AN ELEMENTARY TREATISE ON THE DYNAMICS OF A PARTICLE AND OF RIGID BODIES

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v7 AN ELEMENTARY TREATISE ON THE

DYNAMICS OF A PARTICLE

AND OF

RIGID BODIES

.

BY ^.

L.

LONEY, M.A.

Professor of Mathematics at the Royal

Holloway College

(University of London), sometime Fellow of

Sidney Sussex College, Cambridge

Cambridge at the

University Press

1913

O

}

Published, November. 1909.

Second Edition, 1913.

PREFACE

TN -*-

the following work I have tried to write an elementary class-book on those parts of

Dynamics

of a Particle

and

Rigid D3'naraics which are usually read by Students attending

a course of lectures in Applied Mathematics

for

a Science or

Engineering Degi-ee, and by Junior Students

for

Mathematical

Within the

Honours. I hope I

it will

limits with

be found to be

which

fairly

it

professes to deal,

complete.

assume that the Student has previously read some such

course as

is

included in

my Elementary Dynamics.

I also

assume

that he possesses a fair working knowledge of Differential and Integral Calculus

;

the Differential Equations, with which he

will meet, are solved in the Text,

find a

summary

of the

and in an Appendix he

will

methods of solution of such equations.

In Rigid Dynamics I have chiefly confined myself to twodimensional motion, and I have omitted

all

reference to moving-

axes.

I have included in the book a large

number

of Examples,

mostly collected from University and College Examination Papers; I have verified every question, and hope that there will not

be found a large number of serious

For any

corrections, or suggestions for

errors.

improvement, I shall

be grateful. S. L.

eoyal holloway college, Englefield Green, Surrey. October 23, 1909.

LONEY.

CONTENTS DYNAMICS OF A PAETICLE CHAPTER I.

II.

III.

Fundamental Definitions and Principles Motion in a Straight Line Simple Harmooic Motion Motion under the Earth's Attraction axes are given

,

21

....

........

Uniplanar motion referred to Polar Coordinates Revolving Axes

.

.

Stability of Orbits

Uniplanar motion when the acceleration is towards a fixed centre and varies as the inverse square of the distance Kepler's

Time

Laws

of describing

Planetary Motion Disturbed Orbits

VI.

VII.

VIII.

......... .... any arc of the path

........

45 48 54 59 69

75 79 84 89 92 97 100

The Simple Pendulum Motion on a rough curve

105 Ill

Motion in a Resisting Medium Motion when the Mass varies

119 130

Oscillatory Motion

137 143

Oscillations

when

101

Medium

the Forces are Periodic

.

Motion of a pendulum in a Resisting Medium

Motion in Three Dimensions. Polar Coordinates

.

.

.

147

.

.

.

151

Accelerations in terms of

155

.

Accelerations in terms of Cartesian Coordinates

X.

33 36

Tangential and Normal Accelerations Constrained Motion Conservation of Energy

Oscillations in a Resisting

IX.

1

fixed

Central Forces Apses and Apsidal distances

V.

PAGE

10 13

Uniplanar motion where the accelerations parallel to Composition of Simple Harmonic Motions

IV.

....

The Hodograph Motion on a Revolving Cm-ve Impulsive tensions of Chains

.

.

.

164

170 173

ISO

Contents

viii

DYNAMICS OF A KIGID BODY CHAPTER XI.

Moments and Products of The Momental Ellipsoid

PAGE 185

Inertia

193 196 199

Equi-momental Systems Principal Axes

.....

XII.

D'Alembert's Principle The general equations of Motion Independence of the Motions of Translation and Rotation Impulsive Forces

204 205 208

XIII.

Motion about a Fixed Axis

213 219 230

.

.

.

.

.

.

.

.

The Compound Pendulum

.

.

Centre of Percussion

XIV.

Motion

Two

in

Dimensions.

Finite Forces

Two Dimensions Momentum in Two Dimensions

of

Varying Mass

XV. XVI.

.

,

.

.

.

.

.

.

269

.

.

274

261

.

Motion in Two Dimensions. Impulsive Forces Impact of a Rotating Sphere on the ground

.

Instantaneous Centre Composition of Angular Velocities

282 289 295

Finite Rotations

Moment

of

Momentum and

Kinetic

Energy

in

Three

Dimensions General

equations

298 of

motion

of

a

body

in

Three

Dimensions Motion of a Billiard Ball

XVII. XVIII.

Conservation of Linear and Angular Conservation of Energy

Normal Coordinates

Lagrange's Equations for Blows

XIX.

XX.

301

302

Momentum

Lagrange's Equations in Generalised Coordinates Principal or

238 240 242

.

Kinetic Energy in

Moment

211

.

.

.

.

306 313

330 338 339

Small Oscillations Initial Motions Tendency to break

346

Motion of a Top

360

Appendix on

369

Ditierential Equations

351

356

CHAPTER

I

FUNDAMENTAL DEFINITIONS AND PRINCIPLES 1.

so that,

The if

P

velocity of a point

be

its

is

the rate of

position at time

t

its

displacement,

and Q that at time

PQ

the limiting value of the quantity -ry

,

as

At

is

t

+ Lt,

made very

small, is its velocity.

Since a displacement has both magnitude and direction, the velocity possesses both also;

the latter can therefore be

represented in magnitude and direction by a straight is

line,

and

hence called a vector quantity. 2.

A

point

may have two

velocities in different directions

they may be compounded into one velocity by the following theorem known as the Parallelogram at

the same

instant;

of Velocities

If a moving point possess simultaneously and direction hy

represented in magnitude

velocities

which are

a parallelogram dratvn from a point, they are equivalent to a velocity which is represented in magnitude and direction hy the diagonal of the parallelogram passing through the point. the tiuo sides of

velocities AB, AC are equivalent to AD, where AD is the diagonal of the parallelogram of which AB, AC are adjacent sides. If BAChesi right angle and BAD = 6, then AB = ADcosd,

Thus two component

the resultant velocity

AC= AD sin 6, and a velocity v along AD is equivalent to the velocities v cos 6 along AB and v sin 6 along AG.

two component

Triangle of Velocities. If a point possess two velocities completely represented {i.e. represented in magnitude, direction

and sense) by two straight

lines

AB

and BC, their resultant

is

Dynamics of a

2

Particle

completely represented by AC. For completing the parallelogram ABGD, the velocities AB, BG are equivalent to AB, AD whose resultant is A G.

Parallelepiped of Velocities. velocities completely represented

OG

point

If a

possess three

by three straight

lines

OA,

by successive applications of the parallelogram of velocities, completely represented by OD, the diagonal of the parallelopiped of which OA, OB, OG are conOB,

their resultant

is,

terminous edges.

OA, OB and OG are the component velocities of OD. OA, OB, and OG are mutually at right angles and u, v, w are the velocities of the moving point along these directions, the Similarly If

resultant velocity

is s/ ic^

+ v^ + w'^ v, w

cosines are proportional to w,

u

V

M"-'

—

+ v^ + W

Similarly, if

along a line whose direction and are thus equal to

V ,

wu^

OD be

+

v^

w

,

and

+ w-

V 1*2

a straight line whose direction cosines

referred to three mutually perpendicular lines I,

m,

n,

then a velocity

velocities 3.

lV,mV,nV

=r.

+ v^ + w-

V along OD

is

OA, OB, OG

are

equivalent to component

along OA, OB, and

OG

Change of Velocity. Acceleration.

respectively.

If at

any instant

the velocity of a moving point be represented by OA, and at any subsequent instant by OB, and if the parallelogram OABG

be completed whose diagonal is OB, then OG ov AB represents the velocity which must be compounded with OA to give OB, i.e. it is the change in the velocity of the moving point. Acceleration is the rate of change of velocity, i.e. if OA, OB represent the velocities at times t and t + M, then the

AB

limiting value of -r—

{i.e.

the limiting value of the ratio of the

change in the velocity to the change in the time), as At becomes

moving point. As moving point may possess simuldifferent directions, and they may be

indefinitely small, is the acceleration of the

in the case of velocities, a

taneously accelerations in

compounded

into one

by a theorem known

as the Parallclogx'am

of Accelerations similar to the Parallelogram of Velocities. As also in the case of velocities an acceleration may be

resolved into two component accelerations.

Fundamental The

Definitions

and

Principles

3

results of Art. 2 are also true for accelerations as well

as velocities.

When the distance between two either in direction or in magnitude or said to have a velocity relative to the

Relative Velocity.

4.

moving points

is altering,

in both, each point

is

other.

Suppose the velocities of two moving points A and B to be represented by the two lines and Bq (which are not necessarily in the same plane), so that in the unit of time the positions of the points would change from A and B to and Q.

AP

Draw

P

BR

equal and parallel to A P. The velocity BQ is, by the Triangle of Velocities, equivalent to the velocities BR, RQ, i.e. the velocity of B is equivalent to the velocity of A together with a velocity RQ.

The

velocity of

Now

B relative to A is thus represented by RQ. RQ is equivalent to velocities RB and BQ

the velocity

(by the Triangle of Velocities), represented by BQ and PA.

Hence pounding

the velocity

of

B

i.e.

to

relative to

the absolute velocity

of

B

velocities

A

ivith

is

a

completely

obtained by com-

velocity equal

and

opposite to that of A.

Conversely, since the velocities

BR

and RQ,

velocity

BQ

is

equivalent to the

to the velocity of

A

together with the velocity of B relative to A, therefore the velocity of any point B is obtained by compounding together its velocity relative to any other point A and the velocity of A. i.e.

The same results are true for accelerations, since they also are vector quantities and therefore follow the parallelogram law. Angular velocity of a point whose motion 5. one plane. If a point

point and

Ox

is

P

in

be in motion in a plane, and if be a fixed a fixed line in the plane, the rate of increase of

1—2

Dynamics of a Particle

4 the angle

P about

xOP

per unit of time

is

called the angular velocity of

0.

Hence, if at time t the angle be 6, the angular velocity

xOP

about

C/ IS -TT

at

P at

Q be the position of the point time t+ t^t, where A* is small,

and

V the velocity of the point at

If

time

t,

then

PQ If

ZP0Q = A^,

and

OP = r, OQ = r + Ar,

then r (r

+

Ar) sin A^ = 2 APOQ =

PQ

.

F,

where OF is the perpendicular on PQ. Hence, dividing by Af, and proceeding to the limit when is very small, we have

M

upon the tangent at P to where p is the perpendicular from the path of the moving point. Hence, if to be the angular velocity, we have ?•-&) = v .p. The angular acceleration is the rate at which the angular velocity increases per unit of time, and d fv.p\

d

= dt^''^ = di[l^)' Areal velocity. The areal velocity is, similarly, the rate at is which the area XOP increases per unit of time, where It the point in which the path of P meets Ox.

