together by a light elastic band
of tension T, roll with their axes horizontal down a rough plane of inclina-
tion a. Show that their acceleration down the plane is
(^ 2PT \
* q sin all —. — I ,
\ mg sm a/
fjt being the coefficient of friction between the cylinders.
21. A rod AB, whose density varies in any manner, is swung as a pendulum
about a horizontal axis through A. Prove that the couple resisting bending is
greatest at a point P determined by the condition that the centre of mass of
the part PB is the centre of oscillation of the pendulum.
22. A uniform rod of mass m has one extremity fastened by a pivot to the
centre of a uniform circular disk of mass J/, which rolls on a horizontal plane,
the other extremity being in contact with a smooth vertical wall. The plane
of the wall is at right angles to the plane containing the disk and the rod.
Prove that the inclination 6 of the rod to the vertical when it leaves the wall
is given by the equation
9 .A/" cos3 6 + 6m cos d — fan cos a=0,
the system starting from rest in a position in which 6 = a.
23. A smooth circular cylinder, of mass M and radius c, is at rest on a
smooth horizontal plane ; and a heavy straight rail, of mass m and length 2«,
is placed so as to rest with its length in contact with the cylinder, and to have
one extremity on the ground. Prove that the inclination of the rail to the
vertical in the ensuing motion (supposed to be in a vertical plane) is given by
the equation
where a is the initial value of 6.
236 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
24 A circular cylinder, of radius a and radius of gyration k, rolls inside a
fixed horizontal cylinder of radius b. Prove that the plane through the axes
moves like a simple pendulum of length
The second cylinder is free to turn about its axis ; the first cylinder is of
mass m, and the moment of inertia of the second about its axis is MK*.
Prove that the length of the equivalent simple pendulum is (b — a) (! + »)/»,
where n=a2/k'2 + mbz/MK2 ; prove also that the pressure between the cylinders
is proportional to the depth of the point of contact below a plane which is
at a depth 2nbcosa/(l + 3n) below the fixed axis, where 2a is the angle of
oscillation.
25. A uniform circular hoop of radius a is so constrained that it can only
move by rolling in a horizontal plane on a fixed horizontal line ; and a particle
whose mass is I/A of that of the hoop can slide on the hoop without friction.
Prove that, if initially the hoop is at rest, and the particle is projected along
it from the point furthest from the fixed line with velocity v, then the angle
turned through by the hoop in time t will be
where ty is the angle through which the diameter through the particle has
turned in the same interval. Prove also that
26. A uniform rod swings in a vertical plane, being suspended by two
cords which are attached to its ends and to points A, B in a horizontal line.
A B is equal to the length of the rod, and the cords are not crossed. Prove
that, if the cords attached to A and B are of lengths a and a + X respectively,
where X is small, the angular velocity of the cord attached to A, when inclined
to the vertical at an angle 6, is greater than it would be if X were zero by
X (#/2a3)2 (cos Q - cos a)^ (tan2 B - \ sec 6 sec o)
approximately, a being the value of 6 in a position of rest, and not l>eing
nearly equal to a right angle.
27. A uniform rod, which is free to turn about a point fixed in it, touches,
at a distance c from the fixed point, the rough edge of a disk of mass m,
radius a, and radius of gyration k about its centre. The system being at
rest on a smooth horizontal plane, an angular velocity Q is suddenly com-
municated to the rod so that the disk also is set in motion. Prove that in the
subsequent motion the distance r of the point of contact from the fixed point
satisfies the equation
where MX2 is the moment of inertia of the rod about the fixed point, and the
edge is rough enough to prevent slipping.
MISCELLANEOUS EXAMPLES 237
28. A uniform rod has its lower end on a smooth table and is released
from rest in any position. Show that the velocity of its centre on arriving at
the table is J(§gh\ where h is the height through which the centre has fallen.
Prove also that, at the instant when the centre reaches the table, the pressure
on the table is one quarter of the weight of the rod.
29. A wheel can turn freely about a horizontal axis ; -and a fly of mass m
is at rest at the lowest point. If the fly suddenly starts off to walk along the
rim of the wheel with constant velocity V relative to the rim, show that he
cannot ever get to the highest point of the rim unless F is at least as great as
2
where a is the radius of the wheel, and MK2 its moment of inertia about its
axis.
30. A hollow thin cylinder, of radius a and mass M, is maintained at rest
in a horizontal position on a rough plane of inclination a ; and an insect of
mass m is at rest in the cylinder on the line of contact with the plane. The
insect starts to crawl up the cylinder with velocity F, and the cylinder is
released at the same instant. Prove that, if.the relative velocity is maintained
and the cylinder rolls uphill, then it will come to instantaneous rest when the
angle which the radius through the insect makes with the vertical is given by
the equation
F2{1 — cos (6 — a)} + ag (co$a—cos6) = (I+Mlm)aff(0-a) sin a.
31. A rigid square ABCD, formed of four uniform rods each of length 2a,
lies on a smooth horizontal table, and can turn freely about one angular point
A, which is fixed. An insect, whose mass is equal to that of either rod, starts
from the corner B to crawl along the rod BC with uniform velocity F relative
to the rod. Prove that, in any time t before the insect reaches (7, the angle
through which the square turns is
3 . .fVt / 3
32. The corners A, B of a uniform rectangular lamina A BCD are free to
slide on two smooth fixed rigid wires OJ, OB at right angles to each other in
a vertical plane and equally inclined to the vertical. The lamina being in a
position of equilibrium with AB horizontal, find the velocity produced by an
impulse applied along the lowest edge CD.
Prove that, if A B = 2a, BC=4a, then AB will just rise to coincidence with
a wire if the impulse is such as would impart to a mass equal to that of the
lamina a velocity
33. A uniform rigid semicircular wire is rotating in its own plane about
a hinge at one end, and is suddenly brought to rest by an impulse applied at
the other end along the tangent at that end. Prove that the impulsive stress
couple is greatest at a point whose angular distance from the hinge is 0, where
(f> tan ^ (f) = 1 .
238 MOTION OF A RIGID BODY IN TWO DIMENSIONS [CH. VIII
34. A particle of mass m impinges directly on a smooth uniform spheroid
of mass M and semi-axes a, b, the spheroid being at rest, and no energy being
lost in the impact. Prove that, if
1 < M/m < 6 - 10a&/(«2 + fc2),
the point of impact may be so chosen that the particle is reduced to rest.
35. A uniform equilateral triangular board is suspended by three equal
cords, which are attached to its corners and to the corners of a similar fixed
triangle in a horizontal plane ; the plane through any two cords makes an
angle a with the horizontal. Prove that, if one of the cords is cut, the tensions
in the remaining two are diminished in the ratio
3 sin2 a : 2 + 4 sin2 a.
36. A circular ring hangs in a vertical plane on two pegs which are in a
horizontal line, and the line joining the pegs subtends an angle 2a at the
centre. One peg is suddenly removed. Find the pressure on the remaining
peg (1) when it is smooth, (2) when it is sufficiently rough to prevent slipping,
and prove that these pressures are in the ratio 1 : (1 + j tan2a)^.
37. A sphere resting on a horizontal plane is divided into a very large
number of segments by planes through the vertical diameter, and is kept in
shape by a band round the horizontal great circle. Prove that, if the band is
cut, the pressure on the plane is diminished by the fraction 45n-2 '2048 of itself.
38. The lower end of a uniform rod of length a slides on an iuextensible
thread of length 2a whose ends are fixed to two points distant 2 J(a? — 62)
apart in a horizontal line, and the upper end of the rod slides on a fixed
smooth vertical rod which bisects the line joining the two fixed points. Prove
that, if 26 > a, the time of a small oscillation about the vertical position of
equilibrium is
39. In a heavy plane lamina, whose centre of gravity is G, are two narrow-
straight slits DA, AC, such that AG bisects the angle BAG. Through each
slit passes a fixed peg, the pegs, P, Q, being in the same horizontal line. Prove
that the time of a small oscillation of the lamina in its own plane, about a
position of equilibrium in which the vertex A of the triangle APQ is upwards, is
g sin A (±PQ* - AG2 sin2 A ) '
where k is the radius of gyration of the lamina about a line through G perpen-
dicular to its plane.
40. Two equal wheels, each of mass J/, radius a, and radius of gyration k
about its axis, are rigidly connected by an axle of length c and run on a hori-
zontal plane. Two particles, each of mass m, are connected, one to each of the
centres of the wheels, by cords which pass over smooth pegs in the line of
MISCELLANEOUS EXAMPLES 239
centres. Prove that, if the wheels are symmetrically placed between the pegs,
and slightly displaced by rolling on the plane, the time of a small oscillation is
Sir J {Mb (a2 + F)/m^a2},
where 26 +c is the distance between the pegs.
41. A solid circular cylinder, bounded by two planes making given angles
with the axis, is laid on its curved surface on a rough horizontal plane. Find
the position of stable equilibrium, and prove that, if I is the length of the
equivalent simple pendulum for a small oscillation, and d the diameter of the
cylinder, then the ratio of the longest and shortest generators is
CHAPTER IX*
RIGID BODIES AND CONNECTED SYSTEMS
239. Impact of two solid bodies. To investigate the motion
of solid bodies which collide, Poisson "I" introduced a certain
hypothesis as to the motion which takes place while the bodies are
in contact. In this short interval of time the bodies may not be
regarded as rigid, but the deformation that occurs must be taken
into account (Art. 192). Poisson supposed that this interval could
be divided into two periods: during the first period the bodies are
undergoing compression ; during the second period the restitution
of form takes place. Further Poisson supposed that the impulse of
the pressure between the bodies during the period of restitution
bears to the impulse of the pressure during the period of compression
the ratio e, which is the coefficient of restitution.
This hypothesis leads to the following rule for solving the
problem of impact: — First solve the problem on the supposition
that there is no restitution, and find the impulsive pressure between
the bodies. Multiply this pressure by (1 + e). Now solve the
problem again on the supposition that the impulsive pressure has
the value so determined.
Let us apply this method to the problem of the direct impact of two
spheres. With the notation of Art. 195 in Ch. VII, the equations of the
problem, on the supposition that there is no restitution, are
u — w' = 0, mu + m'u'=mU+m'U',
and the impulsive pressure R0 between the bodies is
T7 mm' . T, ,,,.
— m(u-U) or - — ,(U—li}.
m+m
We multiply this by (1 +e). The equations of the problem, on the supposition
that the impulsive pressure between the bodies is (l + e)R0, are
-m(u-U)=
- ,
m+m m+m
and the values of u and u' which are found from these equations are the same
as those found in Art. 195.
* This Chapter may be omitted in a first reading.
f S. D. Poisson, Traiii de Mecanique, 2nd ed., Paris 1833, t. 2, pp. 273 ct seq.
239, 240]
THE PROBLEM OF IMPACT
241
In the case of the direct impact of smooth spheres the results
that can be deduced from Poisson's hypothesis are the same as the
results that can be deduced from Newton's experimental result.
We may show in like manner that, in the case of the oblique
impact of smooth spheres (Art. 197), the results that can be
deduced from Poisson's hypothesis are the same as those that can
be deduced from the "generalized Newton's rule" stated in Art. 196.
We shall show that this result holds for the impact of any two
bodies, whether smooth or rough, provided that the friction is not
great enough to prevent sliding.
240. Impact of smooth bodies. Let two rigid bodies moving in the
same plane corne into contact at a point P. Suppose the bodies to be smooth
at P. Let R be the impulsive pressure between the bodies at P. The direc-
tion of R is the common normal at P to the two surfaces. Let the axis of x
be taken in this direction, the axis of y being any fixed line in a perpendicular
direction.
j
Let m and m' be the masses of the bodies, U, V, Q the velocity system of m
before impact, u, v, « corresponding quantities after impact, and let accented
letters denote similar quantities for TO'. Also let x, y be the coordinates of the
centre of mass of m and x', y' those of in' at the instant of impact, and let £, TJ
be coordinates of P at the same instant. Also suppose that, as acting on m,
the sense of R is the negative sense of the axis of x (Fig. 77).
Fig. 77.
The velocity of P, considered as a point of TO, has components
U— Q(TI- y), V+ a(£-ji) before impact, and
u - m (77 - y\ v + m(£- x) after impact.
The velocity of P, considered as a point of TO', has components
U' - Q' (q - y'\ V' + Q' (£ - xf] before impact, and
u' - to' (77 -y'}, v' + to' (| - x') after impact.
The equation provided by the generalized Newton's rule is accordingly
T M
L. M.
16
242 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
The equations of impulsive motion of the two bodies, obtained by resolving
parallel to the axis of x, are
m (u-U)=- It, m' (u1 - U') = R.
The equations obtained by resolving parallel to the axis of y are
m(v-V) = 0, m'(v'-V')=Q.
The equations of moments about axes through the centres of mass perpen-
dicular to the plane of motion are
where k and kf are the radii of gyration of the bodies about the axes in
question.
On substituting for u, u', <o, «' in the equation containing e, we find
and this equation shows that the impulsive pressure with any value of e is
(1 +e) times what it would be if e were zero.
The result of this Article can be expressed in the statement that the
generalized Newton's rule and the rule derived from Poisson's hypothesis are
equivalent for any two smooth bodies moving in one plane.
241. Impact of rough bodies. The impulsive action between two
rough bodies which come into contact, when there is sliding at the point of
contact, is assumed to be expressible by means of an impulsive pressure, as in
the case of smooth bodies, and an impulsive friction tending to resist sliding,
the friction and the pressure having a constant ratio, the coefficient of friction.
We shall suppose the geometrical condition as regards the relative velocity to
be the same as in the case of smooth bodies, viz. the generalized Newton's
rule.
We shall show that, when there is sliding at the points that come into
contact, the rule deduced from Poisson's hypothesis is equivalent to the
generalized Newton's rule, for the impulsive action between rough bodies.
Writing F for the impulsive friction at the point of contact, and taking
the same notation as in the last Article, we have the equations of impulsive
motion
m(u-U)=-R, m(v-V}=-F,
,\
J "
and
m'(u'-U') = R, m'(v'-V'} = F,\
) '
Also we have the equation of sliding friction
F=?R ....................................... (3),
and the equation provided by the generalized Newton's rule
«-»(•/ -,?)-«' + «' (?-/)= -e{U-Q(i-y)-U'+Q! (•?-/)} . ..(4).
240-242] THE PROBLEM OF IMPACT 243
From these equations we obtain, by elimination of u, u', v, v', a, CD', F, an
equation for R, viz.
This equation shows that R contains (1 +e) as a factor and is otherwise inde-
pendent of e, and thus proves the equivalence of the two rules.
242. Case of no sliding. When the bodies are sufficiently rough to
prevent sliding the problem is more complicated. The effects of the elasticity
of the bodies cannot be so simple as in the previous cases*.
We may obtain a provisional solution by assuming that the generalized
Newton's rule holds good. Then equations (1), (2), (4) of Art. 241 are still
valid, but instead of equation (3) we have the condition that there is no
sliding, viz.
v + u> (%-x}=Lv' + a>' (t--s?} (5).
From equations (1), (2), (4), (5) we can form two equations for R and F, viz.
mk* m'k'*
J/)l
i i
'
yTi
J
It is clear that the solution of these equations will give an expression for R
consisting of two terms, one of them having (1 + e) as a factor and the other
not containing that factor.
Since R is not in general proportional to 1+e, the result which would be
obtained from Poisson's hypothesis is not in general the same as that which
would be obtained from the generalized Newton's rule.
The results would however be the same in any case in which either
The first of these equations expresses the condition that there is no relative
velocity of sliding at the instant of impact, or that the impact is, in an obvious
sense, "direct." The second is satisfied if rj=y=y', that is if the normal at
the point of contact passes through the centres of mass of the two bodies, as
it would if the bodies are spheres or circular disks. It is also satisfied if iy=y
and £=#', which would be the case if one body is a sphere or a circular disk
and the other is a thin rod.
* Poisson himself did not suppose his hypothesis to be applicable to cases in
which there is sufficient friction to prevent sliding. The question is not really of
any practical interest because the motion must depend largely on accideutal
circumstances.
16—2
244 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
243. Examples.
1. A uniform sphere of radius a and mass m, moving without rotation,
impinges directly on a smooth uniform cube of side 2a and mass m ', the line
of motion of the sphere being at a distance b from the centre of mass of the
cube. Prove that, if there is no restitution, the kinetic energy lost in the
impact is to that of the sphere before impact in the ratio
2. A uniform rod, falling without rotation, strikes a smooth horizontal
plane. Prove that, for all values of the coefficient of restitution, the angular
velocity of the rod immediately after impact is a maximum if the rod before
impact makes with the horizontal an angle cos"1 1/\/3.
3. A sphere whose centre of mass coincides with its centre of figure is
moving in a vertical plane and rotating about an axis perpendicular to that
plane when it strikes against a horizontal plane which is sufficiently rough to
prevent sliding. Prove that the sphere will rebound at an angle greater or less
than if there were no friction according as the lowest point of it at the instant
of impact is moving forward or backward-
4. A disk of any form, of mass TO, moving in its plane without rotation
and with velocity V at right angles to a fixed plane, strikes the plane, so that
the distances of the centre of mass from the point of impact and from the
plane are r and p. Prove that, if the plane is sufficiently rough to prevent
sliding, the impulsive pressure is
mV(l+e) (F + j»2)/ (F + r2),
where k is the radius of gyration of the disk about its centre of mass.
5. A ball spinning about a vertical axis moves on a smooth table, and
impinges on a vertical cushion, the centre moving directly towards the cushion.
Prove that, if B is the angle of reflexion, the kinetic energy is diminished in
the ratio
10 + 14tan2<9 : 10e-2 + 49 tan2 (9,
the cushion being sufficiently rough to prevent sliding.
6. A circular disk of mass M and radius c, moving in its own plane with-
out rotation, impinges on a rod of mass m and length 2a which is free to turn
about a pivot at its centre, and the point of impact is distant b from the pivot.
Prove that, if the direction of motion of the centre of the disk makes angles a
and (3 with the rod before and after collision, then
2 (:W&2 + ma2) tan 0 = 3 (3 M 62 - ema2) tan a,
the edges in contact being sufficiently rough to prevent sliding.