X

^ ^ = Lt.

"

area

POQ

^^

,

,

=hr-.o>.

Mass and Force. Matter has been defined to be 6. that which can be perceived by the senses " or " that which

can be acted upon by, or can exert, force." space a primary conception, and hence

It is like it

is

time and

practically im-

Fundamental Definitions and Principles possible to give

it

A

a precise definition,

body

is

5

a portion of

matter bounded by surfaces.

A particle in

its

all

is

a portion of matter which

dimensions.

It

is

is

the physical

indefinitely small

correlative

of a

A body

which is incapable of any rotation, or which moves without any rotation, may for the purposes of Dynamics, be often treated as a particle. The mass of a body is the quantity of matter it contains. A force is that which changes, or tends to change, the state of rest, or uniform motion, of a body. geometrical point.

If to the

7.

same mass we apply

forces in succession,

and

they generate the same velocity in the same time, the forces are said to be equal. If the same force be applied to two different masses, and if produce in them the same velocity in the same time, the masses are said to be equal.

it

assumed that it is possible to create forces of on different occasions, e.g. that the force necessary to keep a given spiral spring stretched through the same distance is always the same when other conditions are It is here

equal

intensity

unaltered.

Hence by applying the same force number of masses each equal

obtain a

in succession

we can

to a standard unit of

mass. units of mass are used under and in different countries. The British unit of mass is called the Imperial Pound, and consists of a lump of platinum deposited in the Exchequer Practically,

8.

different

different conditions

Office.

The French, and

is

or Scientific, unit of

mass

is

called a

gramme,

the one-thousandth part of a certain quantity of platinum,

called a

Kilogramme, which

is

deposited in the Archives.

One gramme = about 15'432

grains

= -0022046 lb. One Pound = 45359 grammes. 9.

The

units of length employed are, in general, either a

foot or a centimetre.

Dy^iamics of a Farticle

6

A centimetre

is

the one-hundredth part of a metre which

= 39-37

inches

= 32809

approximately.

ft.

The unit of time is a second. 86400 seconds are equal to a mean solar day, which is the mean or average time taken by the Earth to revolve once on

The system of

its axis

with regard to the Sun.

units in which a centimetre,

gramme, and

second are respectively the units of length, mass, and time called the C.G.S. system of units.

The density

10.

of a body,

when

uniform,

is

is

the mass of a

unit volume of the body, so that, if

m is

the mass of a volume

F

then

m=

of a body whose density

is

p,

Vp.

When

the

any point of the body is equal to the limiting value of the ratio of the mass of a very small portion of the body surrounding the point to the volume of density

is

variable, its value at

that portion, so that p

=

Lt.

^

,

when

V is taken to be indefinitely small.

The weight of a body at any place is the force with which the earth attracts the body. The body is assumed to be of such finite size, compared with the Earth, that the weights of its component parts may be assumed to be in parallel directions. If m be the mass and v the velocity of a particle, its Momentum is mv and its Kinetic Energy is \mv-. The former is a vector quantity depending on the direction of the velocity. The latter does not depend on this direction and such a quantity is

called a Scalar quantity. 11.

Newton's Laws of Motion.

Law

Every body continues in its state of rest, or of I. uniform motion in a straight line, except in so far as it be compelled by impressed force to change that state.

Law

II.

The

rate of change of

to the impressed force,

momentum

is

proportional

and takes place in the direction

in

which

the force acts.

Law

III.

To every

action there

is

an equal and opposite

reaction.

These laws were first formally enunciated by Frincipia which was published in the year 1686.

Newton

in his

Fimdamental If

12.

P

mid

Definitions

Principles

7

be the force which in a particle of mass

Law

produces an acceleration /, then

P = \-r. (mv),

where \

is

m

II states that

some constant,

If the unit of force be so chosen that it shall in unit mass = mf. produce unit acceleration, this becomes If the mass be not constant we must have, instead,

P

The

unit of force, for the Foot-Pound-Second system,

called a Poundal,

and that

for the C.G.S.

system

is

called a

is

Dyne.

The

acceleration of a freely falling body at the Earth's denoted by g, which has slightly different values at In feet-second units the ^alue of g varies different points. from 32-09 to 32-25 and in the c.G.s. system from 978-10 to 13.

surface

is

98311. For the latitude of London these values are 32*2 and 981 very approximately, and in numerical calculations these are the values generally assumed. If be the weight of a mass of one pound, the previous

W

article gives that

W = 1 .g poundals, so that the

weight of a

lb.

= 32'2 =

So the weight of a gramme

A poundal are the

poundals approximately.

981 dynes nearly.

and a dyne are absolute

units, since their values

same everywhere.

Since, by the Second Law, the change of motion pro14. duced by a force is in the direction in which the force acts, we find the combined effect of a set of forces on the motion of a particle by finding the effect of each force just as if the other This forces did not exist, and then compounding these effects. is the principle known as that of the Physical Independence of Forces.

From

this principle,

Accelerations, Forces.

we can

combined with the Parallelogram of deduce the Parallelogram of

easily

Dynamics of a Pari ids

8

Impulse of a

15.

Suppose that at time

force.

value of a force, whose direction

impulse of the force in time t

constant,

is

is

is

defined to be

t

the

Then the

P.

P

I

.

dt.

Jo

From

Art. 12

it

follows that the impulse

^^^^. = r m-yat= rmvT 1

\

dt

Jo

I

= the momentum

Jo generated in the direction

of the force in time

r.

Sometimes, as in the case of blows and impacts, we have to deal with forces, which are very great and act for a very short We time, and we cannot measure the magnitude of the forces. measure the effect of such forces by the momentum each produces, or by its impulse during the time of its action.

Work.'

16.

The work done by a

force

is

equal to the

product of the force and the distance through which the point of application is moved in the direction of the force, or, by

what

is

the same thing, the product of the element of distance

described by the point of application and the resolved part of the force in the direction of this element.

where ds

is

It therefore

= JPds,

the element of the path of the point of application

of the force during the description of w^iich the force in the

direction of ds If

axes etc.

X, Y,

when

Z

its

was P. be the components of the force parallel to the

point of application

is {x, y, z),

X = P -r

so that

,

then

Lxdx + Ydy + Zdz)=j(P^ÂŁdx +... +

...)

-H^f-m-(l^h-h = the work done by the force P. The theoretical units of work are a Foot-Poundal and an Erg. The former is the work done by a poundal during a displacement of one foot in the direction of its action the latter that done by a Dyne during a similar displacement of one cm. One Foot-Poundal = 421390 Ergs nearly. One Foot-Pound is the work done in raising one pound vertically through one foot. ;

is

Fundamental

Definitions

and

Principles

9

Power. The rate of work, or Power, of an agent is the 17. work that would be done by it in a unit of time. The unit of power used by Engineers is a Horse-Power, An agent is said to be working with one Horse-Power, or h.p., when it would raise 33,000 pounds through one foot per minute. 18.

The

Potential Energy of a body due to a given system

the work the system can do on the body as

passes

of forces

is

from

present configuration to some standard configuration,

its

it

usually called the zero position.

For example, since the attraction of the Earth (considered a and density p) is known to be

as a uniform sphere of radius

7

.

—~-

.

— at

a distance x from the centre, the potential energy

of a unit particle at a distance'

y from the centre

of the Earth,

the surface of the earth being taken as the zero position. 19. From the definitions of the following phj^sical quantities terms of the units of mass, length, and time, it is clear that their dimensions are as stated.

in

Quantity

Volume Density Velocity

Acceleration

CHAPTER

II

MOTION IN A STRAIGHT LINE Let the distance of a moving point P from a fixed be ^ at any time t. Let its distance similarly at time A^ be a; + ^x, so that PQ = Ax. 20.

point f

+

tThe .

velocity of

.

Hence the

P

at time

t

PQ -~

= Limit, when

At

=

= Limit, when

A^

= 0, of -r-^ =

velocity

v=

0,

of

At At

-r-

dt

-j-.

Let the velocity of the moving point at time V

Then the

acceleration of

=

limit,

+

t

+ At

be

Av.

P at

time

when A^ =

0,

t

of -r-

At

_ dv ~di _d^x ~dt'' 21.

tion

Motion in a

straight line with constant accelera-

/.

Let X be the distance of the moving point at time a fixed point in the straight line.

t

from

Motion in a straight

§=/

Tt™ Hence, on integration,

where

A

line

is

v

11

(!>•

= -r=ft-]r A

(2),

an arbitrary constant. we have

Integrating again,

x where

B

is

= \ft' + At-^B

(3),

an arbitrary constant.

Again, on multiplying (1) by 2-77, and integrating with respect to

t,

we have v^~=(J^) =2fa^

+C

(4).

where G is an arbitrary constant. These three equations contain the solution of all questions on motion in a straight line with constant acceleration. The arbitrary constants A, B, C are determined from the initial conditions.

Suppose for example that the particle started at a distance on the straight line with velocity w in a from a fixed point a direction away from 0, and suppose that the time t is reckoned from the instant of projection. We then have that when ^ = 0, then v=u and a; = a. Hence the equations (2), (3), and (4) give u

= A,

Hence we have and

= B, and vJ'^C + ^fa. v = u +ft, £c — a = ut+ ^fP, v- = + 2f{x - a), a

u''

the three standard equations of Elementary Dynamics.

A particle moves in a straight line OA starting from rest and moving with an acceleration which is always directed and varies as the distance from 0; to find the motion. towards at any Let X be the distance OP of the particle from time t and let the acceleration at this distance be fix. The equation of motion is then 22.

at

A

;

d'x

5^ = -^"

... '^^-

Dynamics of a

12

[We have -J—

is

a negative sign on the right-hand side because

the acceleration in the direction of x increasing,

OP

the direction i.e.

whilst fix

;

Multiplying by 2

be

a,

then

is

a;

= a,

so that

is

(2).

put on the right-hand side because

clearly negative so long as

moving towards

= - fia^ + C,

| = -V;iVa?:r^'

[The negative sign is

=-M^+^-

= ^ when

-Tf

.-.

the velocity

towards 0,

and integrating, we have

-j-

di)

OA

the acceleration

is

in

i.e.

PO.]

in the direction

If

Particle

OP

is

positive

and

P

0.]

Hence, on integration, tA^/j,

_ =—

_ = cos~^ X- +

dx

[

,

I

0= cos~-

where if

a

a

+

G,,

i.e.

C,

= 0,

the time be measured from the instant

was

C,

when the

particle

at A. .'.