244. Impulsive motion of connected systems. In illus-
tration of the application of the equations of impulsive motion to
systems of rigid bodies with invariable connexions we take the
following problems. In the first it will be observed that we do not
243, 244]
SUDDEN CHANGES OF MOTION
!45
need to introduce explicitly the reactions between the connected
bodies. The second illustrates the choice of equations; for, although
some of the unknown reactions must be introduced, it is unnecessary
to form equations for each body separately.
I. Three uniform rods of masses proportional to their lengths are freely
jointed together and laid out straight, and one of the end rods is struck at the
free end at right angles to its length. It is required to find how they begin to
move.
Let 2a, 26, 2c be the lengths of the rods, the last being struck, and let xja,
M+x+y
PA
Fig. 78.
#/&, z/c be the angular velocities with which they begin to move, u the velocity
of the centre of mass of the first. Then the system of velocities is as shown
in the figure. Let P be the impulse applied at the end A, and <a, Kb, <c the
masses of the rods.
We take moments about C for the rod CD, about B for the rods BC, CD,
and about A for the three rods, and we resolve for the whole system at right
angles to the rods. We thus obtain the equations
u + x + <2y + z) + Kb(u+x+y) + Kau = .
Subtracting the second and third we get, on dividing by c,
'
and, on simplifying this and the second by using the first, we get
a (a + 46 + 2c) + y (3c + 36) + zc = 0,
and (2b+a)x + by=0.
Hence we have
u *' cz
246 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
II. A rhombus formed of four equal uniform rods freely jointed at the
corners is set in motion by an impulse applied to one rod at right angles to it.
To find how the rhombus begins to move.
Let 2a be the length of each side of the rhombus A BCD, a the angle DAB,
x the distance of the point struck from the middle point of the side AB con-
taining it, P the impulse, m the mass of each rod.
R'
Fig. 79.
The centre of mass of the system is the point of intersection of the lines
joining the middle points of opposite sides. Since the figure is always a
parallelogram, opposite sides have the same angular velocities, and the lines
joining the middle points of opposite sides are of constant length 2a and turn
with the angular velocities of the sides to which they are parallel. Let these
angular velocities be o> and «', and let v be the velocity of the centre of mass.
Then the velocities of the centres of mass of the rods and their angular veloci-
ties are as shown.
Now let the impulsive reaction of the hinge at C be resolved into S parallel
to EC and R at right angles to BC, and the impulsive reaction of the hinge at
D into S', R in the same directions. These impulses act in opposite senses
on the two rods which meet at a hinge. The figure shows the senses in which
we take them to act on the rod CD.
We form two equations of motion by resolving for the system in the
direction of the impulse and by taking moments about the centre of mass.
We thus obtain
| ma2 (to 4- «') = P (x + a cos a),
Again, we can form three equations containing R and R by resolving for
CD at right angles to BC, and taking moments for BC and AD about B and
244, 245] SUDDEN CHANGES OF MOTION 247
A respectively. We thus obtain
m (v cos a - aw') = R + R',
m [(v - aw) a cos a - ^aV] = - 2aR,
m [(v + aw) a cos a - J aV] = - 2aR',
from which, on elimination of R and R', we get
v cos a = § aw'.
Hence v = %P/m, w = fP^/»ia2, w' = |P cos a/ma.
245. Examples.
1. Two equal rods AB, AC freely jointed at A are at rest with the angle
BAG a right angle, and AC is struck at C by an impulse in a direction
parallel to AB. Prove that the velocities of the centres of mass of A .Sand
AC in the direction of AB are in the ratio 2 : 7.
2. Two equal uniform rods freely hinged at a common end are laid out
straight, and one end of one of them is struck by an impulse at right angles
to their length. Prove that the kinetic energy generated is greater than it
would be if the rods were firmly fastened together so as to form a single rigid
body in the ratio 7 : 4.
3. Four equal uniform rods are freely hinged together so as to form a
rhombus of side 2a with one diagonal vertical, and the system falling in a
vertical plane with velocity V strikes against a fixed horizontal plane. Taking
a to be the angle which each rod makes with the vertical and assuming no
restitution, prove that (i) the impulsive action between the two upper rods is
directed horizontally, (ii) the angular velocity of each rod after the impulse
is | ( V/a) sin a/(l +3 sin2 a), (iii) the impulsive action between the two upper
rods is to the momentum of the system before impact in the ratio
sin a (3 cos2 a ~ 1) : 8 cos a (1 -f 3 sin2 a),
(iv) the impulsive action at either of the hinges in the horizontal diagonal
makes with the horizontal an angle tan"1 {(3 cos2 a ~ 1) cot a}.
4. In Example 3, prove that, if the coefficient of restitution between the
rhombus and the ground is e, the angular velocity of each rod after the impulse is
f ( 1 i- e) ( 7/o) sin a/( 1 + 3 sin2 a).
5. A square framework A BCD is formed of uniform rods freely jointed
at B, C, and D, the ends at A being in contact but free. Prove that, if A B
is struck by a blow at A in the direction DA, the initial velocity of A is
79 times that of D.
6. A rectangle formed of four uniform rods, of lengths 2a and 2b and
masses TO and TO', freely hinged together, is rotating in its plane about its
centre with angular velocity n when a point in one of the sides of length 2a
becomes suddenly fixed. Prove that the angular velocity of the sides of length
26 instantly becomes \n (3TO + TO;)/(3TO + 2TO/), and find the angular velocity of
the sides of length 2a.
248 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
246. Initial motions and initial curvatures. The kinetic
reactions of the parts of a connected system of particles and rigid
bodies can always be expressed in terms of a finite number of
geometrical quantities which are unconnected by any geometrical
equations. This can usually be effected by methods similar to those
used in Art. 205.
It may however happen that such methods are difficult of
application. When this is the case we may begin by writing down
the geometrical equations which hold between the coordinates of
the points in any position. If we differentiate these equations
twice with respect to the time, and, in the results, substitute for
every first differential coefficient of a geometrical quantity the
value 0, and for every geometrical quantity the value that it has
in the initial position, we shall obtain the relations between the
initial accelerations of the various geometrical quantities involved.
Thus if x, y are the coordinates of any particle whose acceleration
is required, and 0, <f), . , . are a series of geometrical quantities which
define the position of the system, there will be certain values 60, </>0, . . .
for these quantities in the initial position. Now the geometrical
equations provide the means of expressing the x and y of the
particle in any position in terms of the values of 6, (f>, ... for that
position. Let cc=f(Q, <f>, ...) be the form of one of the equations
we can obtain. On differentiating we have
Reducing in the way that has been explained we obtain
where x0) 00, ... denote the initial values of x, 6, ..., and hfe]
\ovJo
£\ /•
, ... denote the values of ~^, -r , ... when 6 = 00, <i = d>0,
o<p/0 vv o<p
Now this process can be carried further, and arranged as a
process of approximation for expressing the values of x, y, ... as
246, 247] INITIAL CURVATURES 249
series in ascending powers of the time. We have in fact as a first
approximation x — ^x0t2, y = ^y^.
From such series we can deduce the initial curvatures of the
paths of all the particles.
It will be easier to understand how this process is carried out
after studying its application to a particular problem, and it will
at the same time be seen how simplifications may at times
suggest themselves. A complicated problem has been chosen
intentionally.
247. Illustrative problem. Two uniform rods AB, BC of masses
mi, mz an(t lengths a, b are freely kinged at B, and AB can turn about A in
a vertical plane. The system starts from rest in a position in which AB is
horizontal and BC vertical. It is required to determine the initial curvature of
the path of any point of BC.
Let AB make an angle 6 with the horizontal, and BC an angle <£ with the
Fig. 80.
vertical at time t. Since B describes a circle of radius a about A, and since
the centre of mass of BC describes a circle of radius \b relative to B, the
diagram of accelerations is that shown in Fig. 80.
By taking moments about B for BC, and about A for the system, we obtain
the two equations
(<9-f 0) -
- m2a cos
os6+$b sin 0).
Adding the equations, and dividing out common factors, we have
(£mj + m2) a6 - £m260 sin (6 + (/>)- £m2602 cos (6 + 0) =g cos 6 (\ml + m2). . .(1).
250 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
Also the first of the above equations is
=-J#sin0 ............ (2).
In the initial position 0=0, <£=0, 0 = 0, (/> = 0, and we have
g
In any position we have, by Maclaurin's theorem,
also
Now, taking equation (2), we see that if <f)0 were finite, <f> would be of order
23, and 6 of order £2, so that the terms would be respectively of orders 1, 2, 2, 3.
This shows that (/>0 must be zero. Again, if <£0iv is finite the equation can be
reduced, by picking out the terms of order 2 in t, to
^oiv - W>0 ($>) - M2 = 0,
giving $
Again, taking equation (1), and observing that cos 0=1— "- + — — ..., we
see that the lowest power of t in this series is the fourth, and then it appears
from equation (1) that the lowest power of t in 0 is the fourth, so that the
series for 6 begins
6!
Going back now to equation (2), it is clear that <£ contains no term in t3
but there is a term in t4. In fact, picking out the terms in t* in equation (2)
we have
..; f i *A • :— v i i i_ v
41'
giving ,,,=
Now, in the figure, taking as origin the initial position of S, and taking
the axes of x and y horizontal and vertical, we can write for the coordinates
of a point of BC distant r from B,
x= - a (1 —cos ff) + r sin <£, ?/=asin 0 + r cos<£ ;
expanding these we have approximately
giving #=--f<V + ;
y-r =
247-249] INITIAL MOTIONS 251
which are correct as far as t2. Hence the initial path of the point is approxi-
mately a parabola
and the radius of curvature of the path is 2a6/(3r — 26) unless r=$b.
If, however, ?'=§&, in order to get au approximate equation to the path,
we must expand to a higher order. We find
b 480 '
correct as far as tr>, and thus the initial path is given by the approximate
equation
(y - 1 6)3 =
248. Examples.
1. Two equal uniform rods are freely jointed at common ends, the other
end of the first is fixed so that the rods can turn about it, and the other end
of the second is held at the same level as the fixed end of the first, so that
the rods make equal angles a with the horizontal, and this end is let go.
Prove that the initial angular accelerations of the rods are in the ratio
6 - 3 cos 2a : 9 cos 2a - 8.
2. Three equal uniform rods are freely jointed at B and C so as to form
three sides of a quadrilateral A BCD, and the ends A and D can slide on a
smooth horizontal rod. The system is initially held (by means of horizontal
forces applied at A and D] in a symmetrical position with BC lowest and
horizontal, and with A B and CD equally inclined at angles a to the horizontal.
Prove that, when the ends A and D are released, the pressures at A and D are
changed in the ratio 1 + sin2 o : 5 - 3 sin2 a.
3. A uniform rod of length 2a is held at an inclination a to the horizontal
in contact with a smooth peg at its middle point. Prove that, when the rod
is let go, the initial radius of curvature of the path of a particle distant r from
the middle point is («2/r) tan a.
4. Two equal uniform rods AB, BC each of length a are freely jointed
at B, and can turn freely about A. Prove that, if the system ia released from
a horizontal position, the initial radius of curvature of the path of C is fa.
249. Small oscillations. Illustrative problem.
The following problem illustrates the application of the method
of Art. 211.
A uniform rod is supported at its ends by two equal vertical cords suspended
from faced points. It is required to find the small oscillation in which the
middle point moves vertically and the rod, remaining horizontal, turns round
its middle point.
252 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
Let 2a be the length of the rod, / the length of either cord, z the distance
through which the middle point has
risen at time t, 6 the angle through
which the rod has turned in the same
time. The depth of either end A or
B below the corresponding point of
support is I — z, and the distance AM
or BN of an end from the equilibrium
position of the corresponding cord is
2a sin ^d. Hence we have
this equation shows that when z and
6 are small z = ^(a2/l)02 to the second
order, and z — 0 to the first order.
Now, if m is the mass of the rod,
the kinetic energy in any position is
Fig. 81.
and the potential energy is nigz, the lowest position being the standard
position.
Hence, in the small oscillations, the kinetic energy is, with sufficient ap-
proximation,
and the potential energy is, with sufficient approximation,
\mg (a*ll)62.
The motion in 6 is therefore the same as for small oscillations of a simple
pendulum of length $1.
250. Examples.
1. A number of equal uniform rods each of length 2a are freely jointed
at a common end and arranged at equal intervals like the ribs of an umbrella,
and this cone of rods is placed in equilibrium over a smooth sphere so that
the angle of the cone is 2a. Prove that, for small vertical oscillations of the
joint, the length of the equivalent simple pendulum is
\a cos a (1 + 3 cos2 a)/(l + 2 cos2 a).
2. Prove that the length of the equivalent simple pendulum for small
oscillations of the handle of a garden roller rolling on a horizontal walk is
where a is the radius of the roller, M the mass of the roller alone, k its radius
of gyration about its axis, m the mass of the handle, h the distance of the
centre of mass of the handle from the axis of the roller, and I the length of
the equivalent simple pendulum for the oscillations of the handle when the
roller is held fixed.
249-251]
STABILITY OF STEADY MOTIONS
253
3. Four equal uniform rods are freely jointed so as to have a common
extremity, and four other like rods are similarly jointed ; the other ends of
the rods are then jointed in pairs so as to form eight edges of an octahedron.
One of the joints where four rods meet is fixed and the other is attached to it
by an elastic thread, so that in equilibrium the octahedron is regular and the
thread vertical. Prove that the length of the equivalent simple pendulum for
small vertical oscillations of the lowest point is ^ (I — ?0), where I and ?0 are the
equilibrium length and the natural length of the thread.
251. Stability of steady motions.
The principles of energy and momentum may frequently be applied to
problems concerning the stability of steady motions. We shall illustrate the
method by considering the steady motion of a spherical pendulum, that is a
particle moving under gravity on the surface of a sphere so as to describe a
horizontal circle.
Let 0 be the angle which the radius vector from the centre of
the sphere to the particle makes
with the downwards vertical at
time t, a the radius of the sphere,
<j> the angle contained between the
plane through the particle and the
vertical diameter and a fixed plane
through the same diameter.
The energy equation is
±ma*(fa+ sui*d<j>*)+mga (1 - cos 0)
= const.,
and the equation of constancy of
moment of momentum about the
vertical diameter is ma2 sin2 0<fa = const.
We wish to discover the condition that motion in a horizontal
circle, 0 = a, with angular velocity « may be possible. We have
<f> sin2 6 = a sin2 a,
so that the energy equation may be written
Fig. 82.
ia U* + a)2 ^Vzf) -gcos0 = const.
V sin2 ft/
Differentiating with respect to the time we obtain the equation
sin4 a cos 6 g • a
- 0>2 - T-T - + -Sin 0 =
sin3 0 a
254 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
Now the steady motion is possible if to is so adjusted that 0 = 0
when 6 = a. This gives us the condition
aw2 = g sec a ........................... (2).
(Cf. Art. 79.)
If the particle is projected from a point for which 6 is nearly
equal to a, in a nearly horizontal direction, with an angular
momentum wia2&> sin2 a about the vertical diameter, where w is
given by (2), then either it tends to remain always very near the
circle 6 = a, or to depart widely from it. Supposing it to remain
near the circle, we may put 6 = a + %, expand the terms of equa-
tion (1), and reject powers of % above the first. We thus find
[d ( g . „ cos 6 } 1
-^•^sin0-6>Jsm4a-^--^ =0,
do (a sm3#)Je=a
g 1 + 3 cos2 a
or %+Y- - = 0,
* a cos a
showing that the particle oscillates about the state of steady
motion in a period equal to that of a simple pendulum of length
a cos a/(l + 3 cos2 a).
The steady motion is stable if cos a is positive, or the circular
path is below the centre of the sphere.
Note. If the angular momentum (as well as the direction and point of
projection) is slightly altered, the possible steady motion would take place
along a slightly different circle; but the period of oscillation would be un-
changed.
252. Examples.
1. Utilize the method of Art. 251 to show that the motion of a particle
describing a circular orbit under a force f(r) directed to the centre is stable
if [3 + c/'(c)//(c)J is positive, c being the radius of the circle. Deduce the
results in Art. 106.
2. Prove that the steady motion with angular velocity w of a conical
pendulum of length I is stable, and that, if a small disturbance is made,
oscillations take place in time
3. A particle describes a horizontal circle of radius r on a smooth para-
boloid of revolution whose axis is vertical and vertex downwards. Prove that,
if it is slightly disturbed, its period of oscillation is
7r
where 4a is the latus rectum.
251-253] ENERGY AND MOMENTUM 255
4. A circular wire of radius a and of negligible mass rotates freely about
a vertical chord distant c from the centre ; a small heavy ring can slide on the
wire without friction. In the position of relative equilibrium the radius of
the circle drawn through the ring makes an angle a with the vertical. Find
the angular velocity with which the wire rotates, and prove that the length of
the equivalent simple pendulum for small oscillations of the ring is
a cos a(c + a sin a)/{c +« sin a (1 + 3 cos2 a)}.
Prove also that, if the wire is made to rotate uniformly, the period of small
oscillations is the same as for a simple pendulum of length
a cos a (c+a sin a)/(c + a sin3 a).
[In the second case energy is expended in keeping up the angular velocity
of the wire, and an equation of motion of the ring must be formed by resolving
along the tangent to the circle. The angular velocity in relative equilibrium
is the same as before.]
5. An elastic circular ring of mass m and modulus of elasticity X rotates
uniformly in its own plane about its centre under no external forces. Prove
that, if a is the radius in steady motion, and I is the radius when the ring is
unstrained, the period of the small oscillations about the state of steady
motion is
253. Illustrative problem. In further illustration of the principles
of Energy and Momentum consider the following problem :
A uniform rod and a particle are connected by an inextensible thread attached
to one end of the rod, the system is laid out straight, and the particle is projected
at right angles to the thread. It is required to find the motion when there are no
forces.
Let 2a be the length of the rod, I the length of the thread, % the angle
which the thread makes with the line of the rod produced at time t. Consider
first the motion of the particle P relative to the centre of mass M of the
rod AB.
M
A ""
Fig. 83.
Let 6 be the angle which AB makes at time t with its initial direction.