When

x=

acos'\/fit

the particle arrives at 0, x

velocity

= — a V/*-

The

particle

is

zero

(3), ;

and then, by

thus passes through

immediately the acceleration alters

its

(2),

the

and

direction and tends to

diminish the velocity; also the velocity

is destroyed on the was produced on the righthand side hence the particle comes to rest at a point A' such that OA and OA' are equal. It then retraces its path, passes through 0, and again is instantaneously at rest at A. The whole

left-hand side of ;

as rapidly as

it

Simple Harmonic Motion motion of the particle

13

thus an oscillation from

is

A

A' and

to

back, continually repeated over and over again.

The time from

^

to

is

obtained by putting x equal to zero

,—

This gives cos (v/ii)

in (3).

A

The time from complete

to

is

TT

0,

i.e.

= ttt

t

A' and back again,

oscillation, is four

This result

=

times

'

the time of a

i.e.

and therefore

this,

independent of the distance

a,

=

i.e.

~

27r .

is inde-

pendent of the distance from the centre at which the particle It depends solely on the quantity ft which is equal started. to the acceleration at unit distance from the centre. 23. is

called

The

Motion of the kind investigated in the previous Simple Harmonic Motion. time,

—27r-

for

,

a complete

oscillation

is

article

called

the

Periodic OA or OA', to which the particle vibrates on either side of the centre of the motion is called the Amplitude of its motion. The Frequency is the number of complete oscillations that

Time

of the motion, and the distance,

1

the particle makes in a second, and hence

The equation

24.

left-hand side of d'^x

-7-

dv

of motion

when

Periodic time

the particle

is

_ ~

VM; *

27r

on the

is

= acceleration

P'A

in the direction

= fl P'O = fJL(— X)= — .

fix.

Hence the same equation that holds on the

right

hand of

holds on the left hand also.

As

in Art. 22 it

of this equation

is

easily seen that the

most general solution

is

x

=

acoB

[\//x^

+

e]

which contains two arbitrary constants a and This gives (1)

and

(2)

-^'

=—

aVft sin

both repeat when

t

is

(Vyu,i

(1), e.

+ e)

increased by

(2).

27r

-—

,

since the sine

Dynamics of a

14

Particle

and cosine of an angle always have the same value when the angle is increased by 27r, Using the standard expression (1) for the displacement in a simple harmonic motion, the quantity e is called the Epoch, the angle V/i^ + e is called the Argument, whilst the Phase of the motion is the time that has elapsed since the particle was at its

maximum maximum

distance in the positive direction.

where Hence the phase at time at time

V/a^o

to

Clearly

a;

is

a

+ e = 0.

t

_^^t + e

^

-t-t -t\

Motion of the kind considered in this article, in which the time of falling to a given point is the same whatever be the distance through which the particle falls, is called Tautochronous.

In Art. 22 if the be projected from direction, we have 25.

initially,

particle,

A

= fjL(b'-a;%

t\//j,

= — cos~^-r+Gi, .".

From

(1),

^ t

The

t \/fJb

where

where

in the positive

h''

=

a''

+

= — cos~'

y-

— +

(1),

(7i.

= COS~^-r — COS~^j

the velocity vanishes

and then, from

same

V

(^Y =V' + fi (a' - x")

Hence

and

instead of being at rest

with velocity

(2).

when

(2),

J ix = /

particle

cos

,<^ ^

T

•

,

I.e.

^ t

=

b

then retraces

KJii

its

*

1 —1cos

/

V

path, and the motion

as in Art. 22 with h substituted for a.

is

the

Simple Harmonic Motion

15

Compounding of two simple harmonic motions of 26. the same period and in the same straight line. The most general displacements a cos

+

{nt

e)

and

h cos {nt

+

e'),

of this kind are given by

so that

x=a

cos {nt + e) + 6 cos {nt + e') = cos nt {a cos e + 6 cos e') — sin nt {a sin e + 6 sin e'). a cos e + 6cos 6' = J. cos^l Let and a sin 6 + 6 sin e' = J. sin £"1

,

^

^'

so that

^=Va^ + 62+2a6cos(e-e'), Then

a?

and tan

^ = ^^i^^±^^ a cos 6 6 cos 6 +

= J. cos {nt + E),

so that the composition of the two given motions gives a similar

motion of the same period whose amplitude and epoch are known. If we draw OA (= a) at an angle e to a fixed line, and OB {= b) at an angle e' and complete the parallelogram OA CB then by equations (1) we see that 00 represents A and that it is inclined at an angle E to the fixed line. The line representing the resultant of the two given motions is therefore the geometrical resultant of the lines representing the two component motions. So with more than two such motions of the same period.

We

27.

cannot compound two simple harmonic motions

of

different periods.

The

case

when

the periods are nearly but not quite equal,

of some considerable importance.

In

this case

we have x = a cos (ni + e) +

where

n'

Then

—n

is

=\

small,

«=a

+ e) + 6 cos ei' = \t + e.

cos {nt

where

By

6 cos {n't

e'),

[nt

+ e^'J,

the last article

+ E) — e^) = (i' + ¥ + 2ab cos [e - e' - {u -n)i] x=- Acos{nt

where

+

say.

A'^

= a- + b^ + lab

(1),

cos (e

(2),

is

Dynamics of a Particle

16 J

sin 6 + 6 sine/ E = aa cos + 6 cos

X

and

T^

tan

-.

5

e

_

a sin

ej

+ (w' — n) t]

e 4- 6 sin [e'

.

a cos 6 + 6 cos [e' + {n — n)t] The quantities A and ^ are now not constant, but they vary slowly with the time, since n' — n is very small.

The

A

greatest value of

multiple of

The

and then

tt

least value of

multiple of

value

A

is

and then

tt

At any given time

— e' — {n' — n)t = any even + &. when e — e — {n — n) t = any odd value is a — h. when

is

its

its

is

e

a

therefore the motion

may be taken

to

be

a simple harmonic motion of the same approximate period as either of the given

A

and epoch

~27r bemg °

component motions, but with

its

amplitude

minimum

gradually changing from definite

maximum

definite ,

E

to

values, the periodic times of these changes

.

—n

n

.

[The Student who

may compare

acquainted with the theory of Sound

is

the phenomenon of Beats.]

Shew that the resultant

of two simple harmonic vibrations and of equal periodic time, the amplitude of one being twice that of the other and its phase a quarter of a period in advance, is a simple harmonic vibration of amplitude J5 times that of the first and 28.

Ex.

in the

same

1.

direction

whose phase Ex.

2.

in

is

A

advance of the

first

by

tan ~ ^ 2

-^

of a period.

particle is oscillating in a straight line

force 0, towards

which when at a distance r the force

the amplitude of the oscillation

when

;

at a distance

about a centre of m nh', and a is

is

.

^

from

0, the

blow in the direction of motion which generates a If this velocity be away from 0, shew that the new amplitude

particle receives a

velocity na. is

a

/J3.

Ex.

3.

a force

A

«i/x

particle P, of

mass m, moves

Ox under which moves in the Shew that the motion of P

in a straight line

(distance) directed towards a point

Ox with constant

straight line

acceleration

a.

A

2_.

is

simple harmonic, of period

-t--

,

about a moving centre which

is

always

VM at a distance

- behind

An

A.

without weight, of which the unstretched and the modulus of elasticity is the weight of ?i ozs., is suspended by one end, and a mass of m ozs. is attached to the other shew that the

Ex.

length

4.

is

elastic string

I

;

time

of a vertical oscillation is

/ml 2n ng

V

Simple Harmonic Motion Ex.

One end

5,

of an elastic string,

and whose unstretched length is a, horizontal table and the other end is is

The

lying on the table.

extension of the string

complete oscillation

An

(

and then

+ -r-

tt

fixed to a

elasticity is X

point on a smooth

m which where the shew that the time of a

tied to a particle of

mass

particle is pulled to a distance

is b

2

is

is

17

whose modulus of

/

»

j

let

—-

go

;

.

U and 21' masses per unit of length being m and m'. It is placed in stable equilibrium over a small smooth peg and then slightly displaced. Shew that the time of a complete oscillation is Ex.

6.

endless cord consists of two portions, of lengths

respectively, knotted together, their

{m-m)g

A'

Ex. 7. Assuming that the earth attracts points inside it with a force which varies as the distance from its centre, shew that, if a straight frictionless airless tunnel be made from one point of the earth's surface to

any other

would traverse the tunnel

point, a train

in slightly less

than

three-quarters of an hour.

29.

Motion when the motion

is in

a straight line and the from a fixed point

acceleration is proportional to the distance

and is always away from Here the equation of motion is

in the straight line

0.

d'a;

dF = ^^

(!)•

Suppose the velocity of the particle at time zero.

a from The

integral of (1)

to

be zero at a distance

is

+ A,

l-Tij =fiOc-

.-.

where

= /jLa^ + A.

^ = V/.(^^-aO

(2),

the positive sign being taken in the right-hand

the velocity

is

dx

r

••

member

positive in this case. .

^^^ = jvFz9 = ^°g[^ + ^^'-«']+^.-

=

where .

.'.

^ t

.

V/i

,

log [a]

= log

+ B.

x-Y^/'aF^^^ .

since

Dynamics of a

18

.•.

X

+

'^x-

—

a^

x—Nx^-ar —

/.

Particle

=

ae*^".

/ w x^ —

•

,

x-\-

,

= ae~V^

a-

Hence, by addition,

^

As

t

=

increases, it follows

and then from

+ |,-sV-.

|,s/.-..

(3).

from (3) that x continually increases,

(2) that the velocity continually increases also.

Hence the

would continually move along the x and with continually increasing

particle

positive direction of the axis of velocity.

may

Equation (3)

be written in the form

x = a cosh and then 30.

{\/ fxt),

v=a^fM sinh

(2) gives

{"J fit).

In the previous article suppose that the particle were with velocity V; then towards the origin

initially projected

we should have

-tt

equal to

— F when x=a

We may

would be more complicated.

;

and equation

(2)

however take the most

general solution of (1) in the form

x= where C and Since,

when

i

= 0, we

a=C+D, 1

Hence •'•

Ce'^^^

+

De-^'^^

(4),

D are any constants.

/

G= ^{a -

have x

and

V\ ^^)

= a and

— = — F, this gives

dx ^

- V = '^^G-

V]iD.

D ^ ^^{a + V 1 /

^nd

-^

(4) gives

= acosh(Vy[i«)--r-sinh(\/yL4 In this case the particle

will arrive at the origin

(6).

when

Motion in a straight I.e.

gW^ =

when

«

,

,

I.e.

when

t

_—

i/i

K

1 = ^r—r-

19

line,

asjyi,

,

log

F + a V/tA j^

V=a V/w-

In the particular case when

this value of

t

is

infinity.

If therefore the particle were projected at distance a towards

the origin with the velocity

a\//ji,

it

would not arrive at the

origin until after an infinite time.