Then the velocity of B relative to M is aB at right angles to AB, and, since
BP makes an angle 6 + x with a line fixed in the plane of motion, the velocity
of P relative to B is I (0 + x) perpendicular to BP. The velocity of P relative
to M is the resultant of these two velocities. Its resolved parts along and
perpendicular to AB are accordingly
and a6 + l(6 + x)GOS\-
256 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
Now the centre of mass G is always at the point dividing MP in the ratio
of the masses of the particle and the rod ; and, if these masses are p and m
respectively, the velocity of M relative to O has components
aud
m+p v m + p^
along and perpendicular to AB, and the velocity of P relative to G has com-
ponents
m -\-p w. -\~p
in the same directions.
Hence the moment of momentum in the motion relative to G is
[(a + l cos x
also twice the kinetic energy in the motion relative to G is
al0 (6 + ft cos*].
Now the centre of mass moves with uniform velocity in a straight line ;
and thus the kinetic energy of the whole mass placed at the centre of mass
and moving with it is constant, and the moment about any fixed axis of the
momentum of the whole mass placed at the centre of mass and moving with
it is also constant. Also the kinetic energy of the system and its moment of
momentum about any fixed axis are constants. Hence the moment of momentum
in the motion relative to G and the kinetic energy in the same relative motion
are constants.
•
Let V be the velocity with which the particle was initially projected at right
angles to the thread ; then the initial values of the moment of momentum
and kinetic energy in the motion relative to G are
(a + 1) Vmp/(m +p) and ^ V2mp/(m +p).
Hence throughout the motion we have the equations
p)a*d + aQ(a + lcosx) + l(0 + x)(l + aco*x) = (a + l) F, )
m/p) a*6* + a?0* +P(0+ft* + 2al0 (6 + ft cos x = V2.
254. Kinematic al Note. It is sometimes convenient in calculating
the velocities of points in a connected system to use the coordinates of a point
referred to axes which do not retain the same directions. In the problem of
Art. 253 we might have obtained the velocity of P relative to M by taking
as axes lines through M along and perpendicular to AB. When we wish to
calculate the velocity of a point in this way we have to attend to the fact that
the component velocities parallel to the moving axes are not the differential
coefficients (with respect to the time) of the coordinates referred to the same
axes.
253-255]
ENERGY AND MOMENTUM
257
Consider the motion of a particle P whose coordinates at time t are x', y'
referred to rectangular axes rotating in their own plane about the origin ;
let $ be the angle which the axis of x' makes with a fixed axis of x in the
plane at time «, and x, y the coordinates of the particle referred to fixed
rectangular axes of x and y. Also let u, v be component velocities of the
particle parallel to the axes of x' and y'.
We have
whence
Fig. 84.
x = x' cos 0 — y sin 0, y = y' cos 0 + x' sin 0,
I cos 0 + (x' — y' 0) sin c
x — u cos 0 — v sin 0, ?/ = v cos 0 + u sin 0 .
Also
Hence we find u = x' — y'<f>, v=y' + x'$>.
Now, if we write co for 0, « is the angular velocity of the moving axes,
and the resolved parts parallel to the moving axes of the velocity of the
particle whose coordinates are x1, y' are
x' — t&y' and i/' + ax1.
We may prove in precisely the same way that, if a, 3 are the resolved
parts of the acceleration of P parallel to the axes of x1, y', then
In the problem of Art. 253, we take axes through M along and perpen-
dicular to AB. Then the angular velocity of the moving axes is 0, and the
coordinates of P are a + l cos x and I sin x- From these the component veloci-
ties of P relative to M which were obtained otherwise in that Article might be
deduced.
255. Examples.
1. Two uniform rods AB, BO, freely jointed at B, move in one plane
under no forces ; it is required to find the motion.
We may use the figure and notation of Art. 253, taking P to be the
middle point of BC, and writing m and p for the masses of AB and DC and
2a and 21 for their lengths. We have to add to the expression given in that
17
258 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
Article for the moment of momentiiin in the motion relative to G the term
3/>£2(0+x)> and to the expression there given for the kinetic energy in the
motion relative to G the term ^pl2(^ + x)z- The energy equation and the
equation of constancy of moment of momentum determine the motion.
Note. This Example affords a good illustration of the result, to which
attention 'was called in Art. 157, to the effect that the moment of kinetic
reaction about an axis is not in general the same as the rate of increase of the
moment of momentum about that axis when the axis is in motion. In the
present Example the moment of kinetic reaction of either rod about B is zero,
because the resultant force acting on either rod (the reaction at the hinge)
passes through B ; but the moment of momentum of either rod, or of the
system, about B is not constant.
2. Two equal circular rings, each of radius a and radius of gyration k
about its centre, are freely pivoted together at a point of their circumferences,
so that their planes are parallel, and the rings are so thin they may be
regarded as in the same plane. " The system being at rest on a smooth table
with the pivot in the line of centres, the pivot is struck by a blow perpen-
dicular to the line of centres, so that the centre of mass of the system starts
to move with velocity V. Prove that the angle 0, which either radius through
the pivot makes with its initial direction at any subsequent time, is given by
the equation
3. A uniform straight tube of length 2a contains a particle of equal mass,
and, the particle being close to the middle point, the tube is started to
rotate about that point with angular velocity a>. Prove that, if there are no
external forces, the velocity of the particle relative to the tube when it leaves
it is ao)>/f .
4. Two horizontal threads are attached to a circular cylinder of negligible
mass whose axis is vertical, are coiled in opposite directions round it, and carry
equal particles which are initially at rest on two smooth horizontal planes.
One of the particles is struck at right angles to its thread so that it starts off
with velocity V and its thread begins to unwind from the cylinder. Prove
that, if the initial length of the straight portion of the thread attached to the
particle struck is c, its length r at time t is given by the equation
the cylinder being free to turn about its axis.
5. A thread is attached to a rigid cylinder of radius a and moment of inertia
/ about its axis, and carries a particle of mass m which is free to move on a
smooth plane perpendicular to the axis, while the cylinder is free to rotate
about the axis. The particle is projected on the plane at right angles to the
thread with velocity V so that the thread tends to wind up round the cylinder.
Prove that the length r of the straight portion at any subsequent time is given
by the equation
(/+ wia2) ?-2r2 = {/ + m (r2 + a2 - c2)} a2 F2,
255-257] MOVING CHAIN 259
where c is the initial value of r. Hence prove that
where M— I/a2.
6. A cone of vertical angle 2a is free to turn about its axis, and a smooth
groove is cut in its surface so as to make with the generators an angle /3.
A particle of mass m moves in the groove, and starts at a distance c from the
vertex. Prove that, if at any subsequent time the particle is at a distance r
from the vertex and the cone has turned through an angle 6, r and 6 are con-
nected by the equation
(7+ me2 sin2 a) e*e sin a cot *= (7+ mr* sin2 a),
where 7 is the moment of inertia of the cone about its axis.
7. An elliptic tube of latus rectum 2£, eccentricity e, and moment of inertia
7 about its major axis, is rotating freely about its major axis, which is fixed,
with angular velocity 12, and contains a particle of mass m which is attracted
to one focus by a force /*m/(distance)2 and is initially at rest at the end of the
major axis nearest the centre of force. Prove that, if the particle is slightly
displaced, and if pe (l+e)2 < £3Q2, it will come to rest relatively to the tube at
an end of the nearer latus rectum, provided that
8. Four equal uniform rods are freely hinged together so as to form a
rhombus of side 2a and the system rotates about one diagonal, which is fixed
in a vertical position, the highest point of the rhombus being fixed and the
lowest being free to slide on the diagonal. Find the angular velocity in the
steady motion in which each rod makes an angle a with the vertical, and prove
that the period of the small oscillations about this state of steady motion is
the same as for a simple pendulum of length
fa cos a (1 + 3 sin2 a)/(l +3 cos2 a).
MOTION OF A STRING OR CHAIN
256. Inextensible chain. When a chain moves in a straight
line, the condition of inextensibility is that all the particles of it
have at any instant the same velocity. When the chain forms a
curve, and moves so as to be in contact with a given curve, the
condition takes the form :— The velocity of a particle, resolved along
the tangent to the curve at the position of the particle, is the same
for all the particles.
257. Tension at a point of discontinuity. It often happens
that two parts of a chain move in different ways, and that portions
of the chain are continually transferred from the part that is moving
in one way to the part that is moving in the other way. The
17—2
RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
tension at the place where the motion changes is then to be
determined by the principle that the increase of momentum of a
system in any interval is equal to the impulse of the force which
acts upon it during that interval. (Art. 162.) This principle is to
be applied to a hypothetical particle of the chain, supposed to pass
during a very short interval from one state of motion to the other,
and the mass of the hypothetical particle is to be taken to be the
mass of the part of the chain which changes its motion during the
interval. (Cf. Art. 189.) This principle is illustrated in the follow-
ing problems.
It is important to observe that discontinuous motions such as
are considered here in general involve dissipation of energy.
258. Illustrative Problems.
I. A chain is coiled at the edge of a table with one end just hanging over.
It is required to find the motion.
At any time t let x be the length which has fallen over the edge, T the
tension at the edge in the falling portion. There is no tension in the part
coiled up. Let m be the mass per unit length of the chain.
During a very short interval A£ a length of the chain which can be taken
to be x&t is set in motion with velocity x, and the impulse of the force by
which it is set in motion can be taken to be T&t. Hence we have the approxi-
mate equation
T&t = mx&t . z,
which passes over into the exact equation
T=mx\
The equation of motion of the falling portion is therefore
mxx = mxg - inx2.
AVriting v for &•, this is xv -j- + v2 = gx,
or -j- (x2v2) = 2g.v2.
Integrating, and observing that v and x vanish together, we have
v* = %gx.
This equation gives the velocity of the falling portion when its length is x.
The time until the length is x is
fx ^ /6?
Jo V(jw~ v7*
The potential energy lost while the free end falls through x is \rngx*, and
the kinetic energy gained is ^mxvz or \mgx'i ; and the amount of energy dissi-
pated in the same time is
257—259]
MOVING CHAIN
261
II. A chain, one end of which is held fixed, is initially held with the other
end close to the fixed end, and the other end is then let go.
Let 21 be the length of the chain, m the mass per unit length, l + x the
length of the part that has come to rest at time t, T the tension at
its lower end. A *
The free end has fallen through 2x under gravity, so that
and the falling portion is free from tension.
During a very short interval A£ a length approximately equal
to \gt . At passes from motion with velocity gt to rest, so that an
impulse, which is approximately equal to T&t, destroys an amount
of momentum which is approximately equal to ^mg2t2At. Hence we
have the exact equation
T=
Fig. 85.
Thus the motion and the tension at any time are determined.
259. Constrained motion of a chain under gravity. We
shall suppose the chain to be in a rough tube, or in a groove cut
on a rough surface, so that the line of it is a given curve. We shall
take this curve to be in a vertical plane.
Let s be the distance, measured along the curve, of a point P
of the curve from a fixed point, p the radius of curvature of the
curve at P, <f> the angle which the normal to the curve at P makes
with the vertical. Let P' be a point near to P, for which s, <j>
become s + As, $ + A<£. Between P and P' we may imagine a
hypothetical particle of mass raA.9. Let v be the velocity of this
particle, which we may take, with
sufficient approximation, to be
directed along the tangent to
the curve at P. We may regard
the particle as moving under the
tensions T and T + AT, which
we may take to be directed along
the tangents at P and P', the
pressure of the curve, which we
may take to be directed along
the normal at P, and the friction,
which we may take to be directed
along the tangent at P. We
denote the pressure and friction by R&s and pR&s, so that R is
the pressure per unit of length, and p is the coefficient of friction.
T+AT
Fig. 86.
262 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
We form equations of motion by resolving along the tangent
and normal at P. The equations are
sin<£ + (T4- AT) cos A</> — T -
mAs . - = mg As . cos <£ + (T + AT) sin A<£ —
On dividing by As, and passing to the limit, we have the exact
equations of motion
\AjJ-
mv
= mq sin c
, U/J.
6 + ——
— aR
...d)
m —
is l
= mo cos d
ds
k+^-
R
• -v *•/>
...(2)
P
C7 7
P
If the curve is smooth we omit pR from the first equation.
If, further, .the ends of the chain are free, the velocity v can be
determined by means of the energy equation, and the tension can
be found by substituting for v in the equation (1). When the
tension is known the pressure at any point can be found from the
equation (2).
260. Examples.
1. A uniform chain of length a is laid out straight on a smooth table, and
lies in a line at right angles to the edge of the table. One end is put just over
the edge. Prove that, if the edge of the table is rounded off, so that the part
of the chain which has run off at any time is vertical, the velocity of the chain
as the last element leaves the table is \l(ag}.
2. A uniform chain of length I and weight TF is suspended by one end and
the other end is at a height k above a .smooth table. Prove that, if the upper
end is let go, the pressure on the table as the coil is formed increases from
3. A uniform chain AB is held with its lower end fixed at B and its upper
end A at a vertical distance above B equal to the length of the chain. The end
A is released, and at the instant when it passes B the end B is also released.
Prove that the chain becomes straight after an interval equal to three-quarters
of that in which A fell to B.
4. Two uniform chains whose masses per unit of length are »ij and rn.2 are
joined by a thread passing over a fixed smooth pulley. Initially the chains
are held up in coils and they are released simultaneously without causing any
finite impulse in the thread. Prove that, until one of the chains has become
entirely uncoiled, the thread slips over the pulley with uniform acceleration
and that the portions of the chains which have become straight increase during
the interval with uniform accelerations
and
259-261]
MOVING CHAIN
263
5. A uniform chain of length I and weight W is placed on a line of greatest
slope of a smooth plane of inclination a to the horizontal so that it just reaches
to the bottom of the plane where there is a small smooth pulley over which it
can run off. Prove that, when a length x has run off, the tension at the bottom
of the plane is
6. A uniform chain is held with its highest point on the highest generator
of a smooth horizontal circular cylinder, and lies on the cylinder in a vertical
plane, subtending an angle /3 at the centre of the circular section on which it
lies. Prove that, when the chain is let go, the lower end is the first part of it
to leave the cylinder, and that this happens when the radius drawn through
the upper end makes with the vertical an angle $ given by the equation
J/3 cos (<£ + /3) = sin /3 + sin $ - sin (<£ + /3).
261. Chain moving freely in one plane. Kinematical
equations. At any instant the chain forms a curve. Let A be
the position on this curve of a chosen particle, P that of any other
particle, and let s be the arc of the curve measured from A to P.
If the chain is inextensible we may regard s as a parameter speci-
fying the particle which is at the point P at time t. Let ^> be the
angle which the tangent at P to the curve, drawn in the sense
of increase of s, makes with a fixed axis of x in the plane ; </> is
estimated as the angle through which a line coinciding with the
axis of x must turn in the positive sense so as to coincide with the
tangent. Also let p be the radius of curvature of the curve at P.
Fig. 87.
We resolve the velocity of the particle of the chain which is
at P at time t into components u, v, of which u is directed along
the tangent to the curve at P in the sense of increase of s, and v
is directed along the normal. The sense of the normal is taken
264 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
to be such that, if the curve is described in the sense of increase
of s, the normal is drawn towards the left hand. If this sense is
O I
that of the normal drawn towards the centre of curvature, -, - is
ds
positive; otherwise, it is negative. (See Fig. 87.) The absolute
d(h 1
value of -^- , without regard to sign, is - .
OS p
Let x, y be the coordinates of P, i.e. of the position of the
particle specified by s at time t. We have the equations
dx . dy
cos <p = *- , sin q> = ^- .
ds ds
in which the differential coefficients are partial, s and t being
independent variables. From these equations we have
(Py _ d(f> da; d*x _ d<j) dy
~ ~
Further the direction cosines of the normal drawn in the sense
dy i dx
already chosen are — ^ and ^- .
J ds ds
The velocity of the particle specified by s at time t has com-
ponents u and v in the stated directions, and also has components
f/T1 riv/
=- , -J? parallel to the axes of coordinates. We have therefore the
ot ot
equations
dx dx dy dy dy dx dx dy
.y _ i 9_ 3 y C7 I «7
ds dt ds dt ' ds dt ds dt '
0. fdx\* fdy\*
Since { =- + Ur = 1, we have the equation
\08/ \dsJ
\ds
which is the same as
_ .
ds dsdt ds dsdt
d fdx dx dy dy\ d*x dx d'2y dy
Of [ . I «Z iJ I . y_ J _. f)
ds\ds dt ds dt) ds* dt ds2 dt
du dd>
or 5 v ^ = 0.
ds ds
This equation, combined with the statement that s and t are inde-
261, 262] MOVING CHAIN 265
pendent variables, expresses the condition of inextensibility of the
chain.
The angular velocity ^ , with which the tangent to the curve
at the position of the particle specified by s is turning, may be
expressed in terms of u and v. We have the equation
~r = — sin (f) =- (cos <£) + cos 0 »- (sin <£),
dd> dy d*x dx d2y
/")!* ' — — .^ «^ _t I */_
dt ds dsdt ds dsdt
= __
ds\ ds dt ds dt) ds* dt d&dt
_ d_ ( dy_ dx dx dy\ B<£ /dx dx dy dy\
~ds( fo di fodt) + ds~(dsdi + ?)sdi)'
d(b dv dd>
or zT = ^+u^-
dt ds ds
The two equations
du d(k dv dd> d(b
<JI Tl (I I ni T_ I •
d " ^ "» ^ I ** ~r\~ 'TT
us as ds ds ot
are the kinernatical conditions which must be satisfied at all points
of the chain throughout the motion.
Note. If the chain is extensible, and s0 is the natural length of the portion
of it that is contained between a chosen particle A and any other particle P,
the particle P is specified by the parameter s0) and we may take s0 and t as
independent variables. We may then prove in the same way as in the above
Article that the following kinematical equations must hold at all points of the
chain :
du 9d> 3e dv 3d) . 3d>
31 .L — U ii ' — M 4- * \ r
^ " «5 — ^ . 1 ^ ' ^ *^ — V * T e / o j i
3s0 3s0 3« 3s0 3s0 7 dt '
where € is the extension of the chain at the particle P.
262. Chain moving freely in one plane. Equations of
motion. We form the equations of motion by resolving the kinetic
reaction of a small element of the chain in the directions of the
tangent and normal to the curve which it instantaneously forms.
The component accelerations in these directions are obtained by
the method of Art. 254 in the forms
du d<f> dv .90
di~Vdi' di + Udt'
266 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
The resultant of the tensions at the ends of the element is obtained
in the same way as in Art. 259. If 8 and N denote the component
forces per unit mass applied to the chain in the directions of the
tangent and normal to the curve, the equations of motion are
m I -- + u -.
where m denotes the mass per unit of length.
263. Invariable form. Interesting cases of the motion of a chain
arise in which the shape of the curve formed by the chain is invariable, but
the chain moves along the curve. In discussing such cases it conduces to
clearness to imagine the chain to be enclosed in a fine rigid tube, of the shape
in question, and to move along the tube while the tube moves in its plane. The
velocity of any point of the tube is then determined as the velocity of a point
of a rigid body moving in two dimensions, and the velocity of any element of
the chain will be found by compounding a certain velocity w relative to the
tube with the velocity of any point of the tube. The direction of w is that of
the tangent to the line of the tube at the point, and its magnitude is variable
from point to point in accordance with the kinematical conditions.
Taking now the special case of a uniform chain moving under gravity, we
show that the chain can move steadily in the form of a common catenary, the
curve retaining its position as well as its form. The velocity w is in this case
the velocity of an element of the chain, and, with the notation of Art. 261, we
have
The kinematical conditions become
0M?_ d<p 3<£
cs ' dt 9a '
so that the chain moves uniformly along itself.
The equations of motion of Art. 262 are satisfied by
T= mgc sec <£ + mw2,
the curve being the catenary s = c tan 0.
264. Examples.
1. Prove that any curve which is a form of equilibrium for a uniform
chain under conservative forces is a form which the chain can retain when
moving uniformly along itself under the same forces, and thatjthe tension is
greater in the steady motion than in equilibrium by mwz, where m is the'mass
per unit length of the chain, and w is the velocity with which the chain moves
along itself.
262-264] MOVING CHAIN 267
2. A uniform chain moves over two smooth parallel rails distant 2a apart
at the same level and is transferred from a coil at a distance h vertically below
one rail to a coil at a distance k + b vertically below the other. Prove that
the portion between the rails can be a common catenary, provided that the
velocity of the chain along itself is \f(gb).
3. A uniform chain moves in a plane under no forces in such a way that
the curve of the chain retains an invariable form which rotates about a fixed
point in the plane with uniform angular velocity w, while the chain advances
relatively to the curve with uniform velocity V. Prove that the general (p, r)
equation of the curve must be of the form
(p + 2V/<o)r*=ap + b,
where a and b are constants.
4. A uniform chain falls in a vertical plane under gravity. Prove that the
square of the angular velocity of the tangent at any element is
1 /T _ ^T\
m \p2 ds2 J '
5. A uniform chain hangs in equilibrium over a smooth pulley ; one end
is fixed at an extremity of the horizontal diameter, and portions hang vertically
on both sides. Prove that, if the end is set free, the distance y of the lowest
point from the horizontal diameter during the first part of the motion satisfies
the equation
where I is the length of the chain and 2c is the circumference of the circle.
6. A uniform chain of length 2// and mass 2Lfj. has its ends attached to
two points A, C, and passes over a smooth peg B between A and C and in
the same horizontal line with them, the points A, B, C being so close together
that the parts of the chain between them may be considered vertical. Elastic
threads of natural lengths I and I' and moduluses X and X' are fastened to
points P and P' of the chain on opposite sides of B and their other ends are
fixed to points 0 and 0' vertically below P and P'. The system oscillates so
that the threads are always stretched, and the points P and P' are never for
any finite time at rest. Prove that the time of a complete oscillation is
7. A fine elliptic tube is constrained to rotate with uniform angular
velocity <o about its major axis which is vertical, and contains a uniform
chain whose length is equal to a quadrant of the ellipse. Prove that, if
co2 _ 4.gi^ where I is the latus rectum of the ellipse, the chain will be in stable
relative equilibrium with one end at the lowest point.
8. A rough helical tube of pitch a and radius a is placed with its axis
vertical, and a uniform chain is placed within it, the coefficient of friction
between the tube and the chain being tan a cos e. Prove that, when the chain
has fallen a vertical distance ma, its velocity is J(ag sec « sinh 2/i), where p, is
determined by the equation
cot 56 tanh p = tanh (/* sin f + \m cos a sin 2f ).
268 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
265. Initial Motion. When the chain starts from rest in a
position which is not one of equilibrium the initial velocities are
zero, and* the equations of motion are simplified by the omission of
d<f>idt. At the same time the kinematic conditions are altered in
**yn i
form. Since s«? vanishes initially, the result of differentiating
,. du d<t> .
the equation ^— = v^r- is
ds ds
&uL_dvd(f>
dsdt dt ds '
We may write the equations of motion in the form
du I dT
-~-. =o H -- ^- ,
dt m ds
~, — T _ .
dt m ds
Differentiating the first with respect to s, multiplying the second
a^
by -^ , and subtracting, we obtain an equation
d / 1 an IT ds „ ^
•i \ "5~ I -- o — ~^ ' •" ^ •
os \m os / m p2 ds ds
This equation serves to determine the initial tension at any
point of the chain. To determine the arbitrary constants which
enter into the solution of the equation we have to use the condi-
tions which hold at the ends, or at other special points, of the
chain. If, for example, one end of the chain is guided to move on
a given curve, the acceleration of the extreme particle must be
directed along the tangent to the curve.
Cases arise in which this method cannot be applied. In the case of a
heavy chain with an end which moves on a smooth straight wire, not perpen-
dicular to the tangent at the end, the equation of motion of an element at the
end, found by resolving along the wire, cannot be satisfied if the acceleration
of the element is finite (not infinite) and the tension is finite (not zero). The
conclusion in such cases must be that the chain becomes slack at the end,
and it may become slack throughout. In such cases it is usually convenient
to suppose the end of the chain to be attached to a ring which can slide on
the wire, and to take the mass of the riug, at first, to be finite ; when the
problem has been solved with this condition we can pass to the case above
described by supposing the mass of the ring to be diminished without limit.
266. Impulsive Motion. The equations of impulsive motion
of a chain which is suddenly set in motion are obtained at once
265-267] MOVING CHAIN 269
by the method of Art. 262. We have only to regard S and N as
the resolved parts of an impulse, reckoned per unit of mass, applied
to an element, and T as impulsive tension. The equations are
dT
mu = TT- +
ds
mv = T ~
ds
The kinematical conditions are the same as those which were
obtained in Art. 261 for a chain in continuous motion.
In case no impulses are applied to the chain except at its ends,
*S and N vanish, and we can eliminate u and v, obtaining an equa-
tion for T in the form
'ds \m ds J m/32
The solution of this equation subject to the given terminal
conditions gives the impulsive tension at any point of the chain.
267. Examples.
1. In the initial motion of a chain under gravity prove that the tension
satisfies the equation
2. A uniform chain hangs under gravity with its ends attached to two
rings which are free to slide on a smooth horizontal bar. Prove that, if the
rings are initially held so that the tangents to the chain just below them
make equal angles y with the horizontal, and are let go, the tension at the
lowest point is changed in the ratio 2J/' : 2 M' + M cot2 y, where M is the mass
of the chain, and M' that of either ring. [Of. Ex. 5 in Art. 207.]
3. If the ends of the chain of Ex. 2 are free to move on smooth bars along
the normals at the ends, and the chain is severed at its vertex, prove that the
tension at a point where the tangent .makes an angle 0 with the horizontal
immediately becomes
sec $ cos y/(cos y + y sin y).
4. Impulsive tensions Ta, Tp are applied at the ends of a piece of chain of
mass M hanging in the form of a common catenary with terminal tangents
inclined to the horizontal at angles a and j3. Prove that the kinetic energy
generated is
1 tana-tang p^cosa-^cos^)^ WginaOMa_^rinj8oog<9)l .
2 M a-fi i
270 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
MISCELLANEOUS EXAMPLES
1. A rod of length 2a is held in a position inclined at an angle a to the
vertical, and is then let fall on a smooth horizontal plane. Prove that, if there
is no restitution, the end of the rod which strikes the plane will leave it im-
mediately after impact provided that the height through which the rod falls
is greater than
-^g a sec a cosec2 a (1 + 3 sin2 a)2.
2. A sphere of radius a rolling on a rough table with velocity V comes to
a slit of breadth b perpendicular to its path. Prove that, if there is no restitu-
tion, the condition that it should cross the slit without jumping is
72 > l%Q.ga (1 - cos a) sin2 a (14 - 10 sin2 a)/(7 - 10 sin2 a)2,
where 6 = 2a sin a and 1 *iga cos a > F2 -f- lOga.
3. A sphere of mass m falls vertically and impinges with velocity V
against a board of mass M which is moving with velocity U on a horizontal
table. The coefficient of restitution between the sphere and the board is e,
and the friction between the board and the table can be neglected. Prove
that, if the coefficient of friction between the sphere and the board exceeds
2MU/(7M+2m) (l + e)F, the kinetic energy lost in the impact is
4. A ball is let fall upon a hoop, of which the mass is 1 jn of that of the
ball ; the hoop is suspended from a point in its circumference, about which it
can turn freely in a vertical plane. Prove that, if e is the coefficient of resti-
tution, and a the inclination to the vertical of the radius passing through the
point at which the ball strikes the hoop, the ball rebounds in a direction
making with the horizontal an angle tan~ l {(1 +§ri) tan a — e cot a}.
5. A plank of length 2a is turning about a horizontal axis through its
centre of gravity, and a particle strikes the rising half, rebounds, and strikes
the other half, the coefficient of restitution being unity. Prove that, if the
motion indefinitely repeats itself, the inclination of the plank to the horizontal
must never exceed a where / («• + 2a) tan a = ma2, / being the moment of inertia
of the plank about its axis, and m the mass of the particle.
6. Two equal rigid uniform laminae, each in the shape of an equilateral
triangle, rest with two edges in contact. They are struck at the same instant
with equal blows P in opposite directions bisecting the common edge and one
other edge of each, so that they are pressed together and begin to slide one
over the other. Find the velocity v of the point of application of either blow
resolved in its direction and prove that, if p is the coefficient of friction, the
kinetic energy generated in the system is (1 - p *J3) Pv, assuming no restitution.
7. Two lengths 2a and 26 are cut from the same uniform rod of mass M
and freely jointed at one end of each. The rods being at rest in a straight
line, an impulse MV is applied at the free end of a. Prove that the kinetic
energy when b is free is to that when the further end of b is fixed in the ratio
(4a + 36) (3a + 46)/12 (a + b}\
MISCELLANEOUS EXAMPLES 271
8. An equilateral triangle, formed ^of three equal uniform rods hinged at
their ends, is held in a vertical plane with one side horizontal and the opposite
corner downwards. Prove that, if after falling through any height the middle
point of the highest rod is suddenly stopped, the impulsive stresses at the
upper and lower hinges will be in the ratio N/13 : 1.
9. A rectangle, sides 2a and 2b, formed of four uniform rods of the
same material and section, smoothly hinged at the ends, is moving without
rotation on a smooth horizontal plane, when a side of length 2a impinges on
a small rough peg (zero restitution). Prove that, for that side to acquire the
greatest possible angular velocity, the point of impact must be at a distance
a{(3b+a)/(3b + 3a)}% from its centre. Prove also that the rectangle cannot
begin to rotate as a rigid body unless the direction of motion before impact
makes with the impinging side an angle greater than
10. Twelve equal rods each of length 2a are so jointed together that they
can be the edges of a cube, and the framework moves symmetrically through a
configuration in which each rod makes an angle & with the vertical ; prove that,
if u is the velocity of the centre of mass, the kinetic energy is |J/(f a202 + w2),
where M is the mass of the framework, and that, if the frame strikes the ground
when 0 = 0, then u is reduced in the ratio
cosec20).
11. Any number of equal uniform rods are jointed together so as to have
a common extremity and placed symmetrically so as to be generators of a cone
of vertical angle 2a ; the system falling with velocity V strikes symmetrically
a smooth fixed sphere of radius c (no restitution). Prove that the angular
velocity with which each rod begins to turn is
V (c cos a~a sin3 a)/( | a2 sin2 a + c2 cot2 a - ac sin 2a).
12. Two equal uniform rods each of length 2« are freely hinged at one
extremity, and their other extremities are connected by an inextensible thread
of length 21. The system rests on two smooth pegs distant 2c apart in a hori-
zontal line. Prove that, if the thread is severed, the initial angular acceleration
of either rod is
13. A uniform circular disk is symmetrically suspended by two elastic
cords of natural length c, inclined at an angle a to the vertical and attached
to the highest point of the disk. Prove that, if one of the cords is cut, the
initial radius of curvature of the path of the centre of the disk is
36 (6 - c)/(c sin 4a - b sin 2a),
where b is the equilibrium length of each cord.
272 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
14. A uniform rod of length 2a and weight W rests on a rough horizontal
plane with its pressure on the plane uniformly distributed. A horizontal
force P, large enough to produce motion, is suddenly applied at one end
perpendicularly to the length of the rod. Prove that the rod begins to turn
about a point distant x from its middle point, where x is the positive root of
the equation
a?-(\- 2P/M W) a?x - f Pa3//* W= 0,
and /i is the coefficient of friction.
15. A uniform circular disk (mass J/) rotates in a horizontal plane with
angular velocity to. Close round it moves a ring of mass m and radius c
rotating about its centre with angular velocity v(<a>). The ring carries a
massless smooth spoke along a radius, and a bead of mass p can move on the
spoke under the action of a force to the centre of the ring equal to ^/(distance)2,
and the bead is in relative equilibrium at a distance a from the centre. Prove
that, if a slight continuous action now begins between the disk and the ring,
of the nature of friction and proportional to the relative angular velocity, the
distance of the bead from the centre, and the angular velocity of the ring, will
at first increase, and their values after a short time t will be
and
v + 1\ (w - i/)/(?«c2 +pa2) - l\P [X (co - v)l(mci+pa*)'] [2/J/c2+ l/(
where \6 is the frictional couple when the relative angular velocity is P.
16. A series of 2n uniform equal rods, each of mass m, are hinged together
and held so that they are alternately horizontal and vertical, each vertical rod
being lower than the preceding ; the highest rod is horizontal and can turn
freely round its end which is fixed. Prove that, when the rods are let go, the
horizontal component X^r and the vertical component Y2r of the initial action
between the 2rth and the (2r+l)th rods are given by
X*= B ( -5 + 2 v'6r + C ( - 5 -
Y2r= // (- 5 +2 ^6)' + C' ( - 5 -
the constants B, C, B', C' being determined by the equations
17. A chain is formed of n equal symmetrical rods, each of length 2a and
radius of gyration k about its centre of mass. One end is fixed and the whole
is supported in a horizontal line. Prove that, if the supports are simultaneously
removed, the free end begins to move with acceleration
)n + 1sechlog(tanh"£<9)], where 6= log (a/£).
18. A particle of mass M rests on a smooth table, and is connected with a
particle of mass m by an inextensible thread passing through a hole in the
table. Prove that, if m is released from rest in a position in which its polar
MISCELLANEOUS EXAMPLES 273
coordinates are a, a referred to the hole as origin and, the vertical as initial
line, then in the initial motion
(M+m)r0 = mg cos a, a00=-g sin a,
a (J/+w) /•Oiv=3m^2sin2a, a2<90iv=#2 sin a cos a (M+6m)l(M+m).
Also prove that the initial radius of curvature of the path of m is
where £0 = rQ, j/o=a6^ -VT=?Vv-3a002, 3foiT = a0
19. A garden roller is at rest on a horizontal plane which is rough enough
to prevent slipping ; and the handle is so held that the plane through the
axis of the cylinder and the centre of mass of the handle makes an angle
a with the horizontal. Show that, if the handle is let go, the initial radius
of curvature of the path described by its centre of inertia is
cn~2 (sin2 a -f »2 cos2 a)%
where (n - 1 ) M (Kz + a2) = ma2,
c is the distance of the centre of mass of the handle from the axis of the
cylinder, m its mass, and MK2 the moment of inertia of the cylinder about its
axis, the cylinder being homogeneous and of radius a.
20. A rough plank of mass M is free to turn in a vertical plane about a
horizontal axis distant c from its centre of mass, and a uniform sphere of
mass m is placed on the plank at a distance b from the axis on the side
remote from the centre of mass, the plank being held horizontal. Prove that,
when the plank is let go, the initial radius of curvature of the path of the
centre of the sphere is 2l£>#/(5-110), where 0 = (m& — J/e)/(w6+J/a), and Mab
is the moment of inertia of the plank about the axis.
21. An elastic circular ring of which the radius when unstrained is a
rests on a smooth surface of revolution, whose axis is vertical, in the form of
a circle of radius ?•. Prove that the period of the small oscillations in which
each element moves in a vertical plane is the same as for a simple pendulum
of length I, where 1/1= sin a cos a/(r — a) — sec a/p, p being the radius of curva-
ture of the meridian curve at a point on the ring, and a the inclination of the
normal to the vertical.
22. Two equal spheres, each of radius a and moment of inertia / about
an axis through its centre, have their centres connected by an elastic thread
passing through holes in their surfaces, and are set to vibrate symmetrically,
so that the spheres turn through equal angles about their centres and the
thread remains in one plane. Prove (i) that, if in equilibrium the tension of
the thread is T, then the time of an oscillation of small amplitude is
L. M.
274 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
and (ii) that, if the natural length of the thread is 2a and X is its modulus of
elasticity, then the period of a small oscillation of amplitude a is
4 //2A /"a dd
a V W/ Jo v/(l-|sin20)'
[There are no forces besides the tension of the thread and the pressure
between the spheres.]
23. A particle is placed on one of the plane faces of a uniform gravitating
circular cylinder at a very small distance from the centre of the face ; prove
that it will make small oscillations in a period
where a, h, p are the radius of the cylinder, its height, and the density of its
material.
24. A uniform rod rests in equilibrium on a rough gravitating uniform
sphere under no forces but the attraction of the sphere. Prove that, if
slightly displaced, it will oscillate in time
where m is the mass of the sphere, a its radius, and 21 the length of
the rod.