V=a ^//x

Also, putting

X=

ae""^^'^

*,

we have

in (5),

= -y- = ~ a V'/xe"'^'^

and v

'.

The particle would therefore always be travelling towards with a continually decreasing velocity, but would take an infinite time to get there.

A

31.

particle moves in a straight line

tion luhich is

always directed towards

from

the square of its distance

A, find

at rest at

Let at

P

OP

be

accelera-

if initially the particle were

x,

p

+ ,

i

be

an

varies inversely as

the motion.

P'

_

;

OA ivith

and

and

let

1

^

the acceleration of the particle

-^ in the direction

PO.

The equation

when

of motion

therefore

d^x -5-^

=

acceleration along

a

OP — — -^

Multiplying both sides by 2 -^ and integrating,

at/

where

=

—a +

(1),

we have

X

0, from the initial conditions.

2—2

is

Dynamics of a

20

Par-tide

(^) =2/z(-X a

Subtracting,

dx

--JTyJ'^^ V ax

(2).

'^

di

the negative sign being prefixed because the motion of

towards 0,

i.e.

in the direction of

Hence

To

V

f.

—.t = -\ a

j

P

is

x decreasing.

K

V

a

— X dx.

integrate the right-hand side, put x

= a cos^ Q,

and we

have

[CO

/2fi

sm

=a

=a

—

,

2a cos 6 sin B dd

.

d

[ (1

+ cos 2^) de=^a(d+h

cos~^

—

V/ a + A

= a cos-^ (1 ) +

where •••

^=

\/^

\/ax

+

-

C,

a,'2

i.e.

+

sin 2^)

+G

C,

C = 0.

[V^^^^+acos-y^^]

.(3).

Equation (2) gives the velocity at any point P of the path, and (3) gives the time from the commencement of the motion. The velocity on arriving at the origin is found, by putting «;

=

in (2), to be infinite.

Also the corresponding time, from

«

(3), "^

^^

-1 AT = -/,2^[acos..0] ^^. r

The equation of motion (1) will not hold after the particle has passed through 0; but it is clear that then the acceleration, being opposite to the direction of the velocity, will destroy the velocity, and the latter will be diminished at the same rate as was produced on the positive side of 0. The particle will by symmetry, come to rest at a point A' such It will then return, pass again that AO and OA' are equal. and come to rest at A. through it

therefore,

Motion under the EartKs Attraction The from

A

=

to

By

32.

time of the oscillation

total

27r

—

=

=

four

21

times the time

.

the consideration of Dimensions only

we can shew that the

I

time X

a and

^ fi.

Since

.

For the only quantities that can appear in the answer are

Let then the time be .^ r^ is (distance)^ , ,.

the dimensions of

Since this .-.

33.

is

/*

an

a^/x'.

whose dimensions are [Zl [TY^,

acceleration,

are [X]^

{Ty^

;

hence the dimensions of a^^i are

and -2q = \.

a time, we have p-\-Zq =

q= -- and

As an

jo

= -.

Hence the required time x

-j-.

illustration of Art. 31 let us consider the

of a particle let

a point outside

motion

towards the earth (assumed at rest) from It is shewn in treatises on Attractions that

fall

it.

the attraction on a particle outside the earth (assumed to be a homogeneous sphere), varies inversely as the square of its distance from the centre.

the earth at distance x

The

may

acceleration of a particle outside

therefore be taken to be

—

If a be the radius of the earth this quantity at the earth's

surface

is

equal to g, and hence

For a point

P

— = g, ^

i.e.

/j,

= ga^.

outside the earth the equation of motion

is

therefore

(l^^_gar df

^

a?

^'

X If the particle started from rest at a distance h from the

centre of the earth, this gives

(r:T--'(i-F)

••••

(^).

and hence the square of the velocity on reaching the surface of the earth

= 2(/afl —

-,)

(3).

Dynamics of a Particle

22

Let us now assume that there is a hole going down to the admit of the passing of the

earth's centre just sufficient to particle.

On

a particle inside the earth the attr-action can be shewn

to vary directly as the distance from the centre, so that the

acceleration at distance x from its centre

is

where

fji,-^x,

/ija

= its

value at the earth's surface =^g.

The equation therefore

of motion of the particle

inside the earth

a

d'x

and therefore

dv

a

/dx\^

a

\-^A \dtj

Now when x = a, since

when

is

= -- x" a

->r

G..

the square of the velocity

is

given by

(3),

there was no instantaneous change of velocity at the

earth's surface.

2,.(l-|) = -|.a= + a, dx\^

_ g ^)=-^' + Âťa[3-T]dtj a

On

reaching the centre of the earth the square of the

velocity is therefore

^a (3

b

A

shew particle falls towards the earth from infinity on reaching the earth is the same as it would have acquired in falling with constant acceleration g through a distance equal Ex.

34.

that

1.

;

its velocity

to the earth's radius.

Ex.

Shew

2.

that the velocity with which a body falling from infinity

reaches the surface of the earth (assumed to be a homogeneous sphere of radius 4000 miles)

is about 7 miles per second. In the case of the sun shew that it is about .360 miles per second, the radius of the sun being 440,000 miles and the distance of the earth from

it

92,500,000 miles.

Ex.

If the earth's attraction vary inversely as the square of the

3.

its centre, and g be its magnitude at the surface, the time from a height h above the surface to the surface is

distance from of falling

/a + hfa + h where a

is

.

,

/

A

,

//il

the radius of the earth and the resistance of the air

is

neglected.

Motion in a straight

and

It is clear that equations (2)

35.

be true after the particle has passed 0;

23

line

If h be small compared with a, shew that this result

is

approximately

(3) of Art. 31 cannot

on giving x negative

for

values these equations give impossible values for v and

When is

tJ^

,

the particle

i.e.

—

„

is

Now ^— means the accelera-

towards the right.

,

Hence, when P'

tion towards the positive direction of x.

the

left of 0,

t.

at P', to the left of 0, the acceleration

the equation of motion (Px

is

on

is

fl

'dt^'^x-'

giving a different solution from (2) and

The general be

tion

/x

when the

(S).

case can be easily considered.

towards

(distance)"

0.

Let the acceleraof motion

The equation

on the right hand of

particle is

is

clearly

d-x

When P'

is

on the

-y-

df

left

of 0, the equation

= acceleration

in direction

is

OA

=fiiP'or=fi{-xr. These two equations are the same

- /j,.x''= /x{- xY, i.e.

n be an odd integer, or

if

p and q

if it

if

if (

- 1)" = -

;

1,

be of the form

otherwise

it

^—-

,

where

same equation holds

are integers; in these cases the

on both sides of the origin 36.

i.e.

does not.

A small bead, of mass m, moves on a straight rough wire under of a force equal to mfi times the distance of the bead from a fixed Find the outside the wire at a perpendicular distance a from it. JEx.

the action

point

A

motion if the bead start from perpendic^dar from A upon the

Let

P be the position

rest at

a distance

c

from

the foot, 0, of the

toire.

of the bead at

OP=x

and

any time

AP=y.

t,

where

Dynamics of a Particle

24 Let

R be the normal reaction of the wire and ;xi the coefficient of friction. Resolving forces jjerpendicular to the wire, we have R=m\i.y sin

Hence the

The

friction

OPA =mfxa.

fiiR=mfifiia.

resolved part of the force wi/iy along the wire

= m/xy cos OPA = vifijs. Hence the total acceleration = /iptia-/n.r. The equation of motion is thus d'\v

^^=liHia-iix=-fi{x-iiia) so long as [If

motion

P

be to the left of

and moving towards the in the direction

= fMfiia+fi{ — x:), this is the

Integrating,

«''^

sides of 0.]

= (^^y = M[(^-Mi«)--(-^-Mi«n

therefore, as in Art. 22,

J

u.t

—^ +

= co%~'^

o=cos-i^^^^^ + e,,

where

.'.

and

OC

0=-fi{c- fiicc)'^ + G, •••

(2)

the equation of

as in the last article,

same as (1) which therefore holds on both we have

where

and

left,

is

-TY= acceleration and

(1),

P is to the right of 0.

(3) give the velocity

From

(2)

i.e.

C'i,

Ci=0.

Vm< = 003-1-:^:^^ and time

any

for

position.

when x- fiia= ±(c- fiia), x=c=OC\ and when x = - {c - 2ayi{),

the velocity vanishes

I.e.

when

i.e.

at the point C", where

and then from

(3)

0C' = c — 2afii,

the corresponding time

=-7- cos

1

^-^

= -7-

cos

"'•.(^^.

M — 1) = -7-.

(3),

Motion in a straight

Examples

line.

25

The motion now reverses and the particle comes to rest at a point C" on the right of where OC" = OC' - 2fX]a=0C- A^na. Finally, when one of the positions of instantaneous rest is at a distance which is equal to or less than /nja from 0, the particle remains at rest. For at this point the force towards the centre is less than the limiting friction and therefore only just sufficient friction will be exerted to keep

the particle at

rest.

o It will be noted that the periodic

time -r-

but the amplitude of the motion

friction,

is

is

not affected by the

altered

by

it.

Ex. A particle, of mass m, rests in equilibrium at a point N, 37. being attracted by two forces equal to m^i^ {distanceY and m^'" {distance)^ towards two fixed centres and 0'. If the particle be slightly displaced

from

and if n

iV",

be positive, shew that

it oscillates,

and find

the time of

a

small oscillation.

O

N

Let 0(y = a,

ON=d and

NO' = d', /x".C^Âť

O'

so that

= /.'".C^'''

(1),

since there is equilibrium at N. .-.

^=^=

^,

Let the particle be at a distance x from The equation of motion is then

(2).

N towards 0',

^=-^^.OF>' + ,x\PO'^=-^Âťid+xy+lx'^'{d'-x)^ If

X

is positive,

the right-hand side

is

negative

;

if j? is

(3).

negative,

it is

towards iV in either case. Expanding by the Binomial Theorem, (3) gives

positive

;

the acceleration

=

- nx

is

[/ix"e/"

~^

+ m' "<^'" ~ ^]

+ terms involving

higher powers of

^

= -n.m"-' y^"^\ -i-...by(2). If

X be

so small that its squares

and higher powers may be neglected,

this gives

df^-

^^^-

'\,.+^r-^''

Dynamics of a

26

Hence, as in Art.

If

n be

motion

is

Particle

time of a small oscillation

22, the

member

negative, the right-hand

of (4) is positive

EXAMPLES ON CHAPTER 1.

A particle moves towards a centre

at a distance

a from the centre

*/

from the centre vary as

A

—

;

^^

,

of attraction starting from rest

velocity

if its

II.

when

law of

find the

at

any distance x

force.

from rest at A and moves towards a centre of varies inversely as the distance the time to any position varies as the cube moved through, shew that the attraction towards

2.

force at

AP

and the

not one of oscillation.

particle starts

P

if

;

of^P. 3.