25. A particle can move in a smooth plane tube, which can rotate about
a vertical axis in its plane. In a position of relative equilibrium <B is the
angular velocity, a the distance of the particle from the axis, p the radius of
curvature of the tube at the point occupied by the particle, and a the angle
which the normal at this point makes with the vertical. Prove that the
period of a small oscillation is
— /( p sin ° ^ r ?^ // p sin a \
<a \/ \a - p sin a cos2 a/ ' <» \/ \a + 3p sin a cos2 a/ '
according as the angular velocity is maintained constant, or the tube rotates
freely.
26. A thread of length I has its ends attached to two points distant c
apart on a vertical axis, and a bead can slide on the thread ; the system
rotates freely about the vertical axis with angular velocity o>. Prove that, if
the time of a small oscillation about a position of relative equilibrium is
where A = 2^2/«2 (I2 - c2).
27. A particle describes a circle uniformly under the influence of two
centres of force which attract inversely as the square of the distance. Prove
that the motion is stable if 3 cos dcos (p < 1, where 0, (p are the angles which
a radius of the circle subtends at the centres of force.
MISCELLANEOUS EXAMPLES 275
28. A uniform rod of length 26 can slide with its ends on a smooth
vertical circular wire of radius a and the wire is made to rotate about a
vertical diameter with uniform angular velocity o>. Prove that the lowest
horizontal position is stable if
29. One end of a rigid uniform rod of length 2a formed of gravitating
matter is constrained to move uniformly in a circle of radius c with angular
velocity <o, and the rod is attracted to a fixed particle of mass m at the centre
of the circle. Prove that the rod can move steadily projecting inwards
towards the centre, and that this steady motion is stable if
ym > 6>2c (c — 2a)2.
30. A bead is free to slide on a rod of negligible mass whose ends slide
without friction on a fixed circle. Prove that, if there are no external forces,
the bead moves relatively to the rod as if repelled from the middle point
with a force varying inversely as the cube of the distance.
31. Two equal uniform rods AB, BC each of mass m and length 2a are
freely jointed at B and have their middle points joined by an elastic string,
and the system moves in one plane under no forces. Prove that, if 6 is the
angle between the string and either rod at any time, 0 the angle which the
string makes with a fixed line, and V the potential energy of the stretched
string, then throughout the motion
( J -f cos2 6} <j> = const.,
ma2 {( J + cos2 <9) 02 + ( J + sin2 ff) <92} + V= const.
32. Two equal uniform rods AC, CB, hinged at (7, and having their
extremities A, B connected by a thread so that ACB is a right angle, are
revolving in their own plane with uniform angular velocity about the angle
A which is fixed. Prove that, if the thread is severed, the reaction at the
hinge is instantaneously changed in the ratio J5 : 4.
33. A uniform rod of mass m, and length 2« moves at right angles to
itself on a smooth table, and impinges symmetrically on a uniform circular
disk of mass m' and radius a spinning freely about its centre. Prove that,
if there is no restitution, and the edge of the disk is rough enough to prevent
slipping, the bodies will separate after an interval in which the unmolested
disk would have turned through an angle whose circular measure is
34. A uniform cube, of mass M, and radius of gyration k about an axis
through its centre, rests on a smooth horizontal plane, and a smooth circular
groove of radius a is cut on the upper face and passes through the centre 0
of that face. A particle of mass m is projected along the groove from 0 with
18—2
276 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
velocity V. Prove that, if a<p is the arc traversed by the particle, and 6 the
angle turned through by the block in any time,
where 82 = & (M+ wi)/4ma2.
35. Two rough horizontal cylinders each of radius c are fixed with their
axes inclined to each other at an angle 2a ; and a uniform sphere of radius a
rolls between them, starting with its centre very nearly above the point of
intersection of the highest generators. Prove that the vertical velocity of its
centre in a position in which the radii to the two points of contact make
angles <p with the horizontal is
sin a cos (f> J{IOg(a + c)(l — sin <£)/(? — 5 cos2 a cos2 <£)}.
36. A particle of mass m is placed in a smooth straight tube which can
rotate in a vertical plane about its middle point, and the system starts from
rest with the tube horizontal. Prove that the angle 6 which the tube makes
with the vertical when its angular velocity is a maximum and equal to « is
given by the equation 4(mr2+-0 <»4 — 8mcfra>2cos 6 + my"2 sin2 0=0, where / is
the moment of inertia of the tube about its middle point, and r is the distance
of the particle from that point.
37. A square formed of four similar uniform rods freely jointed at their
extremities, is laid on a smooth horizontal table, one of its corners being fixed ;
show that, if angular velocities o>, a> in the plane of the table are communicated
to the rods that meet at this corner, the greatest value of the angle between
them in the subsequent motion is
lcos-l{-f(o)-a,')2/(«2+.«'2)}.
38. Two equal homogeneous cubes are moving on a smooth table with
equal and opposite velocities V in parallel lines, and impinge so that finite
portions of opposing faces come into contact ; show that, so long as they
remain in contact, the line joining their centres meets the opposing faces at
a distance x from the centres of the faces which satisfies the equation
a* (x2 +f a2) (*02 + fa2) = F%02 (a2 +^'2 - #02),
where 2a is a side of either cube, and #0 is the initial value of x.
Prove further that, if the line joining the centres at the instant of impact
cuts the opposing faces at an angle Jrr, then while the faces are in contact they
slip with uniform relative velocity, and separate after an interval (1 + *J3) a/ V
after turning through an angle
39. A string without weight is coiled round a rough horizontal uniform
solid cylinder of mass M and radius a which is free to turn about its axis.
To the free extremity of the string is attached a uniform chain of mass m
and length I ; if the chain is gathered close up and then let go, prove that the
angle 0, through which the cylinder has turned after a time t, before the chain
is fully stretched, satisfies the equation Mla6 =
MISCELLANEOUS EXAMPLES 277
40. A great length of uniform chain is coiled at the edge of a horizontal
platform, and one end is allowed to hang over until it just reaches another
platform distant h below the first. The chain then runs down under gravity.
Prove that it ultimately acquires a finite terminal velocity F, that its velocity
at time t is Ftanh (Vtjh\ and that the length of chain which has then run
down is h log cosh ( Vtjh).
41. Two buckets each of mass M are connected by a chain of negligible
mass which passes over a fixed smooth pulley. On the bottom of one of them
lies a length I of uniform chain, whose mass is pi, one end of which is attached
to a fixed point just above the bottom of the bucket. Prove that, if the system
starts to move from rest, the velocity of the bucket when there remains upon
it a length y of chain is F, where
4Mg. ZM+td
-- - --
42. Two scale-pans each of mass M are supported by a cord of negligible
mass passing over a smooth pulley, and a uniform chain of mass m and length I
is held by its upper end above one of the scale-pans so that it just reaches the
pan. Find the acceleration of the pan when a length x of chain has fallen
upon it, and prove that the whole chain will have fallen upon it after an
interval
43. A chain of length I slides from rest down a line of greatest slope
on a smooth plane of inclination a to the horizontal, the end of the chain
hanging initially just over the edge. Prove that the time of leaving the plane
is \f{ljg (1 —sin a)} log (cot Ja).
44. One end of a uniform chain of length I and mass m is fixed to a
horizontal platform of mass (2k- 1) m ; the chain passes over a smooth fixed
pulley, and is coiled on the platform. As the platform descends vertically,
the chain uncoils, rises vertically and passes over the pulley. Prove that, at
any time t before the chain is completely uncoiled, the depth x of the plat-
form satisfies an equation of the form x*=a+px+ye-wl, where a, ft y are
constants.
45. A chain whose density varies uniformly from p at one end to 3p at
the other end is placed symmetrically on a small smooth pulley and is then
let go. Prove that it leaves the pulley with velocity J J(lllg), where 21 is its
length.
46. An elastic string (modulus A, mass ma, unstretched length a) is
confined within a straight tube to one end of which it is fastened, and the
tube rotates about that end with uniform angular velocity « in a horizontal
plane. Show that the length of the string in equilibrium is
fm
where 6 = at
278 RIGID BODIES AND CONNECTED SYSTEMS [CH. IX
47. A uniform chain falls in a vertical plane with uniform acceleration
f retaining an invariable form, while the chain advances along itself with a
velocity which at any instant is the same for all points of the chain. Prove
that the angle <p which the tangent at any point of the chain makes with the
horizontal, considered as a function of the time t and of the arc s measured up
to this point from some definite point of the chain, satisfies the two partial
differential equations
j £2<i -. • j /d<i\2l . c2d> c(b cd> 32d>
) } + w TS - Tt aJr0'
_ 92^_n
cs csdt dt cs2
48. The ends of a chain of variable density are held at the same level,
and the chain hangs in the form of an arc of a circle subtending an angle
26 (< TT ) at the centre. Prove that, if equal tangential impulses are applied at
the ends, the initial normal velocities at the lowest point and at either end are
in the ratio 1 : cos Q.
49. An endless uniform chain, lying in the form of a circle, receives a
tangential pluck at one point A, which gives it an impulsive tension T0 at that
point ; prove that the impulsive tension at any point P is
T0 sinh (2rr - ff) cosech £77-,
6 being the angle which AP subtends at the centre. Prove also that P starts
to move in a direction making an angle <£ with the tangent, where
50. A uniform chain is suspended from two points in the same horizontal
line so that the tangents at the ends make angles a with the horizontal ; and
the ends can slide on fixed straight wires which are at right angles to the
tangents at the ends. Prove that, if the wire supporting one end is removed,
that end starts to move in a direction making with the horizontal an angle 0,
where
tan 6 = ( 1 + sin2 a + 2a tan a)/sin a cos a.
Prove also that the tension at the other end is diminished in the ratio
1 : l+a~l cot a.
CHAPTER Xt
THE ROTATION OF THE EARTH
268. IT is a fact of observation that there is a relative motion
of the Earth and the stars by which every star moves relatively to
the Earth continually from East to West, or, what is geometrically
the same thing, by which any part of the Earth's surface moves
relatively to the stars continually from West to East. This motion
can be precisely described by saying that the Earth rotates about its
polar axis. The time in which the Earth turns through four right
angles is called a " sidereal day." The rotation is such that, if the
polar axis is supposed to be drawn from South to North, the sense
of this axis and the sense of the rotation are related like the senses
of translation and rotation of a right-handed screw.
269. Sidereal Time and Mean Solar Time. This process of
relative rotation has for ages been accepted as a "time-measuring process,"
that it to say it has been regarded as taking place uniformly. Time measured
by this process is called " sidereal time."
Now we have said (Article 3) that the process used for measuring time
is the average rotation of the Earth relative to the Sun. To explain this
statement, consider in the first place the motion of the Sun relative to
a frame whose origin is the centre of the Earth and whose lines of reference
go out thence to stars so distant as to have no observable annual parallax.
The path and motion of the Sun relative to this frame are the same as the
motion (in a planetary orbit) of the Earth, relative to a frame whose origin is
in the Sun and whose lines of reference go out thence to the same stars
(cf. Ex. 3 of Art. 44). The Sun's path relative to this frame of Earth and
stars is very nearly the same as if his motion were an elliptic motion about
a focus at the centre of the Earth. The sense in which the Sun describes
his orbit is the same as the sense in which any particular meridian plane
of the Earth turns about the polar axis, that is to say, the Sun is always
moving from stars which have a more westerly position towards stars which
have a more easterly position in the plane of his path. The elements of the
elliptic orbit are not quite constant ; in particular the apse line has a small
progressive motion in the sense in which the orbit is described, and the line
of intersection of the plane of the orbit with the plane of the Earth's equator
t Articles in this Chapter which are marked with an asterisk (*) may be omitted
in a first reading.
280 THE ROTATION OF THE EARTH [CH. X
(known as the line of nodes) has a small progressive motion in the opposite
sense. The Sun passes the line of nodes at the Equinoxes, and the periodic
time in the orbit is a year (technically a "tropical year"). Now it is to be
observed that, relatively to a frame fixed in the Earth, the Sun makes about
365j revolutions round the Earth in a year, and the stars make about
366j revolutions, but the time of revolution of the Sun is not a constant
multiple of the time of revolution of the stars. The variability arises in
the first place from the fact that the motion of the Sun in his path, relative
to the frame of Earth and stars, is much more nearly elliptic motion about
a focus than uniform circular motion, and in the second place from the fact
that the plane of the Sun's path is inclined to the equator. To define the
measurement of time by the average rotation of the Earth relative to the
Sun, we imagine a point to move (relatively to the frame of Earth and stars)
in the Sun's path, with a uniform angular motion about the centre of the
Earth (i.e. so that the time of describing any angle is a constant multiple of
the time in which the Earth turns through the same angle), and at such
a rate as always to coincide with the Sun at the nearer apse of his path;
then we imagine a second point to move in the plane of the Earth's equator
with a uniform angular motion about the centre of the Earth, and at such
a rate as always to coincide with the first point at the node corresponding to
the Vernal Equinox. This second point is called the Mean Sun. We may
determine a frame of reference by taking the centre of the Earth as origin,
the line joining the origin to the Mean Sun as a line of reference, and the
plane through this line and the polar axis as a plane of reference. Relatively
to this frame the Earth rotates about its polar axis in an interval called
a mean solar day ; this rotation can be used instead of the rotation
relative to the stars as time-measuring process, and time so measured is
mean solar time. The unit of time is the time in which the Earth rotates
relatively to this frame through an angle equal to 1/86400 of four right angles,
and this unit is the mean solar second.
270. The law of gravitation. When we say that the Earth
is rotating, we imply that a body at rest relative to it is moving
round the polar axis. Any particle of the body is describing a circle
about a centre on the axis, and therefore has an acceleration directed
towards the centre of this circle. If we refer the motion to axes
which rotate with the Earth the particle has no such acceleration.
The specification of the acceleration of the particle, and therefore
of the forces acting on the body, depends upon the axes to which
the motion is referred.
The law of gravitation is a statement concerning the forces
that act upon the particles of bodies. It implies that the motion
is referred to some axes or other. For a complete statement of
the law the origin and axes to which the motion is referred ought
269-271] GRAVITATION 281
to be specified. In other words, the law implies that a frame of
reference has been chosen ; and a complete statement of the law
would involve the specification of this frame of reference.
When the law is applied to the motions of bodies within the
Solar System an adequate frame of reference can be specified by
the statements: (i) The origin is the centre of mass of the system,
(ii) The axes are determined by stars so distant as to have no
observable annual parallax.
Relatively to this frame the Earth as a whole has certain
motions. Of these the most conspicuous are the orbital motion
about the Sun and the rotation about the polar axis.
271. Gravity. The acceleration denoted by g, and described
as the " acceleration due to gravity," is specified by reference to
axes fixed in the Earth. It may be precisely defined as the initial
acceleration, relative to such axes, of a particle starting from rest,
relative to such axes, in a position near the Earth's surface.
This acceleration is not identical with the acceleration produced
in the particle by the field of the Earth's gravitation. The latter
is denoted by g'. (Cf. Ch. VI)
Let O denote the angular velocity of the Earth's rotation, so
that 27T/H is the number of mean solar seconds in a sidereal day.
Let p denote the distance of .a particle from the polar axis. Let/
denote the acceleration of the Earth's centre of mass referred to
the frame specified by the centre of mass of the solar system and
the "fixed" stars. The acceleration of a body, treated as a particle,
which is at rest relatively to the Earth, is compounded of the
accelerations f and pfl2; the acceleration pfl2 is directed towards
the point where the polar axis cuts a plane which is at right
angles to it and passes through the position of the particle.
Let m be the mass of the body, as determined by the law of
gravitation (Ch. VI). The forces acting upon it are the force mf
due to the field in which the Earth moves, the force mg' due to
the Earth's gravitational field, and a force W which keeps the
particle in relative equilibrium.
We disregard in this statement the difference in the values of
the intensity / of the external field at the centre and surface of
the Earth. (See Art. 274.)
282 THE ROTATION OF THE EARTH [CH. X
The direction of W is that of a plumb-line at the place; in
other words it is the "vertical" at the place. The sense of W is
upwards.
The kinetic reaction of the particle is compounded of mf in the
direction of the acceleration f and mp£l2 in the direction of the
acceleration p£l2.
Hence the resultant of W and mg' is equal to mpQ? in the
direction of the acceleration pQ2.
If the particle is released, its initial acceleration is compounded
of/, pQ? and g. The forces acting upon it are then those" specified
by mf and mg'. Hence W = mg; and the line of action of W is
directly opposed to that of the acceleration g.
In obtaining the relation W = mg in Chapter III we neglected
the rotation of the Earth. It now appears that, when g is defined
as above, the relation is unaffected by taking account of this
rotation.
272. Variation of gravity with latitude. Let I be the angle
which the vertical at a place makes with the plane of the equator.
Then I is the (Astronomical) latitude of the place.
Let \ be the angle which the direction of the Earth's gravita-
tional field at the place makes with tjie plane of the Equator.
Consider a body at rest relative to the Earth. Its kinetic
reaction consists of vectors mf, mpfl2; and the forces acting upon
the body are mf, mg', W. The directions and senses of all these
vectors have been specified.
Form an equation of motion by resolving in the direction of
the polar axis. The equation is
0 = mg' sin \ — W sin I.
Form an equation of motion by resolving parallel to the
direction of the acceleration p£l2. The equation is
mpQ? = mg' cos \ — W cos I.
Since W = mg, we have
g g ^
sin I sin \ sin (I — \)
Of the quantities in these equations g, O, I are known by
271-274] GRAVITY AND GRAVITATION 283
observation and p is known in terms of I when the figure of the
Earth is known. The equations determine X and g '.
If the Earth is regarded as spherical, and as made up of con-
centric spherical strata of equal density, the line of action of the
force mg passes through the centre, and we have
where R is the radius of the Earth, and E is its mass.
Hence
sin(7-X) J?O2 E sin X
_
sinXcosX"^"' ^"^"slnl ............
Now RWjg is a small fraction equal to ^ nearly, and therefore
I — X is a small angle, approximately equal to -^ sin I cos I radians.
This angle is called the "deviation of the plumb-line." Also g is
approximately equal to
vE
^(1- irk cos2/)-
,With the above assumptions as to the figure and constitution
of the Earth, X becomes the "geocentric" latitude of the place.
The assumptions enable us to account for the variation of g with
latitude. There is a small correction to the formula for g on account
of the spheroidal figure of the Earth.
273. Mass and weighing. When two bodies are found to be of
the same weight, by weighing them in a common balance, it is verified that
the forces required to support them in equilibrium relative to the Earth are
equal at the same place. Hence the product mg is the same for both. Now
the ratio g : g' is sin X : sin I, where I is the Astronomical latitude of the place,
and X is the angle which the direction of the Earth's gravitational field at the
place makes with the plane of the Equator. It follows that the product mg'
is the same for the two bodies. But the ratio of two masses, as determined
by the law of gravitation, is the ratio of the forces with which they are
attracted by a gravitating body when they occupy, successively, the same
position with respect to that body. Hence the masses of the two bodies, as
determined by the law of gravitation, are equal.
The determination of the mass of a body by weighing it in a common
balance may therefore be regarded as a particular case of the determination
of mass by means of mutual action, on the basis of the law of gravitation, as
was stated in Chapter VI.
274. Lunar deflexion of gravity. In the above discussion we
have treated the external field as uniform, or as having the same intensity at
the centre of mass of the Earth and at any point on its surface.
284 THE EOTATION OF THE EARTH [CH. X
The external field arises from the gravitational attractions of the Sun,
Moon and Planets. Its intensity varies slightly from centre to surface.
This variation is most marked in the case of the Moon on account of its
comparatively small distance from the Earth.
Let / denote, as before, the intensity of the external field at the Earth's
centre of mass, and let /' denote the intensity at a point on the surface.
A force compounded of mf, in the sense of /', and mf, in the sense of
/ reversed, is available for producing motion of the body m relative to the
Earth.
The effect of this force is to make the direction of the plumb-line at
a place deviate slightly from the direction which it would take if /' were
the same as/. Since the difference between / and /' arises mainly from the
attraction of the Moon, this effect is generally referred to as the " lunar
deflexion of gravity."
The direct measurement of this effect is extremely difficult*. The
theoretical value can, however, be determined. Cf. Ex. 5 in Art. 275.
The force which produces the lunar deflexion of gravity is the same as
that which produces the tides, at least in so far as these depend upon the
Moon. The force which arises, as above, from the difference between /and/'
is the tide-generating force.
275. Examples.
[In these examples the Earth is regarded as a homogeneous sphere.]
1. If the Earth were to rotate so fast that bodies at the equator had no
weight, prove that, in any latitude, the plumb-line would be parallel to the
polar axis.
2. If the acceleration due to gravity at the Poles is g0 and at the Equator
ge, prove that in (geocentric) latitude X the value of g is
and that the deviation of the plumb-line from the (geometrical) vertical is
tan"1 {(ffo-ffe} sinAcos A/(<ft)Sin2^+,<7eCos2A)}.
3. Prove that a pendulum which beats seconds at the Poles will lose
approximately 30m cos2 1 beats per minute in latitude I, where 1 + m : 1 is
the ratio of the values of g at the Poles and at the Equator.
4. A train of mass m is travelling with uniform speed v along a parallel
of latitude in latitude I. Prove that the difference between the pressures on
the rails when the train travels due East and when it travels due West is
4mvQ cos I approximately.
5. Assuming that the mass of the Moon is ^ of that of the Earth, and
that the Moon's distance is 60 times the Earth's radius, prove that, owing to
the Moon's attraction, a seconds' pendulum at the Eai-th's surface will be
losing at a rate -± fo (3 sin'2 a — 1) seconds per day, where a is the altitude of the
Moon at the place of observation.
* See G. H. Darwin, The Tides and kindred phenomena in the Solar system,
London, 1898.
274-276] MOTION RELATIVE TO THE EARTH 285
*276. Motion of a free body near the Earth's surface.
We form first the equations of motion of the body referred to axes.
fixed in the Earth. As in Art. 272 we take the Earth to be
spherical. We take the origin to be at the centre of the Earth, the
axis of z to be the polar axis (from South Pole to North Pole), the
axis of x to be the intersection of the plane of the equator and the
meridian plane near which the motion takes place, the positive
sense of the axis of x being from the centre to the meridian in
question ; also we take the axis of y to be at right angles to this
meridian plane and directed towards the East. This system is a
right-handed system. By the results of Art. 254, the component
velocities of the body parallel to these axes are not x, y, z, but.
they are
& -fly, y + flao, z,
and the component accelerations are
Hence the equations of motion of the body are
m(x- 2fly - OV) = - (ymE/R*) cos \,~\
m (y + 2fU - &y) = 0,
mz = — (ymE/R2) sin X, J
where X is the angle which the radius of the Earth drawn through
the body makes with the plane of the equator. Now, as the body
remains near a place, we may take X to be constant, and we may
in the terms containing fl2, put x = R cos X and y = 0. Then, using
equations (2) of Art. 272, we find
x — 2£ly = — gcosl,
y + 2flA = 0,
z = — g sin I.
Since these equations contain only differential coefficients of x, y, z
with respect to the time, we may, without making any alteration,
suppose the origin to be on the Earth's surface in the latitude and
longitude near which the motion takes place.
We shall now, taking the origin as just explained, transform to
the horizontal drawn southwards as axis of x, the horizontal drawn
eastwards as axis of y', and the vertical drawn upwards as axis of
z. We have
x' = x sin I — z cos I, y' = y, z' = zs'ml + x cos I
286 THE ROTATION OF THE EARTH [CH. X
We thus obtain the equations
(1),
z -
these equations determine the motion of the body relative to the
axes at the place of observation.
*277. Initial motion. Suppose the body to fall from rest
relative to the Earth. Then the initial velocities relative to the
axes at the place of observation are given by the equations
a?' = 0, y' = 0, z=0,
and we shall suppose that the initial value of the coordinate y'
is zero. The motion is determined by equations (1) of Art. 276.
Integrating the first of these, we have
£' = 2%' sin/ ........................ (1),
and integrating the third equation, we have
_/=££_ 2%' cos J ..................... (2),
where t is the time from the beginning of the motion. Substi-
tuting in the second equation, and neglecting H2;/, we have, on
integration,
y = flgt2 cos I,
so that y' = ^£lgt3cosl ........................ (3).
Substituting in equations (1) and (2), and neglecting terms of the
same order as before, we have, on integration,
x' = ac0',
where #„' and ZQ' are the initial values of x and /.
In the beginning of the motion the acceleration relative to
axes fixed on the Earth is directed vertically downwards, and it
is what we have called g. To the order of approximation here
adopted the vertical component of acceleration remains constant
throughout the motion.
It appears that the body falls a little to the East of the starting
point, the eastward deviation in a fall through a height h being
very approximately
f O \/(2A3/#) cos I
This result accords well with observed facts.
276-278] MOTION RELATIVE TO THE EARTH 287
*278. Motion of a Pendulum. Let a simple circular pendulum
of length L be free to move about its point of support, which is
fixed relatively to the Earth, and let T be the tension of the
suspending fibre.
Let of, y, z be the coordinates of the bob referred to the system
of axes described in Art. 276, the origin being at the equilibrium
position ; then the line of action of T makes with the axes angles
whose cosines are
-x'lL, -y'lL, (L-z')/L,
and we have the relation
x* + y'* + (L-zJ = L* ..................... (1).
Now the equations of motion are, by Art. 276,
mx — 2mfly' sin I = — T (x'/L), \
my" + 2mO (as sin I + z cos 1) = - T (y'/L), I ...... (2).
m'z - 2m£ly cos / = - mg + T(L- z')/L. J
We shall integrate these equations on the assumption that the
pendulum makes small oscillations. On this assumption we have
approximately
S-KaP + ML ........................ (3).
Multiply the equations (2) in order by x, y', z', and add. The
terms containing T vanish identically by (1), the terms containing
fl also vanish identically, and the equation can be integrated.
Omitting z2 in the integral equation, and substituting for z' from
(3), we have
\m (x* + y'2) = const. - \mg (x'z + y'*)/L ......... (4).
Again, multiplying the first of equations (2) by — y', and the
second by x', adding, and omitting the term in x' z', we have on
integration
x'y - y'x' = - H sin I (x2 + y'2) + const .......... (5).
Introducing polar coordinates in the horizontal plane given by
x' = T cos 6, y' = r sin 0,
from equations (4) and (5) we obtain equations of the form
and, if we put
4> ........................... (6),
288 THE ROTATION OF THE EARTH [CH. X
we shall have
=(A+ 20B sin Z) - r2 \(gjL) + O2 sin2 1],
These equations determine the motion. It is to be noticed
that r and <£ are polar coordinates referred to an initial line which
rotates about the vertical from East to West with an angular
velocity H sin I.
*279. Foucault's Pendulum. When the pendulum can turn
freely about its point of support and is set oscillating so as to pass
through its equilibrium position, the system is known as a Foucault's
Pendulum.
Since r can vanish, it follows by the second of equations (7) of
the last Article that B must vanish, and thus 0 vanishes through-
out the motion. Hence the pendulum oscillates so that its plane
of vibration turns round the vertical relatively to the Earth with
angular velocity H sin I from East to West.
The first of equations (7) of the last Article then becomes, if
we neglect C2 sin2 1 in comparison with g/L,
r2 = A- r2 (g/L),
showing that the horizontal motion in the plane of vibration is
simple harmonic motion of period 2?r \f(L/g).
If a is the amplitude of the simple harmonic motion, so that
the pendulum has no velocity in the plane of vibration when
r = a, it will not move as here described unless its angular velocity
relative to the Earth is O sin I from East to West. To start the
pendulum, therefore, it is not sufficient to hold it aside from its
equilibrium position; it must be projected at right angles to the
vertical plane containing it with velocity «O sin I. When thus set
going it moves like a simple pendulum of the same length in a
plane which turns about the vertical from East to West with
angular velocity H sin I.
This result accords well with observed facts.
*280. Examples.
1. A projectile is projected from a point on the Earth's surface with
velocity V at an elevation a in a vertical plane making an angle £ with
the meridian (East of South). Prove that after an interval t it will have
278-280] MOTION RELATIVE TO THE EARTH 289
moved southwards through #, eastwards through y, and upwards through z,
where
x = Vt cos a {cos /3 + Q.t sin I sin /3}, ^
y=Vt {cos a sin /3 - Qt (sin I cos /3 cos a + cos I sin a)} + ^Slgt3 cos £, l
z = Vt {sin a+Qt cos £ sin /3 cos a} - §gt2,
approximately, Q2y being neglected.
2. Prove that, if the bob of a pendulum of length L is let go from
a position of rest relative to the Earth when its displacement from its
equilibrium position is a, and the vertical plane through it makes an angle j8
with the meridian (East of South), its path is given approximately by the
equation
03 - 6) = a*J(Llg) sin I y (a* - r*)/r - cos - J (r/a)},
powers of LQ?jg above the first being neglected.
3. A particle is observed to move, relatively to a certain frame, with
a simple harmonic motion of period 2ir/n in a line, which turns uniformly
about the mean position of the particle in a plane fixed relatively to the
frame with angular velocity CD ; prove that the acceleration of the partfcle
when at distance r from its mean position is compounded of a radial
acceleration (%2+o>2)r. and a transverse acceleration 2a>r in the sense in
which the line turns.
L. M.
CHAPTER XI
SUMMARY AND DISCUSSION OF THE PRINCIPLES OF
DYNAMICS
GALILEO discovered by experiment that the velocity of a falling
body is proportional to the time during which it has been falling,
and he was thus led to the notion of acceleration. He recognized in
the motion of a body on a very smooth horizontal plane that a body,
which could be regarded as subject to no forces, moved uniformly
in a straight line ; and he was thus led to connect the existence of
force with the production of acceleration.
* Newton found that the notion of acceleration, thus introduced
by Galileo, availed for the description of the motions of the bodies
of the Solar System equally with the motion of falling bodies near
the Earth's surface, and he made the idea of force, as that which
produces acceleration, the cardinal notion in his philosophy. Newton
also introduced the notion of mass, as distinct from weight, and
stated that the mass of a body is the quantity of matter which it
contains. He formulated his theory in a series of definitions, in the
three celebrated Laws of Motion, which he called Axiomata sive
Leges Motus, and in the Scholia attached thereto. We give here a
translation of the three Laws of Motion :
"First Law. Every body remains in its state of rest or of uniform
" motion in a straight line, except in so far as it is compelled by
" impressed forces to change its state."
"Second Law, Change of motion is proportional to the impressed
" moving force, and takes place in the direction in which that force
" is impressed."
" Third Law. Reaction is always equal and opposite to action ;
" or the actions of two bodies one on the other are always equal and
" oppositely directed."
The definitions preceding the laws introduce the notions of mass,
and of impressed moving force as an action on a body by which its
state of motion is changed, and as proportional to what we now call
LAWS OF MOTION 291
momentum generated in a given interval. The scholia attached to
the laws contain a demonstration of the theorem of the parallelo-
gram of forces, and an account of the' determination of masses by
direct experiment with the ballistic balance. The latter is given
as a verification of the Third Law.
In the course of this book the theoretical aspect of the science
has been developed from two principles which are essentially the
same as Newton's laws of motion, but are expressed in a form that
is more convenient for application. They are
I. The kinetic reaction of a particle has the same magnitude,
direction and sense as the resultant force acting on the particle
(Art. 64).
II. The magnitude of the force exerted by one particle on
another is equal to the magnitude of the force exerted by the
second particle upon the first, the lines of action of both the forces
coincide with the line joining the particles, and the forces have
opposite senses (Art. 142).
These principles correspond precisely to the second and third
of Newton's laws. The first law may be regarded as a particular
case of the second ; for, if there is no impressed force, there is no
change of motion, and the motion goes on unchanged. In Newton's
time this particular principle was so subversive of current ideas
that it was necessary to state it explicitly.
The first step in the formulation of the principles of Mechanics*
is the recognition of the vectorial character of such quantities as
velocity and acceleration. The statement that velocity is a vector
is the proposition that is often called the "parallelogram of velo-
cities." It is not a physical law, nor is it a mathematical proposition
capable of mathematical proof from definitions, postulates and
axioms, but it is a definition arrived at by gradually increasing the
precision of a notion already formed. This notion is the notion of
velocity as rate of displacement per unit of time.
The discussion, given in many books as a "proof," by means of the motion
of a ball in a moving tube, is valuable as an illustration ; but the process that
* Discussions of the principles of Mechanics will be found in the works cited
on p. 300 below, and also in H. Hertz's Principles of Mechanics, Translation,
London, 1899.
19—2
292 SUMMARY AND DISCUSSION OF THE PRINCIPLES [CH. XI
it illustrates is not the composition of two velocities relative to the same
frame, but the composition of a velocity relative to one frame with the
velocity of that frame relative to another frame. The analytical formulation
of this latter process is very simple (see Art. 27).
We make a step which has physical significance when we recog-
nize the existence of & field of force. The establishment of this notion
was one of the services rendered to science by Galileo. He showed
that we could say of a free body near the Earth's surface that it
has such and such an acceleration, no matter how its motion is
started. In Newton's hands the principle was carried further. It
was found to be possible to say of a body anywhere in the Solar
System, and free from contact with other bodies, that it had a
definite acceleration.
It is hardly necessary to say that neither Galileo nor anyone else has ever
experimented upon a free body. Galileo found how to isolate the effect
which we now call the " acceleration due to gravity," and he demonstrated
the existence and nature of this effect conclusively.
It is inferred that there is some action of the Earth upon bodies
in its neighbourhood, or of one body of the Solar System on another,
by which the acceleration is produced. This hypothetical action is
called force.
When we draw this inference we go beyond the facts. The occurrence
of definite accelerations in definite places is a physical fact. The inference
that some " action " or "force" produces them may, or may not, he legitimate.
In so far as the analytical formulation of the facts is concerned it is un-
necessary. In our Chapter II it has not been introduced. In our Chapter IV
it is introduced merely for the purpose of stating results in the same terms
as in subsequent Chapters.
We make another step which has physical significance when we
recognize that the motion of bodies in a field of force is modified
when they are in contact with other bodies. A book placed on a
table rests on the table, instead of falling through to the floor. A
ball thrown into the air does not move in a parabolic path, but the
trajectory is steeper in falling than in rising. It does appear to be
a legitimate inference that there is an action of some sort, due to
the table, or due to the air, whereby the acceleration that a free
body would have is modified. When we infer such action we assert
the existence of force.
The existence of pressure between bodies in contact seems obvious to
common sense. Nevertheless it is to be noted that the pressure is just as
FORCE 293
much inferred from an observation about the motion of the bodies as the
action between gravitating bodies is inferred from the motions of these
bodies. Yet action at a distance appears to common sense to be absurd. We
shall make a mistake if we suppose that the existence of any action between
bodies is verified by our muscular sensations, although it was from these
sensations that the notion of such action grew up. In like manner it is not
verified, nor is its measure determined by the use of the spring balance
(Art. 58). The result that, under suitable conditions, the extension of the
spring, by a body hung on to it, is proportional to the weight of the body (as
determined by the common balance) is a fact about the elasticity of the
spring.
Another point to be noted is that the notion of force is not really
necessary to the analytical formulation of those parts of the science in which
we pay attention to the motion of one body at a time. For example, in our
Chapters III and V (as well as in II and IV), nearly all the questions dis-
cussed could be expressed without using the notion of force. We might, for
instance, discuss the motion of a particle which moves in a given field of
force, and has, in addition to the acceleration of a free body, an acceleration
directed along the tangent of its path, whatever that tangent may be, in the
sense opposite to the velocity, and proportional to a power of that velocity,
whatever the magnitude and sense of the velocity may be. We should have
the method and results of Art. 138. In the same parts of the science the
notion of mass is irrelevant. We have introduced it in Chapter III solely in
order that the statement of the results may take the same form as in the
subsequent parts of the theory.
It would appear from this discussion that the action of one body
on another is a concept — something conceived by us — in terms of
which we describe the motions of bodies. We infer the existence
of the action from observed accelerations, which we regard as pro-
duced by the actions. It would appear also that we are at liberty
to define " force " in the way that we find most convenient. We
define it as a particular measure of the action of one body on
another, and we state how it is to be measured.
The definition can be given in most precise terms when the body
acted upon can be treated as a particle. We define the magnitude
of any force, acting on a particle, as the product of the mass of the
particle and the acceleration that is produced in it by the corre-
sponding action.