Prove that

its velocity

it is

move from rest so that from the commencement of

impossible for a particle to

varies as the distance described

the motion. If the velocity vary as (distance)",

shew that n cannot be greater

than 4.

A point moves in a straight line towards /

M

|(distance)^J

starting from rest at a distance

a centre of force

1 '

a from the centre of

force

;

shew that the

time of reaching a point distant h from the centre of force

and that 5.

A

its

velocity then

particle falls

is

—^

Ja^ —

6.

its velocity is infinite

A particle

moves

varying as (distance) ~ infinity to a distance a

,

V^.

from rest at a distance a from a centre of

where the acceleration at distance x

shew that

is

s^

is

nx~

is

a distance a to a distance

;

when

and that the time

in a straight line ;

"^

it

it

has taken

in falling

equal to that acquired in falling .

2a^ is

-t^=

.

under a force to a point in

shew that the velocity

-^

force,

reaches the centre

it

from rest at from rest at

Motion in a straight 7.

A particle,

whose mass

towards the origin

A

moves

particle

a distance a

-

line,

which

oscillates

it

and that

,

acted upon by a force

rest at a distance a,

is

toju (

x-\- -^ \

shew that

it will

its

an acceleration towards

equal to

x from the given point

shew that

;

—

distance

is

in a straight line with

a fixed point in the straight particle is at a distance

w,

from

27

——-

arrive at the origin in time

8.

is

if it start

;

Examples

line.

^ — ;^ when

the

from rest at between this distance and the ;

it

starts

periodic time is

9. A particle moves with an acceleration which is always towards, and equal to fi divided by the distance from, a fixed point 0. If it start from rest at a distance a from 0, shew that it will arrive at in time

a

10.

A

*/ ^

particle

Assume

.

by a

attracted

is

that

e~=^

1

dx =

^--

force to a fixed

inversely as the nth. power of the distance

;

if

point varying

the velocity acquired by

a from the centre to the velocity that would be acquired by it in falling from in falling from

an

infinite distance to a distance

is

it

equal

rest

at

a distance a to a distance j , shew that n = -.

11.

A particle

rests in equilibrium

under the attraction of two centres

of force which attract directly as the distance, their attractions per unit of

mass at unit distance being towards one of them

12.

fi

and

n'

;

the particle

shew that the time of a small

is slightly

oscillation is

displaced . ,

A uniform

chain, of length 2a, is hung over a smooth peg so that on the two sides are a + b and a-b motion is allowed shew that the chain leaves the peg at the end of time

the lengths of to ensue

;

;

it

;

- log-

v/;g

A particle

moves

°

b

an acceleration equal to in the straight line. nth power of the distance from a fixed point If it be projected towards 0, from a point at a distance a, with the velocity it would have acquired in falUng from infinity, shew that it will reach 13.

in a straight line with

/x-rthe

in time

1

n+l

\/ V

-?:

—

2fx

.

a

2

,

Dynamics of a Fartide

28

In the previous question if the particle started from rest at in time it would reach

14.

distance a, shew that

according as w

A

is

>

or

<

unity.

whose mass

is 50 lbs., is fired from a gun, 4 inches in diameter and 8 feet in length. The pressure of the powder-gas is inversely proportional to the volume behind the shot and changes from an initial value of 10 tons' weight per square inch to 1 ton wt. per sq. inch as the shot leaves the gun. Shew that the muzzle velocity of the shot is approximately 815 feet per second, having given logÂŤ 10 = 2-3026.

15.

shot,

If the Moon and Earth were at rest, shew that the least velocity 16. with which a particle could be projected from the Moon, in order to reach the Earth, is about 1^ miles per second, assuming their radii to be 1100

and 4000 miles miles,

between their centres 240,000

respectively, the distance

and the mass of the Moon to be -^ that of the Earth. oi

A

small bead can slide on a smooth wire AB, being acted upon by a force per unit of mass equal to /ii-=-the square of its distance from which is outside AB. Shew that the time of a small oscillation a point 17.

about

its

position of equilibrium

distance of 18.

-^b^ where

is

,

h

the perpendicular

is

from AB.

A solid attracting sphere,

of radius

a and mass

J/,

has a

fine hole

bored straight through its centre a particle starts from rest at a distance b from the centre of the sphere in the direction of the hole produced, and moves under the attraction of the sphere entering the hole and going through the sphere shew that the time of a complete oscillation is ;

;

^[V2a'^sin- x/g^.+Z'^cos- ^/^Waft (6 - a)] where y 19.

is

A

the constant of gravitation. circular wire

of riidius

according to the Newtonian law y

a and density p ,.

^—^g

'>

^^

attracts a particle

^^^ particle be placed on

the axis of the wire at a distance b from the centre, find its velocity when it is at any distance x. If it be placed on the axis at a small distance from the centre, shew that the time of a complete oscillation

is

a

a/

—

Motion 20.

and the from

a straight

ill

Examples

line.

29

In the preceding question if the wire repels instead of attracting, particle be placed in the plane of the wire at a small distance

shew that the time of an

its centre,

oscillation is

a/ -

2a

yp

A

21.

moves

particle

towards, and equal to

/x

an acceleration directed

in a straight line with

times the distance from, a point in the straight

and with a constant acceleration / in a direction opposite to that of initial motion shew that its time of oscillation is the same as it is

line,

its

;

when / does not

A

22.

exist,

P

particle

a force mft

was at a distance distance of

P

c

from -

line OOP being attracted by C moves along DC with was C at rest at the origin 0, and P and moving with velocity V, prove that the

moves in a straight

PC always

.

constant acceleration /.

directed towards C, whilst If initially

from at anv time

t is

+c\cos^at+-7=

sin

Vm t-

- -k--^ t\

M

Two bodies, of masses and 31', are attached to the lower end 23. an elastic string whose upper end is fixed and hang at rest M' falls off; shew that the distance of Af from the upper end of the string at time t is of

;

Wi')' where a is the unstretched length of the string, and b and c the distances and M' respectively. by which it would be stretched when supporting

M

A point is performing a simple

harmonic motion. An additional acceleration is given to the point which is very small and varies as the cube of the distance from the origin. Shew that the increase in the amplitude of the vibration is proportional to the cube of the original amplitude if the velocity at the origin is the same in the two motions. 24.

One end of a light extensible string is fastened to a fixed point 25. and the other end carries a heavy particle the string is of unstretched length a and its modulus of elasticity is n times the weight of the particle. The particle is pulled down till it is at a depth b below the fixed point and then released. Shew that it will return to this position at the end of ;

time 2 /v/

—

^ + cosec -^p + Jp- - 1

^ ng\_J,

that jo

is

,

J

where p =

a

('i

+ 1),

provided

not >\/l+4?i.

If jt)>\/i

+ 4Âť,

shew how to

find the corresponding time.

An endless elastic string, whose modulus of elasticity is X and 26. natural length is 2ttc, is placed in the form of a circle on a smooth horizontal plane and is acted upou by a force from the centre equal to

Dynamics of a

30 ft.

Particle

times the distance per unit mass of the string.

will

mean

vary harmonically about a

mass of the

string,

Examine the

An

27.

length ^

Shew

^^

,

that

its

m

where

radius is

the

assuming that 27rX>m^c.

case

when

27rX

= mfj.c.

mass

elastic string of

m

and modulus of

elasticity

X rests

imstretched in the form of a circle whose radius is a. It is now acted on by a rejDulsive force situated in its centre whose magnitude per unit mass of the string is (distance)^

Shew

that

when the

circle

next comes to rest

its

radius

is

a root of the

quadratic equation ttA

A

smooth block, of mass M, with

its upper and lower faces is free to move in a groove in a parallel plane, and mass m is attached to a fixed point in the upper face by an elastic string whose natural length is a and modulus E. If the system starts from rest with the particle on the upper face and the string stretched parallel to the groove to (n + 1) times its natural length, shew

28.

horizontal planes,

a particle of

'

that the block will perform oscillations of amplitude ^ periodic time 2

29.

a

(

tt

A particle

^

^ M+m )

^^

j^ ^j^q

Mm

+ - j */ ElM+mj' is

attached to a point in a rough plane inclined at an originally the string was unstretched and lay

angle a to the horizon

;

along a line of greatest slope the coefficient of friction

is

;

shew that the

particle will oscillate only

if

< o^ tan a.

30. A mass of m lbs. moves initially with a velocity of u per sec. A constant power equal to If horse-power is applied so as to increase its ft.

velocity

;

shew that the time that elapses before the acceleration

reduced to

31.

mass

M

-th

Shew fired

of its original value is -

_

-,

^^^

is

/,—

that the greatest velocity which can be given to a bullet of

from a smooth-bore gun

/2Y\V \/ ^ , (m log m + 1— m}, where *"

is

H

in front changes of temperature are neglected, and the pressure of the bullet is supposed constant, the volume V of the powder in the cartridge being assumed to turn at once, when fired, into gas of pressure

will

and of volume

V,

Motion Two

32.

m

masses,

a strength that when

both be

they will

free,

m-i^

make n

31

by a spring of such performs n complete vibrations

are connected

held fixed

m^ be held

if

Examples

straight line.

and

rtix

nix is

Shew that

per second.

a

m.<i

fixed, wij will

w—

make n \/ "^

,

and,

if

vibrations per second, the

'-

vibrations in each case being in the line of the spring.

A

33. body is attached to one end of an inextensible string, and the other end moves in a vertical line with simple harmonic motion of

amplitude a and makes n complete oscillations per second.

A

Shew 9

that

compressed by the action of a given force suddenly reversed prove that the gi-eatest subsequent extension of the spring is three times its initial contraction, 34.

light spring is kept

the force

Two

35.

;

is

;

masses,

M

and m, connected by a

vertical line with the spring unstretched until

Shew

that

if

M

the height through which

light spring, fall in a

M strikes an inelastic

falls is

table.

^L^^

greater than

i^

M

will after an interval be lifted from the table, I being the the mass length by which the spring would be extended by the weight of M.

Two uniform spheres, of masses »ii and m^ and of radii aj and ao,., 36. are placed with their centres at a distance a apart and are left to their mutual attractions shew that they will have come together at the end of ;

time

where If

time

R is the radius, mi = m2 = 4

is

and

lbs., ai

B the mean

= a2 = l'5

about 3 hom-s, assuming

density of the Earth.

inches,

^=4000

and a = one

foot,

miles and Z) = 350

shew that the per cubic

lbs.

foot.

[When tion of -y

-4

'

x^

.

the spheres have their centres at a distance x, the accelera-

mi due

Hence the

to,

due to m^

—

acceleration of iiu' relative to nu is v ~^—^ ' x^

equation of relative motion

37.

y -J and that of

to the attraction of m<i is

is

x= —y

Assimaing the mass of the

^

Moon

?

is

and the

^ „

to be

.]

— that of

the Earth, that

and 4000 miles, and the distance between their centres 240,000 miles, shew that, if they were instantaneously reduced to rest and allowed to fall towards one another under their mutual attraction only, they would meet in about 4j days. their radii are respectively 1100

Dynamics of a Particle

32

A

38.

end of the axis of a thin attracting shew that its kinetic energy ;

particle is placed at the

cylinder of radius a and of infinite length

when

it

AB is a uniform

39.

of

has described a distance x varies as log^

it is

string of

repelled with a force,

AB produced

mass

.

M and length 2a

;

every element

.distance, acting from a point

=/Lt

in the

shew that the acceleration of the string is the same as that of a particle placed at its middle point, and that the tension at any point P of the string varies as AP PB. direction of

;

.

Shew that the curve which

40.

each of

its

whose axis

such that a particle will slide down tangents to the horizontal axis in a given time is a cycloid is

is vertical.

41.

Two

string

whose

particles, of

masses

m

and m\ are connected by an elastic they are placed on a smooth

coeflBcient of elasticity is X

;

between them being a, the natural length of the string. The particle m is projected with velocity V along the direction of the string produced find the motion of each particle, and shew that in the subsequent motion the greatest length of the string is a+ Vp, and that the string is next at its natural length after time irp, where table, the distance

;

2_ mm' a ^

Two

~m + m'\'

each of mass m, are attached to the ends of aii ; to one of them. A, another particle of mass 2m is attached by means of an elastic string of natural length a, and modulus of elasticity 'img. If the system be supported with the elastic string just unstretched and be then released, 42.

particles,

inextensible string which hangs over a smooth pulley

shew that

43.

A

A

will

descend with acceleration a sin^

WM-

weightless elastic string, of natural length

has two equal particles of mass

m

I

and modulus

X,

ends and lies on a smooth horizontal table perpendicular to an edge with one particle just hanging over. Shew that the other particle will pass over at the end of time t given by the equation at

its

CHAPTER HI UNIPLAXAR MOTION WHERE THE ACCELERATIONS PARALLEL TO FIXED AXES ARE GIVEN Let the coordinates of a particle referred to axes Ox 38. and Oy be x and y at time t, and let its accelerations parallel and Y. to the axes at this instant be The equations of motion are then

X

»<»

%-

a).

§=^

(2,

Integrating each of these equations twice, equations containing four arbitrary constants. are determined from the initial conditions,

dx

From

,

These

latter

the initial values

dy

the two resulting equations

we then

obtain a relation between x and y which path. 39.

viz.

we have two

is

eliminate

t,

and

the equation to the

Parabolic motion under gravity, supposed constant, the

resistance of the air being neglected.

Let the axis of y be drawn vertically upward, and the axis Then the horizontal acceleration is zero, and

of X horizontal.

the vertical acceleration

Hence the equations d'x

is

— g.

of motion are

^

^,=0,

.

and

d^y

J = -g

(1).

Dynamics of a

34

Integrating with respect to

g

= ^,

Particle

we have

t,

%=-9t + G

and

(2).

Integrating a second time,

x

= At + B, and y = -gtj^Ct + D

(3).

If the particle be projected from the origin with a velocity u at an angle a with the horizon, then when i = we have

X

dx = y = 0, -y- = u COS a, and dv = w sin a.

^ dt

.

r,

"^

dt

Hence from (2) and (3) we have usma = C,0 = B, and = D. .'.

(3) gives

Eliminating

= M cos

a;

is

at

—

^gt\

we have

t,

y

which

= u sin

and y

at,

uqosol=A,

initially

= X tan a — ^r —

—-

the equation to a parabola.

is

A particle describes a path with an acceleration which 40. always directed towards a fixed point and varies directly as

the distance

Let

from

it;

to

find the path.

A

be the centre of acceleration and

the point

Take OA as the X and OY parallel

projection.

axis of

to the direction of the initial velocity, V, of projection.

Let

P

be any point on let be the ordinate of P.

MP

the path, and

The along

acceleration, fM.PO,

PO

is

equivalent,

by

the triangle of accelerations, to

accelerations

and

MO

along

PM

equal respectively to

Hence the equations

of

fj,

.

PM

d'x

and

and

fi

.

MO.

motion are ._.

^=-^"

<^>'

= ~^2/

(2).

dl

of

Composition of Simple HarmoniG Motions The

solutions of these equations are, as in Art. 22,

and

The

a;

=^

2/

= (7 cos [\/yu,i + D]

cos [V/Ii

initial conditions are

= OA =

c

^=

a,

dt

+ £]

when

that

y=

0,

0,

'^

=

and

and

?/

x

(4) give

=

—

1-

a^

The

locus of

P

is

=-

then

%=V. dt J. sin

5.

7-

V/t

= a cos (\//it)

•

OF as

= 0,

and

C=

i'—

=

(5),

—-sin(V/xO

V/i«+^ =

7- cos

^

V= - C ^f/lsin D.

i)=?r, and 2

.'.

(3)

and

cos D,

(3),

(4).

'

Hence, from (3), a = ^ cos 5 and These give B = and A = a. Also, from (4), similarly we have

.*.

35

(6).

1

V-

therefore an ellipse, referred to

OX and

a pair of conjugate diameters.

Also,

F = Vy"-

if

the ellipse meet

OY

OB =

in B, then

V -r-

i.e.

,

Since any point semi-diameter conjugate to OA. on the path may be taken as the point of projection, this result will be always true, so that at any point the velocity ><

= \/fi [This

may

X semi-conjugate diameter.

be independently derived from

(5)

and

(6).

For

(Velocity at P)^ = x^ + y^ + 2xy coso)

= a^ix sin2 = ix\

a2

L

= fj.[a'^ =

iu

{Jixt)

+

+ V^ cos2

a2 cos2

^

(

J

{Jfj.t)

/It)

M

- 2a V^/x sin sin2 Jj^t

(Jfj.t)

- -^- sin

cos (V/"0 ^°^ {J]j.t)

V/*

+ — -x^- y2-2xy cos J] =/x{a^ + — -0P2\

X square of semi-diameter conjugate to OP.]

"

cos {Jjit} cos

w |

-J

Dynamics of a

36

From

equations (5) and (6)

X and y are the same at time

i

Particle clear that the values of

it is

+ -^

as they are at time

t.

2_ Hence the time

of describing the ellipse

-r-

is

If a particle possess two simple harmonic motions, in 41. perpendicular directions and of the same period, it is easily

seen that

path

its

an

is

ellipse.

we measure the time from the time when the has its maximum value, we have

a;-vibration

If

x = a cos wi and

2/

where

a,

= 6 cos

(n^

(1),

+

e)

(2),

h are constants.

(2) gives y

= cos nt cos e — sin nt sin e =

pa/ = cos

,

I.e.

e

sin^

- cos

e

— sin e a/

1

— —

,

•

el V

x^

—

Ixy

y^

a^

ah

TT,

^ cos e +

ci-,

.

,_,

= sm^ e „

(3).

h'

This always represents an ellipse whose principal axes do general, coincide with the axes of coordinates, but

not, in

which

is

The

always inscribed in the rectangle x figure

drawn

If

6

= 0, equation

If

e

= TT, it

is

an

ellipse

(3) gives

gives -

+ r = 0,

h.~^' i.e.

In the particular case when ^/-vibration

at zero time

is

where

e

e is

*'"^"

= ±a,

,

i.e.

=

±h.

^^® straight line

the straight line

=^

y

equal to about

-^

AG.

BD.

when the phase

of the

one-quarter of the periodic time,

equation (3) becomes

i.e.

the path

is

an

ellipse

whose principal axes are in the direction

Composition of

Siinjjle

Hat^monic Motions

37

of the axes of x and y and equal to the amplitudes of the component vibrations in these directions. If in addition a = b, i.e. if the amplitudes of the component

vibrations are the same, the path B

is

a

circle.

Dynamics of a

38

The dotted curve

when the phase

in the figure

Particle the path

is

when

of the ^/-vibration at zero time

is

IT

e

,

..^.

negative and

equal to one-quarter of the period of the 2/-vibration.

When time

= TT,

6

i.e.

when the phase

of the y- vibration at zero

one-half of the y-period, the path becomes

is

^=-|(y-6). i.e.

the parabola

When

= Oj

e

GEB. the path

is

similarly the parabola

^=g(j/+6). For any other value of

e

the path

is

more complicated.

Curves, such as the preceding, obtained by compounding

harmonic

simple

Lissajous' figures.

motions in two directions are known as For other examples with different ratios of

the periods, and for different values of the zero phases, the

may refer to any standard book on Physics. These curves may be drawn automatically by means of a pendulum, or they may be constructed geometrically. student

43. Ex. 1. A point moves in a plane so that its projection on the axis of x performs a harmonic vibration of period one second with an amplitude of one foot; also its projection on the perpendicular axis of y performs a harmonic vibration of period two seconds with an amplitude of one foot. It being given

that the origin

the path, find

is

its

the centre of the vibrations, and that the point equation and draw it.

(1, 0) is

on

Ex. 2. A point moves in a path produced by the combination of two simple harmonic vibrations in two perpendicular directions, the periods of the components being as 2 3 find the paths described (1) if the two vibrations have zero phase at the same instant?, and (2) if the vibration of greater period be of phase one-quarter of its period when the other vibration is of zero phase. :

;

Trace the paths, and find their equations.

44.

If in Art. 40 the acceleration be always from the fixed

point arid varying as the distance from

X = a cosh .'.

— — p- =

1,

V/A<,

and y =

so that the path

it,

we have

-j- sinh V/ii.

is

a hyperbola.

similarly

Accelerations parallel to fixed axes

A

39

a catenary under a force which acts find the law of the force and the velocity at any point of the path. Taking the directrix and axis of the catenary as the axes of 45.

parallel to

cc

and

y,

'particle describes

its accis;

we have

as the equation to the catenary c /

y Since there

is

£

_£\

= 2\^" +

6

(1).

'')

no acceleration parallel to the

—=

.' •

.*.

•

•

df'

-77

=

const.

=M

_^\

dx

(2).

we have

Differentiating equation (1) twice,

lft£

directrix,

1 /

?

--IN

dt

Also

(velocity)'

=

(|y+(|J W"

so that the velocity

Hence the

—

~

velocity

/

'1

= «= +

_i'\-

f (.!-.-!)'

IV

y.

and acceleration at any point both vary

as the distance from the directrix.