The definition is incomplete until we state what the nature of
the dependence of force upon direction is to be taken to be. We
define the force acting on a particle as a vector localized at a point.
294 SUMMARY AND DISCUSSION OF THE PRINCIPLES [CH. XI
From this point of view the "parallelogram of forces" becomes part of a
conventional definition. The "proofs" and "verifications" given in most
books may be regarded as verifications that the definition is, as a matter of
fact, convenient. One way in which the definition may be arrived at has
been sketched in Art. 61.
The definition of force remains incomplete until we explain what
is meant by the " mass " of a body, or of a particle.
To do this we must introduce the Law of Reaction. As has been
explained in Chapter VI, this Law is equivalent to the statement
that the accelerations, which are produced in two bodies by their
mutual actions, have a ratio which is always the same so long as
the bodies remain the same. The reciprocal of this ratio is the ratio
of the masses of the two bodies.
There are two quite distinct sets of circumstances in which we
can observe accelerations or changes of velocity ; and in accordance
with our concept of force, these changes of velocity are regarded
as produced by mutual actions. We may consider, in the first place,
the mutual actions of the bodies and the Earth ; and we are thus
led to the mass-ratio of two bodies, as the ratio of their weights
when weighed in a common balance. In the second place, we may
let the bodies collide, and determine their mass-ratio by the method
of the ballistic balance. The fact that the result is the same,
sometimes expressed by the phrase " identity of gravitational and
inertial mass," seems to the present writer to be the central fact
of Mechanics.
As has been already pointed out, the notions of force and mass
are not essential to the analytical formulation of those parts of the
science in which we study the motion of one body at a time (the
body being treated as a particle). They are essential as soon as we
begin to discuss the motions of several bodies forming a connected
system.
In the course of this discussion we introduce two subsidiary
principles, both of which were introduced by Newton : the law of
gravitation, and the conception of a body as a system of particles.
We have already worked out in considerable detail the consequences
of these principles, when applied to bodies which may be treated
as rigid. It may be stated here that no new principle is required
for the more complete discussion of the motions of rigid bodies, or
STRESS 295
for the discussion of the motions of deformable solid bodies or of
fluids.
The conception of bodies as made up of particles, and the con-
ception of the mutual actions of bodies, as made up of forces between
particles, are, as a matter of historical fact, the two conceptions upon
which the existing science of Mechanics is based. They possess
further the advantages, (1) that it is possible to found upon them
a strictly logical deductive theory, (2) that this theory provides an
adequate abstract formulation of the rules obeyed by the motions
of the bodies of the Solar System, and of matter in bulk under
ordinary conditions. They have thus historically developed into a
scheme which successfully coordinates an immense number of dis-
connected observations concerning matters of fact. Accordingly
this theory constitutes a science — a logically valid and practically
valuable method of representing observed facts by abstract formulas.
We must be on our guard against identifying the "particles" of the
mechanical theory with the atoms and molecules of chemistry and the kinetic
theory of gases, or with the electrons and corpuscles of modern physical
speculation. The mechanical conception of the constitution of bodies is
independent of the chemical and electrical conceptions ; and the problem of
bringing the various conceptions into harmony with each other has not been
solved. There is no reason for thinking that it is incapable of solution*.
It appears to be desirable to explain how it may be possible for internal
forces between the hypothetical particles of a body, or a set of bodies, to be
adjusted so that the motion of the particles may represent the motions of the
bodies.
It has been already explained in Chapter VI how the masses of the hypo-
thetical particles can be assigned.
In the case of a free body, the external forces are gravitational attractions
between the particles of the body and the particles of other bodies, and so
they can be regarded as known.
A body which is not free is in contact with some other body. We regard
all the bodies which are thus in contact as forming a single "system of
bodies."
Let the body, or the system of bodies, be replaced by a system of particles.
The masses of the particles, and the external forces acting on them, are
known.
* See the remarks on the 'Beneke Preis-stiftung' in Gottingen Nachrichten,
1901 ("Geschaftliche Mitteilungen"), and cf. H. M. Macdonald, Electric Waves,
Appendix B, Cambridge, 1902, and J. G. Leathern, Volume and surface integrals
used in Physics, Cambridge, 1905.
296 SUMMARY AND DISCUSSION OF THE PRINCIPLES [CH. XI
To make the motion of the particles represent the motion of the body, or
system of bodies, each particle must have a suitable acceleration. Thus the
kinetic reactions of the particles can be regarded as known.
Let there be n particles in the system. The 3» components of kinetic
reaction can be regarded as given. The magnitudes of the internal forces
between them are ln(n-\} quantities. The ^n(n — 1) unknown quantities
are connected with the known quantities by 3/i equations, which are the
equations of motion of the particles.
The 3n equations are of the form
in which mlxl and Xl are known, and AY is of the form
where the angles $12, ... are those which the lines joining the particles make
with the axis of #, and FiZ denotes the force exerted on the particle m^ by the
particle ra2, and so on.
These quantities are such that, if #21 is the same as #i2, then F2i= — Fn,
and therefore the equations of the types
are satisfied.
But the equations of the types
2m.i? = 2 JT, 2m (tfz - zi/) = 2 (yZ- z T}
also are satisfied identically, since the accelerations and the external forces
are supposed to be adjusted correctly.
We conclude that, if the particles are sufficiently numerous, the \n(n— 1)
quantities /\2 can be adjusted in an infinite number of ways so that the
3n equations may be satisfied.
It appears that the forces between the hypothetical particles are largely
indeterminate. This result offers no difficulty so long as we do not attempt
actually to assign these forces. We conclude that the motion of the body, or
system of bodies, can be represented by the motion of a system of particles.
The method that is actually adopted involves a restriction upon the hypo-
thetical forces, which, nevertheless, leaves them largely indeterminate. The
method involves the introduction of the notion of stress.
Consider a body resting on a horizontal plane in the field of the Earth's
gravity. Let the body be imagined to be divided into two parts by a hori-
zontal plane. When we represent the body by a system of particles we may
suppose that none of the particles are in the plane. Consider the forces
acting upon those particles which are above the plane. Those forces which
are due to the Earth's gravity act vertically downwards. Those which are
due to the mutual gravitation between the particles below the plane and
those above it have horizontal components and vertical components, but the
vertical components are directed downwards. If these were all the internal
forces the centre of mass of the particles which are above the plane would
STRESS 297
have an acceleration, of which the vertical component would be different from
zero and would be directed downwards. Since the centre of mass of the
particles does not move, the particles below the plane must be regarded as
exerting upon those above the plane forces which, on the whole, counteract
the 'gravitational attractions, and thus the internal forces between the two
sets of particles must be regarded as consisting of other forces besides these
attractions.
Since the law of gravitation is assumed to hold for all distances that are
measurable by ordinary means (Art. 146), we must regard the additional
forces as being exerted only between particles which are very near together.
In general let a plane surface pass through a point 0 of a body, and
draw on the plane a closed curve C of area S containing the point 0. Some
of the lines of action of forces between neighbouring particles on the two
sides of the plane cross the plane within the curve C. We consider the forces
thus exerted upon the particles which lie on a chosen side of the plane. Let
£, T), g denote the sums of the components of these forces parallel to the axes.
Then £, ij, £ are the components of a vector quantity, which is called the
"resultant stress" or "resultant traction" across the area S of the plane.
The quantities g/S, rj/S, £/S are the components of a vector quantity which is
called the "average stress" or "average traction" across the area S of the
plane. We suppose that as the area S is diminished, by contracting the
curve C towards the point 0, the components of the average stress tend
to definite finite limits ; then these limits are the components of the "stress"
or the "traction" across the plane at the point 0.
Let S now denote any closed geometrical surface drawn in the body,
Xv, Yv, Zv the components of the stress or traction across the tangent plane
at any point of S. Then the part of the body within S is to be regarded
as a system of particles which move under forces, and the sums of the
components parallel to the axes, and the sums of the moments about the
axes, of the forces which arise from actions between neighbouring particles
on the two sides of S are expressed by such formulae as
jj(
(yZv-zYv)dS,
where the integration extends over the surface.
This specification of the internal forces by means of stress is found to be
adequate for the description of the motions of extended bodies.
The stress across a plane at a point of a body is a measurable quantity
which can sometimes be determined theoretically and in some cases measured
practically. The simplest examples are pressure in a fluid and tension in
a string or chain. This tension is the resultant of the tractions across a
plane which is normal to the line of the chain.
The introduction of the notion of stress carries with it a distinction
between two classes of forces : — body forces and surface tractions.
Gravitational forces are proportional to the masses of the particles on
298 SUMMARY AND DISCUSSION OF THE PRINCIPLES [CH. XI
which they act. The sum of the components, parallel to any fixed direction,
of all the gravitational forces which act upon the part of a body within any
small volume is proportional to the volume. For theoretical purposes we
regard such forces as examples of a possible class of forces which we call
" body forces." They may be specified by the force per unit of volume, or
per unit of mass.
The resultant traction across a portion of a geometrical plane, drawn
through a body, is an example of another class of forces, which we call
"surface tractions." These forces act across surfaces, and are proportional
to the areas of the surfaces across which they act, when these areas are
small enough. They may be specified by the force per unit of area, or, what
is the same thing, by the traction across a plane at a point.
In the course of this book the energy equation has been re-
garded as one of the first integrals of the equations of motion of a
conservative system. This mode of treatment appears to the writer
to be the most natural when the science is based upon Newton's
laws of motion, or any equivalent statements ; but modern Physics
would assign to the energy equation a much more important role.
This comes about through the doctrine of the conservation of
energy. The energy equation in Mechanics is seen to be but an
example of a general principle applicable to all kinds of physical
processes.
Attempts have been made to discard the notion of force, and
to develope the theory of Mechanics from the notions of mass and
energy. It has been proposed also to discard the conception of
bodies as made up of particles at the same time as the notion of
force. One difficulty in the way of this method of formulation is
the difficulty of giving any account of the retained notion of mass.
In the Newtonian Mechanics we have, on the basis of the Law
of Reaction, a clear and definite meaning for the term "mass."
Another difficulty in the way of the " energetic " method of formu-
lation is the difficulty of giving any adequate account of potential
energy, or of work. These difficulties may perhaps be overcome in
the future. In the present state of science we may make a com-
promise between the two methods, by taking the notions of kinetic
energy and work from the Newtonian system, and destroying the
scaffolding offerees and particles by which they are reached. The
masses that occur in this intermediate method of formulation are
then regarded as coefficients in the expression for the kinetic energy.
ENERGY 299
The possibility of this intermediate method depends upon an analytical
transformation of the equations of motion, as developed in accordance with
the Newtonian method. This analytical transformation proceeds by way
of generalization of the principle of virtual work. Just as all the equations
of equilibrium of a system can be deduced from an equation of the form
as has been explained in Art. 208, so all the equations of motion of the system
can be deduced from an equation of motion of the form
2 [m(^'+yy + z2')] = 2 [(X+X'} x' + (Y+ Y')y' + (Z+Z')z'], ...... (A)
which may be obtained by the method of Art. 208. The important result
is that the terms of the equations of motion which, in the Newtonian method,
represent what have been called in this book "kinetic reactions*" are ex-
pressible in terms of the kinetic energy.
To explain this statement we consider the case in which the position
of the system at any time can be expressed in terms of a finite number of
independent geometrical quantities. Let these quantities be denoted by
6, & . . . . Then the kinetic energy T can be expressed as a homogeneous
quadratic function of the corresponding velocities 0, <£, ... ; and the left-hand
member of equation (A) can be expressed in the form
(d
in which 6', <£',... represent any set of velocities with which the system
might pass through the position denoted by 0, 0, .... This result is due to
Lagrange.
It appears from this discussion that, if we can find, for any
system, an expression for the kinetic energy and an expression for
the rate at which work is done, we can obtain the equations of
motion of the system without introducing any considerations of
" forces " or " particles."
The formulation of the principles of Mechanics implies that
choice is made of the frame of reference and of the time-measuring
process. This statement remains true whether the formulation is
carried out in terms of mass and force, or in terms of kinetic energy
and work ; for the two methods require the specification of accele-
rations and velocities. When we say that a particle at a certain
place has a certain acceleration, the place and the acceleration must
be specified by reference to some frame or other, and the specifica-
tion of the acceleration involves also the use of some method or
other of measuring time. A similar statement holds for velocities.
* In some books they are called "effective forces."
300 SUMMARY AND DISCUSSION OF THE PRINCIPLES [CH. XI
For many theoretical purposes it is unnecessary to specify either
the frame of reference or the time-measuring process; it is suffi-
cient to suppose that they have been chosen. But in any problem
concerning observable motions of actual bodies, the description of
the motion is incomplete until the reference system, both for space
and time, is specified. We may ask two questions: (1) How is the
system specified ? (2) How ought the system to be specified ? It
is a little difficult to answer briefly either of these questions ; and
it is comparatively easy to answer the slightly different question :
What reference-systems are inadmissible ? The answer is that no
system ought to be admitted which conflicts with the principles
of Mechanics, or the law of gravitation, or the principle of the con-
servation of energy.
A system of reference which satisfies the conditions of this
question and answer may be described as " kinetic *." A frame of
reference which satisfies the conditions will be called a " kinetic
frame," and time measured in accordance with the conditions will
be called " kinetic time."
To illustrate this question, and the answer, let us consider the
motion of the Earth. The principles of Mechanics require that
the Earth should be regarded as a body having a certain mass and
a certain centre of mass. Observations of falling bodies and Astro-
nomical observations lead us, in accordance with the concept of
* W. H. Macaulay, in the Article ' Motion, Laws of ' in Ency. Brit. 10th Edition,
vol. 30 (1902), describes what is here called a "kinetic frame" as a "Newtonian
base." In regard to the general question of the relativity of motion, reference
should be made to Newton's original argument in the Principia, Lib. 1, ' Scholium'
attached to the 'Defim'tiones,' and to the following more recent works: — J. C.
Maxwell, Matter and Motion (London, 1882), new edition by J. Larmor (1920),
Thomson and Tait, Natural Philosophy, Part I (Cambridge, 1879), E. Mach, The
Science of Mechanics, Translation (Chicago, 1893), C. Neumann, Ueber die Prin-
cipien der Galilei-Newton 'sche Theorie (Leipzig, 1870), K. Pearson, The Grammar
of Science (London, 1900), H. Poincare", La science et I'hypothese (Paris, N.D.), the
Article by W. H. Macaulay cited above and the Article by A. Voss in Ency. d.
math. Wiss. Bd. iv, Teil 1, Art. 1 (Leipzig, 1901). In regard to the reference
system of Astronomy see the Article by E. Anding in Ency. d. math. Wiss. Bd. vi,
Teil 2, Art. 1 (Leipzig, 1905). It need hardly be said that the view adopted from
Newton by Maxwell and by Thomson and Tait, viz. that we have knowledge of
absolute direction but not of absolute position, differs from that stated in the text.
Since the question is not of practical importance, it has seemed to the present
writer to be desirable to set forth, as clearly as may be, a view which seems to him
to be logically defensible, rather than to emphasize the divergence of this view
from those held by others.
SYSTEMS OF REFERENCE 301
force, to regard the Earth as exerting forces on other bodies, and
the law of reaction states that these bodies exert forces on the
Earth, and, therefore, that the centre of mass of the Earth has
certain component accelerations. Thus we cannot choose as a frame
of reference axes fixed in the Earth, and at the same time main-
tain the law of reaction. The change from the geocentric astronomy
of Ptolemy to the heliocentric astronomy of Copernicus may be
regarded as an instance of the discarding of an unsuitable frame
of reference.
As an illustration of the restrictions limiting the choice of the
time-measuring process we may consider the forces that can affect
the rotation of the Earth. The system of Earth and Moon, with
the fluid ocean on the Earth, executes various internal relative
motions, among which the tides are conspicuous. Such internal rela-
tive motions generally involve dissipation of energy* in a system,
for they do not take place without friction. We are thus led to
expect that the kinetic energy of the Earth's rotation is being
dissipated at a finite rate, or that the period of the diurnal rota-
tion (the length of the day) is gradually increasing. On the basis
of the law of gravitation and the principle of the conservation of
energy, but without fixing beforehand what the time-measuring
process is to be, astronomers have shown that one of the inequalities
in the motion of the Moon could be explained by the supposition
that such a gradual slackening in the speed of the Earth's rotation
is taking place. This result implies that the time-measuring pro-
cess is not the rotation of the Earth, or, in other words, that sidereal
time is not kinetic time.
The result is usually stated in the form that the Earth as a
time-keeper is losing at the rate of so many seconds in a century f .
The processes by which we reach a kinetic frame of reference
and a kinetic time-measuring process are approximative. It has
always proved to be possible to correct a choice previously made
so as to harmonize the observations of the motions of actual bodies
with the principles of Mechanics — at least very approximately ;
* That is to say a change in the form of the energy by which less of it is
rendered available, as, for example, in the conversion of kinetic energy into heat.
+ The rate is variously estimated. Two estimates are 22 seconds per century
and 8-3 seconds per century. See Thomson and Tait, Nat. Phil. Part II, Appendix
G (contributed by G. H. Darwin).
302 SUMMARY AND DISCUSSION OF THE PRINCIPLES [CH. XI
there are certain small discrepancies. By means of the law of
gravitation we can determine, to a certain order of approximation,
the masses of the bodies which compose the Solar System and the
position relative to these bodies of the centre of mass of the system.
It has proved to be sufficient to take this centre of mass as origin,
and to take, as lines of reference, lines drawn to "fixed" stars which
have no appreciable proper motion or annual parallax.
In regard to the measurement of time we have no natural system
of reference such as the " fixed " stars provide for the determina-
tion of direction ; but we can proceed in a different fashion by
means of the familiar process of changing the independent variable.
Let t denote sidereal time, that is to say time determined by the
rotation of the Earth relative to the stars ; t is, of course, measured
from some particular epoch, the instant of the occurrence of some
assigned event, and we may take the interval t to denote t sidereal
days. During this interval the Earth turns through 2irt radians.
Let the Earth as a time-keeper be losing at the rate of e seconds
per day. We know that e is a very small fraction. Let a new
variable r be introduced by the equation
86400 2 '
If we measure time by r instead of t, the quantity r measures
kinetic time so far as it has been necessary as yet to determine its
measure.