A

particle moves in one plane ivith an acceleration always toivards and. perpendicidar to a fixed straight in the plane and varies inversely as the cube of the distance

46.

which line

from

is

it; given the' circumstances

Take the

Then the equations

-•J (1) gives

of projection, find the path.

fixed straight line as the axis of x.

of motion are

S=«

w-

%-^f

(^)-

x==At-\rB

(3).

Dynamics of a Particle

40

Multiplying (2) by -^ and integrating, we have

ydy

1

Let the particle be projected from a point on the axis of y component velocities u and v parallel to the axes. Then when ^ = we have distant h from the origin with

x=

r^

v

0,

^

'

A=u, B = 0,

.'.

.'.

and

(3)

x

=

Eliminating

t,

[t V

/Lt

is

an

= ¥v',

+ j— — /xj b-v

¥v

Y

ellipse or a

then

Hence the path

The path as v to

A /t"j.

dxi -77

= v.

dt

0^

\u~ /M- bv) If

,

= u, and

dt

G^v"--^^, and

we have

fx

This

-T-

'

i)

""

=-

,,

f—

b^v-

.

fj,

(4) give

and

tit,

dx

i = o,

is

C=

*-6-

[b-v-

— fi)- + jA b^v — fi z.

.

as the equation to the path

_ - b-'v' ^ y"b'~

"*" yu,

fj,¥

ifM- b'u-y

hyperbola according as

and equation

in this case

thus an

=7TT-:,

is y'^

(4)

— b^ = 2^//^ -

ellipse, parabola, or

/j,<

6V.

becomes

,

i.e.

a parabola.

hyperbola according

according as the initial velocity perpendicular

the given line

is less,

equal

to,

or greater than the velocity

that would be acquired in falling from infinity to the given point with the given acceleration.

For the square of the Cor. X

axis of

If the particle it will

latter

=—

1

2^t/2/=p^

=,^.

describe an ellipse and meets

the

not then complete the rest of the ellipse since

Accelerations parallel to fixed axes

41

the velocity parallel to the axis of x

the same direction;

another equal

If

47.

ellipse.

and

the velocities

particles m^, w^,

space he Vi,v^

is always constant and in proceed to describe a portion of

will

it

...

m^

...

dindf,f

any instant of

accelerations at

parallel

any

to

straight line fixed in

...to find the velocity

and

acceleration

of their centre of mass. If 0?!, x^, ... be the distances of the particles at any instant measured along this fixed line from a fixed point, we have

_

TOja?! 7?li

+ m^X2 + ???2 +

+

•

Differentiating with respect to _ 1} zzz

dx —dt

t,

.

•

.

•

we have

+ W2V2+ ... =• m^ + mo + ... Wit'i

/iv V-l),

/=g = -"'/ ++ '"'.^++

and

•^

where

v

dt^

nia

«'2

•

•

(2),

•

and /are the velocity and acceleration required.

Consider any two particles, m-^ and m^, of the system and the mutual actions between them. These are, by Newton's Third Law, equal and opposite, and therefore their impulses

The changes

resolved in any direction are equal and opposite. in the

momenta

of the particles are thus,

by

Art. 15, equal

opposite, i.e. the sum of their momenta in any direction thus unaltered by their mutual actions. Similarly for any other pair of particles of the system.

and is

of the momenta of the system parallel to and hence by (1) the momentum of the centre of mass, is unaltered by the mutual actions of the system. If Pj, P2, ... be the external forces acting on the particles nil, ^2 ••• parallel to the fixed line, we have

Hence the sum

any

line,

'mifi

+ vi2f2 +

...

= (Pi-h P2+ ') + (i^& sum

of the com-

ponents of the internal actions on the particles)

= P, + P2+..., since the internal actions are in equilibrium taken selves.

by them-

Dynamics of a Particle

42

Hence equation

(2) gives

+ m. +

(mi

.

. .

7= Pi + Po +

)

the motion of the centre of

i.e.

. .

.,

mass in any given

direction is the

same as if the whole of the particles of the system were collected at it, and all the external forces of the system applied at it parallel to the given direction.

Hence

also

If

the

sum of

the external forces acting on

any

given system of particles parallel to a given direction vanishes, the motion of the centre of gravity in that direction remains unaltered, and the total momentum of the system in that direction remains constant throughout the motion. This theorem is known as the Principle of the Conservation of Linear Momentum.

As an example, of its centre of

if

mass

a heavy chain be falling freely the motion

is

the same as that of a freely falling particle.

EXAMPLES ON CHAPTER

A particle describes an ellipse with

1.

the centre

;

shew that

proportional to

its

III

an acceleration directed towards

angular velocity about a focus distance from that focus. its

is

inversely

A particle is describing an ellipse under a force to the centre if and V2 are the velocities at the ends of the latus-rectum and major and minor axes respectively, prove that v^v^=Vi^{2v^^v-^). 2.

;

V, Vi

The

3.

and

v

path

is

x and y are u + ay and a are constants shew that its

velocities of a point parallel to the axes of

+ ta'x respectively,

where

m, v,

a>

j

a conic section.

A particle moves in a plane under a constant force, the direction of 4. which revolves with a uniform angular velocity find equations to give the coordinates of the particle at any time t. ;

A

small ball is projected into the air ; shew that it appears to an 5. observer standing at the point of projection to fall past a given vertical

plane with constant velocity.

A man

and walks, or runs, with a constant starts from a point u along a straight road, taken as the axis of x. His dog starts at a distance a from 0, his starting point being on the axis of y which is 6.

velocity

perpendicular to Ox, and runs with constant velocity r in a direction which

Accelerations parallel to fixed axes. is

Shew

always towards his master.

If X

= 1,

shew that the path

[The tangent at any point point where the

man

then .'.

d r

^ds

dx'X

is,

that the equation to his path

the curve

is

P of

Examples

-y^

—X

=

-^ ut —

ax

dx dy

^

is

2f«4--j = ^-alog-. Ox

the path of the dog meets

so that

43

Also -n

.

at

=7

at the

•

K

= ut — x = \s-x, —y-^ ^ dx

.

.

d^x

.

/,

.

,

(dx\^

-,

A particle is fastened to

one end, B, of a light thread and rests on the other end, A, of the thread is made to move on the plane with a given constant velocity in a given straight line shew 7.

a horizontal plane

;

;

that the path of the particle in space

[Shew that

AB turns round A

a trochoid. with a constant angular velocity.] is

move with a velocity v relative to the water and both cross a river of breadth a running with uniform velocity V. They start together, one boat crossing by the shortest path and the other in the shortest time. Shew that the difference between the times of arrival is 8.

Two

boats each

either

-1 according as F or t> is the greater. [The angle that v makes with IS

a.

—4--i

V'F^jhJHTTjTcos^ V sin 6

condition for a

minimum path (y

9.

A

particle

moves

V

being

B,

the length of the path

, ,, JXand the corresponding ° time '

«

•

is

.

.

TV,^ Ine

.

V sin o

gives

cos ^ + F)

Fcos ^ + v) = 0.]

(

in one plane with

an acceleration which

is

always

perpendicular to a given line and is equal to /x-j- (distance from the line)^. Find its path for different velocities of projection. If it be projected from a point distant 2a from the given line with

a velocity a/- parallel to the given

line,

shew that

its

path

is

a cycloid.

If a particle travel with horizontal velocity u and rise to such 10. a height that the variation in gravity must be taken account of as far as small quantities of the first order, shew that the path is given by the

equation

_

a-y = {a-k) cosh |^>/|

-^J>

where 2a is the radius of the earth the axes of x and y being horizontal and vertical, and h, k being the coordinates of the vertex of the path. ;

Dynamics of a Particle

44

A particle

11.

to the axis of

moves in a plane with an acceleration which is parallel y and varies as the distance from the axis of x shew that ;

the equation to is

its

path

is

of the iovm.y = Aa'+ Ba~^,

when the acceleration

a repulsion. If the acceleration is attractive,

y=A

A

then the equation

cos \ax

is

of the form

+ B\

moves under the action of a repulsive force perpenand proportional to the distance from it. Find its path, and shew that, if its initial velocity be parallel to the plane and equal to that which it would have acquired in moving from rest on the 12.

particle

dicular to a fixed plane

plane to the point of projection, the path

A

13.

particle

is

a catenary.

describes a rectangular hyj^erbola, the acceleration

being directed from the centre shew that the angle 6 described about the t after leaving the vertex is given by the equation ;

centre in time

tan ^ = tanh(\//x<),

where 14.

to the

/i

is

acceleration at distance unity.

A particle

moves

bounding diameter

freely in a semicircle ;

shew that the

under a force perpendicular

force varies inversely as the cube

of the ordinate to the diameter. 15.

Shew that a rectangular hyperbola can be described by a

under a force

parallel to

particle

an asymptote which varies inversely as the cube

of its distance from the other asymptote.

A particle is 16. m —^ towards the axis

moving under the influence of an of x.

Shew

attractive force

that, if it be projected

from the point

with component velocities U and V parallel to the axes of x and y, will not strike the axis of x unless \i> V^k^, and that in this case the

(0, k) it

distance of the point of impact from the origin

is

-^

.

A plane has two smooth grooves at right angles cut in it, and two 17. equal particles attracting one another according to the law of the inverse square are constrained to move one in each groove. Shew that the centre of mass of the two particles moves as if attracted to a centre of force placed at the intersection of the grooves and attracting as the inverse square of the distance.

CHAPTER IV UNIPLANAR MOTION REFERRED TO POLAR COORDINATES CENTRAL FORCES In the present chapter

48.

we

shall

consider cases

of

motion which are most readily solved by the use of polar coordinates. We must first obtain the velocities and accelerations of a moving point along and perpendicular to the radius vector drawn from a fixed pole.

Let

and

Velocities

49.

perpendicidar

to the

of a particle along and

accelerations

radius vector

to it

from a fixed

origin 0.

P

be the position of the particle at time t, and Q its position at time t

e

+

At.

Let XOP = e, XOQ = + AO, OF = r, OQ = r

+ Ar, where OX is

a fixed

line.

Draw

QM

/

perpendi-

OP. Let u, V be the velocities of the moving point along and perpendicular to OF. Then cular to

= Lt

Distance of particle measured along the line OP at time ^ + A^ — the similar distance at time t

At

= Lt At=Q

OM-OP At

J - Lit A^=0

{r

+ Ar) cos, A6 - r T—

At

small quantities above the

first

order

being neglected,

_dr 'It

.....(1).