This discussion suggests also a method by which we might dispense with
the " fixed " stars in the choice of a frame of reference. We may construct a
frame, of which the origin is the centre of mass of the Sun, by means of three
lines drawn from the origin. We may take these lines arbitrarily ; for
instance, we may draw two of them to the centres of mass of the Earth
and Jupiter, and the third, in a chosen sense, at right angles to the plane
of these two. This frame does not, of course, continue to be a kinetic frame ;
but we can take it to coincide with a kinetic frame at some instant. It will
then move relatively to the kinetic frame, and the kinetic frame will move
relatively to it. If the relative motion of the two frames were known, we
could determine the position of the kinetic frame in the system after a short
interval of time ; and thus we might by a continued approximation, determine
the position of the kinetic frame at any time. This method has no practical
value ; but it appears to have some theoretical interest. This interest will
be more apparent if we reflect that, according to the law of universal gravita-
tion, there are gravitational forces acting between the bodies of the Solar
System and the stars. However small the forces which thus act upon the
SYSTEMS OF REFERENCE 303
bodies of our system may be, it remains true that the centre of mass of
the system cannot, in the long run, be a proper origin for a kinetic frame.
The frame which we now adopt, with origin at the centre of mass of the Solar
System, and axes pointing to fixed stars, may be taken to coincide with a
kinetic frame at some instant. Then we are able to state that the relative
motion of the two frames is so small that it has not been detected by any
observations.
Finally it must be said that the choice of a kinetic frame and
of kinetic time, instead of any other frame and time, is a conven-
tion. We have set out to describe the motions of bodies ; and we
wish to utilize the results that have been accumulated during
three centuries by scientific investigators who, for the most part,
paid little attention to the question of systems of reference. To
achieve our object we must state, as precisely as we can, what our
system of reference is, and how actual bodies move with reference
to it. We do this when we say that the system of reference is what
we have called "kinetic," and when we explain how a kinetic frame
can be found and how kinetic time can be determined, with, at any
rate, sufficient approximation for our purpose.
APPENDIX
MEASUREMENT AND UNITS
(a) Measurement. The mathematical theory of measurement rests on
the assumed possibility of dividing an object into an integral number of
parts which are identical in respect of some property. Thus, to measure
the length of a segment of a line, we must suppose the segment divided into
a number of equal segments, where the test of equality of length is con-
gruence ; to measure the mass of a body we must suppose it capable of
division into a number of bodies of equal mass, where equality of mass is
tested by weighing ; to measure an interval of time we measure the angle
turned through by the Earth in the interval ; this requires the division of
an angle into a number of equal angles, and the test of equality of angles
is congruence.
The measurement of an object in respect of any property requires (1) a
unit or standard of comparison, and (2) a mode of referring to the standard.
The standard must be an object which possesses the property in question.
The mode of referring to the standard must be such that it determines a
positive number (integral, rational but not integral, or irrational) which is
the measure of the object in respect of the property. The number is deter-
mined by the following rules : —
(a) When the object can be divided into an integral number n of parts,
each of which is identical with the standard in respect of the property in
question, the measure of the object in respect of that property is n.
(/3) "When the object and the standard can be divided into p and q parts
respectively (p and q being integers), such that all the parts are identical in
respect of the property in question, the measure of the object in respect of
that property is the rational fraction p/q.
Here it is to be noted (1) that the rule (a) is the case of the rule (/3) for
which q = l, and (2) that in practice the integer q may be taken so large that
an integer p may be found for which the fraction pjq measures the object
within the limits of experimental error.
In the mathematical theory of measurement the case where no rational
fraction p/q can measure the object may not be so simply dismissed. It may
happen that however great q is taken there is no corresponding number p,
but that, while the fraction p/q would measure an object somewhat smaller
than that to be measured, the fraction (p + l)/q would measure an object
somewhat greater than that to be measured. When this is the case the
measure sought is an irrational number. We may in fact separate all rational
numbers into two classes — a "superior" class and an "inferior" class — so
MEASUREMENT AND UNITS 305
that all the numbers in the superior class are too large to be the measure of
the object, and all those in the inferior class are too small. Every rational
number without exception falls into one or other of the two classes, and the
separation between them is marked by an irrational number which is the
measure of the object.
Suppose, for example, that we wish to measure the diagonal of a square
whose side is the unit of length. We may separate all rational numbers
into two classes — those whose squares are greater than two, and those
whose squares are less than two. Every rational number without exception
falls into one or other of the two classes. The separation between the two
classes is marked by the irrational number *J2, and this irrational number is
the required measure.
(b) Number and Quantity. When the unit is stated the magnitude of
an object is precisely determined by its measure in terms of the unit, and
this measure is always a number. The "object" may be anything which
we can think of as measurable in respect of any property, and the phrase
"magnitude of an object" is thus coextensive in meaning with the word
" quantity." The quantity does not change when the unit chosen to measure
it changes, and thus the quantity is not identical with the number express-
ing it.
A number can express a quantity only when the unit of measurement
is stated or understood. When the unit is stated or implied the number
expresses the quantity.
Mathematical equations, and inequalities, are relations between numbers,
expressing that a certain number which has been arrived at in one way is
equal to, greater than, or less than, a certain number which has been arrived
at in another way.
Mathematical equations, and inequalities, between numbers expressing
quantities are valid expressions of relations between the quantities, as distinct
from the numbers, only if they hold good for all systems of units.
(c) Fundamental and derived Quantities. The fundamental Physical
quantities are lengths, times, and masses. In Dynamics, as considered in
this book, all the other quantities which occur are derived from these. Thus,
velocity is measured by a fraction of which the numerator is a number
expressing a length and the denominator is a number expressing an interval
of time : acceleration is measured by a fraction of which the numerator
is a number expressing a velocity and the denominator is a number ex-
pressing an interval of time ; force is measured by the product of a number
expressing a mass and a number expressing an acceleration ; and all the
other magnitudes that occur are in similar ways dependent upon lengths,
times, and masses.
(d) Dimensions. A number which expresses a quantity is said to be of
one "dimension" in that quantity. If the unit of measurement is altered
so that the new unit is a certain multiple x of the old, the number expressing
20
L. M.
306 APPENDIX
the quantity in terms of the new unit is the quotient by x of the number
expressing the quantity in terms of the old unit.
The number expressing a derived quantity is, in every case, the product of
three numbers A, B, C, of which A is a homogeneous expression of some degree
p in numbers expressing lengths, B is a homogeneous expression of some degree
q in numbers expressing intervals of time, and C is a homogeneous expression
of some degree r in numbers expressing masses. We say that the quantity
is of p dimensions in length, q dimensions in time, and r dimensions in mass.
We express this shortly by saying that the dimension symbol of the quantity
is [L]p [T]<i[M]r. The numbers p, q, r may be positive or negative, integral
or fractional, or zero.
If the units of length, time, and mass are changed so that the new units
are respectively x, y, z times the old, the measure of any quantity in terms
of the new units is obtained from its measure in terms of the old units by
dividing by xpy9zr, where [L]p [T]*[M]r is the dimension symbol of the
quantity.
The condition that a mathematical equation or inequality between numbers
expressing quantities may be a valid expression of a relation between the
quantities is that every term in it must be of the same dimensions.
(e) Physical Quantities. We give here a list showing the principal derived
quantities that occur in Dynamics and their dimension symbols.
Velocity [LJ[T\~l.
Acceleration [Z]1 [ T] ~ 2.
Momentumj
Impulse J
Moment of Momentum! .. ™ .._, rj/]1
Impulsive Couple j
Kinetic Reaction) f m,
Force J
Kinetic Energy|
Power [L^[T]~S[M]\
Density [Z]-3[Jfp.
Constant of Gravitation [L]3[T]-*[M]-1.
(/) Method of Dimensions. We can frequently determine the form of
a result by consideration of the dimensions of the quantities involved.
This will be made clear by the consideration of some examples. Thus, if
we assume that the period of oscillation of a pendulum can depend only on
its mass, its length, and the acceleration due to gravity, we can prove that
it is proportional to the square root of the length. Since the quantity to be
expressed is an interval of time its expression cannot involve any power of a
mass, and we have assumed that no mass but the mass of the body can
enter into the expression ; the period is therefore independent of the mass
of the body. Now g ha.s dimension symbol [ZJ^T7]"2, and therefore
MEASUREMENT AND UNITS 307
has dimension symbol [77]1 [L] ~ i, hence the only way in which the expression
of the period can contain the length I of the pendulum is by being pro-
portional to its square root. This argument would prove that the period is a
numerical multiple of v/(%)- Again, to take another example, consider the
ellipticity of the Earth supposed to depend on the angular velocity of
rotation o>, the mean density p, and the constant of gravitation y. The
product yp has dimension symbol [T7]"2, and thus o>2/yp is a number (angles
being measured in radians) ; the ellipticity being a number, must be a function
of w2/yp.
The method of dimensions supplies also a useful means of verification.
In any piece of mathematical reasoning where the numbers represent
quantities all the terms in each equation must be of the same dimensions.
INDEX
The numbers refer to pages
Acceleration, Definition of, 21 ; Measure-
ment of, 23; along normal to plane
curve, 33; along principal normal of
tortuous curve, 90; along normal to
surface, 121; uniform, 22; central,
40; radial and transversal, 38; re-
ferred to polar coordinates in three
dimensions, 90; initial, 187; relative
to rotating frame, 257, 285
Anding, E., 300
Angular momentum, 145
Angular velocity, 24, 205
Apses, 95
Areas, Equable description of, 37
Attractions, Theory of, 141
Atwood's machine, 68; Correction for
inertia of pulley, 216
Axes, Right-handed, 4; Principal, 206;
Rotating, 257
Ball, R. S., 218
Ballistic balance, 137
Bodies, Constitution of, 158
Boys, C. V., 141, 142
Central forces, 38, 63, 95 ; Motion under
several, 99
Central orbits, 38-48, 95
Centre, Instantaneous, 211, 229; of
mass, 143, 159 ; of oscillation, 215
Chain, Tension of, 159, 166, 297;
Motion of, 166, 259
Collision, 178, 240
Conic, Construction of, from certain
conditions, 47
Conservative forces, 81, 152 ; Motion of
a particle under, 84, 91
Constraint, Definition of, 85; one-sided,
72, 109
Coordinates, Rectangular, 4; Relative,
24, 144 ; Polar, 38, 99 ; Polar in three
dimensions, 90
Couples, Theory of, 171-175
Cox, H., 2
Curvature, Initial, 189, 248; of path of
a particle of a rigid body, 213
Curve, Motion on a plane, 32, 63, 108,
116; Motion on a tortuous, 89, 117
Cycloid, Isochronism of, 65
D'Alembert's Principle, 149
Darwin, G. H., 284, 301
Density, 139
Descent, Line of quickest, 61
Dimensions, 19, 23, 60, 66, 140 ; Theory
of, 305
Displacement, 7 ; Components of, 15
Dyne, 59
Earth, Mean density of the, 141 ; Rota-
tion of the, 279 ; Motion relative to
the, 285, 288 ; the, as a time-keeper,
301 ; Ellipticity of the, 307
Elasticity, Modulus of, 92
Elliptic motion, 41; disturbed, 101; of
two bodies, 156
Energy, Kinetic, 65, 146, 211 ; Potential,
84, 152 ; Internal, 163 ; Conservation
of, 298; Dissipation of, 108, 177, 260,
301
Energy and momentum, 91, 195, 214,
253, 255
Energy equation, 83, 153, 160
Envelopes, of trajectories, 31, 40, 50
Equations of motion, of a particle, 62;
of a system of particles, 147; of a
body in general, 159 ; of a rigid body,
213; of a chain, 262, 265; Lagrange's,
299
Equilibrium, 191
INDEX
309
Erg, 66
Extension, 92; at a point, 164
Field of force, 27
Foot-pound, 66
Force, Definition of, 57-61; Vectorial
character of, 60, 293 ; Primitive notion
of, 58; Resultant, 61; of simple har-
monic type, 93; Transmissibility of,
161
Forces, Central, 38, 63, 95, 99; Con-
servative, 81, 152 ; External, 147, 159 ;
Internal, 147; Body, 298; Effective,
299
Foucault's pendulum, 288
Frame of reference, 5, 281
Friction, 61, 67, 161; Coefficient of, 67,
162; on plane, 67; on curve, 116,
118; on surface, 120; in rolling and
sliding, 217-219; impulsive, 242
Galileo, 27, 30, 61, 64, 290, 292
Gramme, 59
Gravitation, 42, 50; Law of, 141, 280;
Constant of, 141 ; Work done by, 153 ;
Motion of two bodies under, 155
Gravity, 27, 50, 281, 286; Force of, 58;
Work done by, 66, 163 ; Free motion
under, 28 ; Corrections of, 50, 282, 283
Gyration, radius of, 207
Heat, generated in collision, 179
Hertz, H., 291
Hodograph, 128
Horsepower, 66
Huygens, Ch., 214
Impact. See Collision
Impulse, 74, 150; Internal, 150, 154,
228 ; Effect of, on elastic system, 181
Impulsive motion, 74, 150, 177, 227,
240, 244, 268
Inertia, 139; Moment of, 205; Ellipse
of, 207
Inflexions, Circle of, 213
Initial motion, 187, 229, 248, 268, 286
Inverse square, Law of, 42, 45, 48
Kepler, J., 37, 140
Kinematic conditions, 220, 263
Kinematic formulae, 17, 22, 33, 39, 89,
91, 210, 257
Kinetic energy, 65, 146, 211; Change
of, 66, 84, 153 ; Produced by impulses,
154, 228; Lost in collision, 179
Kinetic frame and kinetic time, 300
Kinetic reaction, of a particle, 62; of a
system of particles, 144; of a rigid
body, 211; Moment of, 77, 146
Lagrange, J. L., 299
Larmor, J., 300
Laws of motion, 290
Leathern, J. G., 295
Line of action, 61
Macaulay, W. H., 300
Macdonald, H. M., 295
Mach, E., 2, 300
Machines, 192, 216
Mass, Notion of, 58, 138, 294; Deter-
mination of, 58, 139, 141, 142;
Measurement of, 59, 283; Centre of,
143, 159
Mass-ratio, 138
Maxwell, J. C., 300
Measurement, Theory of, 304
Moment, of localized vector, 19, 75 ; of
momentum, 77, 145, 210; about a
moving axis, 78, 146
Momental equivalents, 207
Momentum, of a particle, 62; of a system
of particles, 143 ; of a rigid body, 210 ;
Conservation of, 75, 150; Change of,
74, 151
Neumann, C., 300
Newton, I., 114, 137, 141, 155, 178, 291,
300
Notation, for velocities, &c. , 23
OscUlations, 35, 70, 110, 193, 215, 229,
251 ; Free and forced, 94, 124
Osculating plane, of path of particle,
121
Parabolic motion, 28, 91
Parallelogram, of localized vectors, 18 ;
of velocities, 291 ; of forces, 294
Particle, Notion of, 2 ; Dynamics of a,
88, 108
Path, 17
Pearson, K., 300
Pendulum, Simple, 70, 110; Conical,
310
INDEX
73; Revolving, 112; Equivalent
simple, 194; Rigid, 214; Spherical,
253 ; Foucault's, 288 ; Period of, found
by method of dimensions, 306
Perpetual motion, 85
Planetary motion, 37, 101, 140, 154, 157
Plumb-line, 282, 283
PoincarS, H. , 300
Poisson, S. D., 240, 243
Position, Determination of, 3
Potential, 81; Potential function one-
valued, 86
Potential energy, 84, 152; Localization
of, 164; of gravitating system, 153;
due to gravity, 163; of stretched string
or spring, 163
Pound, 59 ; Force of one, 59
Poundal, 59
Power, 66, 165
Pressure, 58, 161, 297; on a curve, 63,
108, 118, 261; on a surface, 121
Problem of two bodies, 155
Projectile, 28, 123, 126
Pull, of a locomotive, 68, 218
Quantity, 305; of matter, 139
Range, of a projectile, 30, 125
Reaction, Law of, 138, 294 ; of bodies in
contact, 161 ; initial, 187
Relative motion, 24, 25, 39, 113, 115
Resistance, 108
Resisting medium, 123
Restitution, coefficient of, 178, 240
Rigid body, 159 ; Motion of, 160 ; Energy
of, 163 ; in two dimensions, 204, 214
Rolling, 162, 218, 220
Rotation, of frame, 115; of rigid body,
205 ; of the Earth, 279, 301
Rough curve, Motion on a, 116, 118
Screw, right-handed, 4, 20, 76
Second, Mean solar, 280
Seconds' pendulum, 71
Simple harmonic motion, 34 ; Composi-
tion of, 35; Production of, 92; of
pendulum, 71; of oscillating system,
194 ; Damped, 124
Sliding, 60, 162, 218
Speed, 17
Spheres, Impact of, 137, 179, 180, 240 ;
Attraction of, 141, 142
Spring, 91 ; Potential energy of, 164
Stability, of circular orbit, 98; of equili-
brium, 194; of steady motion, 253
Stress, 296; in a rod, 226
String, 91; Motion of two bodies con-
nected by a, 109 ; Potential energy of,
163
Surface, Motion on a, 118, 120
Tension, of a string or chain, 159, 166,
297 ; at a place of discontinuity, 259
Thomson and Tait, 300, 301
Thread, 91
Time, Measurement of, 2
Tisserand, F., 158
Traction, Surface, 298
Train, Motion of a, 218
Trajectory, 17
Translation and Rotation, Independence
of, 150
Tycho Brahe, 37
Uniformity of Nature, 1
Unit, of time, 2; of velocity, 19; of
acceleration, 23; of mass, 59; of
force, 59; of work, 66; of power, 66
Vectors, Definition of, 8; Composition
and Resolution of, 10; Localized, 17;
Moment of, 19, 75; Reduction of a
system of, 171-176
Velocity, Definition of, 15-19 ; Terminal,
123, 128 ; of rigid body, 209
Virtual work, 191
Voss, A., 300
Weight, 58
Work, Definition of, 65, 79 ; of internal
forces, 151, 154, 163
Work function, 80, 152
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