Dynamics of a Farticle

46 Also

Distance of particle measured perpendicular to the OF at time ^ + A^ —the similar distance at time t

line

At

QM-0

= Lt

Ai

M=o

= Lt

(r

= Lt

+ Ar) sin A^ A^

At=0 -T~^

,

on

neglecting

small

quantities

of the

second order,

=

-^-

r*

The

,

in the limit

(2).

velocities along

perpendicular to

OP

and

being

u and V, the velocities along and perpendicular to OQ are u + Au and v + Av. Let the perpendicular to

OQ

at

Q

u+Au

be produced to

meet OF at L. Then the acceleration of the moving point along OF velocity

its

along

OF

time

at

+ At —

t

velocity at time

= Lt

its

similar

t

A^

A< = Ol

(u

+ Au) cos Ad — (v +

A6 — u

Av) sin

A^ ^^ \{u

Au) 1 - (y + Ay) A ^ - w' + Au).l-{v+Av).Ae-u'] .

.

= lt\.

A^

'

J

on neglecting squares and higher powers of A0,

du ^^Au-vAd =—1

= Lt

r

cW v^r.

.

,

'i^n

^,

,.

.,

the limit,

(3),

by

(1)

and

(2).

Polar Coordinates

47

Also the acceleration of the moving point perpendicular in the direction of 6 increasing

OP

to

OP

perpendicular to

its velocity

at time

similar velocity at time

Lt

t+ ^t —

its

t

Af

_ J Uu + ~ A^=o

A^)

sin /^9 ->r(v-\-

,

r(tt

Ay) cos

A^ - ^

A^

L

+ A?0 A^ + (y + .

J At')

.

1

- v1 '

^t

^ifoL

"1

J

on neglecting squares and higher powers of A^,

dd

dv

d* cfi+*

Cor.

,.

,,

.

dr^dd

.^

rf^^~rd«L

If r

= a,

= — a62

and

the tangent

dd\

,,.

,

,

,^,

^*^-

a constant quantity, so that the particle

= a^,

(4)

PQ

(

dt\

and radius

describing a circle of centre (3)

d

a,

so that the accelerations of

and the radius

PO

is

the quantity

P

along

are a^ and a6^.

50- The results of the previous article may also be obtained by resolving the velocities and accelerations along the axes of x and y in the directions of the radius vector and perpendicular to it.

For since x = r cos

d

and y = rsva. dx •••

9,

dr

ndO\

^

Tt=dt'''''-'''^'di\ ^^^

| = |sin^ + rcos4^j

and Also d'^x

d^r

^

^drde

.

^

n

/dd\^

.

„d^0

M=dfi''''-^dtdt''^'-''''\Tt)-'''^'^^ .(2).

Dynamics of a Particle

48 The component

acceleration along

OP •(S)^^^(^).

and perpendicular

to

OP

it rf2.r

rdtL

By

51.

moving point

manner about the

By and X

we can

in their

own

-j-

Ox

along

line fixed in space, and, at

to

OA

Let

.

P

;

M

MP, and

the velocities of

N

along

OM

~ along

ON

are

PN produced.

velocity of

the velocity of relative to

=

are -^ along

ctt

Hence the

=

Ox

plane.

at

Ja

and y

the

which revolve in any

fixed in space, but

origin

obtain

referred to rectangular axes

Art. 49 the velocities of the point

-J-

,„,

time t, let 6 be the be the moving point draw perpendicular to Ox and Oy.

be a

inclination of

PM and PN

^

dtj

and Oy, which are not

OA

d^

^drde

the use of Arts. 4 and 49

accelerations of a

Let

^

.

vel.

of

dO

P parallel

N parallel to

to

Ox

Ox +

the velocity of

P

N.

N parallel to

Ox

4-

the

vel. of

M along OM

dx

= ~'^di^di

.(1).

Revolving Axes So the velocity of

P parallel

to

Oy

M parallel to Oy the of M parallel to Oy + the

= vel. = vel.

of

49

4-

vel. of

P

vel. of

N along ON

relative to

M

-4-t

(^)-^^ ^;

M are, by Art. 49, -^ ~ ^

Again, the accelerations of

along

X dt\ of

N

7/7

the accelerations

dt

PN

ON, and^|(2/^|) along

g-2,(^y along

are

MP, and

(

jDroduced.

Hence the

acceleration of

= acceleration

of

P

parallel to

N parallel

to

Ox

Ox + acceleration

of

P

N = acceleration of N parallel to Ox + acceleration of M relative to

along

OM

d

^de\

1

(

^

(de\^

d'x

.„.

(^>-

--y-dt{yw^dt^-''\jt) Also the acceleration of

P parallel to

Oy

M parallel to Oy + acceleration of P relative to M = acceleration of M parallel to Oy + acceleration of N = acceleration

along 1

d

of

ON

,de\^d'y

fddy

( = ^dt[^dt)+-^-y[di)

Cor.

In the particular case when the axes are revolving

with a constant angular velocity

component

velocities

dd

oo,

so

become

^and

^^>-

-^

2/(y

+ xfjo

along Ox,

along

Oy

;

that -^ =

o),

these

Dynamics of a Particle

50 also the

component accelerations are

J^-

and 52.

-Eo;. 1.

velocities

which

is

u and

perpendicular

whose focus

is

With the

and

2/0)"

+

2ft,

of lohich

P which possesses Uoo constant and the second of from a fixed point 0, is a conic

in a fixed direction

is

ivhose eccentricity is

first figure

along Oy.

J"

OP drawn

radius

to the

Oo.,

of a point

«/ia< t/je ipath

5/(ejt)

v, the first

1 along

^-0,^-20,

of Art. 49, let

-.

u be the constant

velocity along

OX

and

V the constant velocity perpendicular to OP.

Then we have dr ;j-

at

n =M cos 6,

J and

''^^

^

-^ = v -

it

sin d.

.:

at

logr= -log fi;-M

.'.

sin ^)

-

I

dr = —

r

ad

u cos 6 -.

v

—

— us\nd

;,

+ const.,

r{v-usin6)=const. = lv,

i.e.

if the

path cut the axis of

at a distance

a;

I.

Therefore the path

is

I

—

1

sin ^

V

.

i.e.

a conic section whose eccentricity

Ex.

2.

A smooth

is

u V

straight thin tube revolves with uniform angular velocity

w

in a vertical plane about one extremity which is fixed ; if at zero time the tube be horizontal, and a particle inside it be at a distance a from the fixed end, and be

moving with velocity

V along

a cosh

(wt)

the tube, shew that its distance at time

+

~j sinh

(

{«()

+

.

-g

sin wt.

At any time t let the tube have revolved round its fixed end angle wt from the horizontal line OX in an upward direction;

OP=r, be the position By Art. 49, d'^r ^-^i

of the particle then.

- rw^ z= acceleration of

P

in the direction

OP

dt-

= The

-^

sin wt, since the tube is

solution of this equation

smooth.

is

:

^{-g&mut) 2w'

where A and B, and so

L and

BI, are arbitrary

t is

constants.

through an P, where

let

Motion referred The

initial conditions are that r

.-.

polar coordinates

to

=a

and

f

V=Mw + S-

a = L, and

51

= F" when « = 0. •

2(1}

r = a cosh

.:

If

R

wt+

-^

\

sinh (ut)+

~

sin ut.

be the normal reaction of the tube, then gicos £ot=the acceleration perpendicular to

= ^~

(r2w),

OP

by Art. 49, =2rw

= 2au' sinh (wt) + (2Vu- g) cosh {ij}t)+g cos tat.

EXAMPLES 1. A vessel steams at a constant speed v along a straight line whilst another vessel, steaming at a constant speed F, keeps the first always exactly abeam. Shew that the path of either vessel relatively to the

other

a conic section of eccentricity

is

y..

A

boat, which is rowed with constant velocity «, starts from on the bank of a river which flows with a constant velocity nu points always towards a point B on the other bank exactly opposite to find the equation to the path of the boat. If n be unity, shew that the path is a parabola whose focus is B. 2.

a point it

A

A

;

;

3.

An

insect crawls at a constant rate

of radius a, the cart

and perpendicular 4.

from a

The

moving with

velocities of a particle along

fixed origin are Xr

and

with velocity

Find the acceleration along

fi6

;

find

and perpendicular to the radius the path and shew that the

and perpendicular to the radius

XV-^ A point

v.

to the spoke.

accelerations, along

5.

u along the spoke of a cartwheel,

velocity

starts

from the

and

/x^

oi'igin in

vector, are

fx + the direction of the

— and moves with constant angular

velocity

initial line

w about the

(I)

and with constant negative radial acceleration -/. Shew that the rate of growth of the radial velocity is never positive, but tends to the limit zero, and prove that the equation of the path is a)V=/(l — e^e). origin

6.

A

point

P describes a curve

velocity about a given fixed point

with constant velocity and

its

angular

varies inversely as the distance from

shew that the curve is an equiangular spiral whose pole is 0, and that the acceleration of the point is along the normal at and varies inversely ;

P

as OP.

4—2

Dynamics of a Particle

52

A

P

describes an equiangular spiral with constant angular velocity about the pole shew that its acceleration varies as OF and is 7.

jJoint

;

making with the tangent

in a direction

OP

at

P the same constant angle

that

makes.

A point

8.

velocity

T',

moves in a given straight line on a plane with constant and the plane moves with constant angular velocity <u about an

axis perpendicular to itself through a given point

from the given straight line be is given by the equation

distance of

a,

of the plane.

If the

shew that the path of the

point in space

^=V»-^-a- + - C0S-1-, as pole.

referred to [If 6 be

measured from the

at zero time, then r2 =

a2+

which the given

line to

V^.t"^

and ^ = w« + cos-i -

1 -

r

A straight smooth

9.

tube revolves with angular velocity w in a horiis fixed if at zero time a particle be at a distance a from the fixed end and moving with velocity V

zontal plane about one extremity which inside

line is perpendicular

it

along the tube, shew that

its

;

distance at time

t

is

a cosh

a)t

+—

sinh at.

O)

A

thin straight smooth tube is made to revolve upwards with 10. a constant angular velocity w in a vertical plane about one extremity when it is in a horizontal position, a particle is at rest in it at a distance a from the fixed end ; if to be very small, shew that it will reach in ;

(

—

)

nearly.

11. A particle is at rest on a smooth horizontal plane which commences to turn about a straight line lying in itself with angular velocity if a be the distance of the particle from the axis of rotation at zero 0) time, shew that the body will leave the plane at time t given by the ;

equation

a sinh a)t+ --—^ cosh wi =

A

12.

particle falls

from

^ cos

a>t.

rest within a straight

smooth tube which

in revolving with uniform angular velocity w about a point being acted on by a force equal to nifi (distance) towards 0.

the equation to

r

= acosh| If

/I

13.

.

=0)2,

its

path in space

/ —^

-

^

A particle

is

is

length,

Shew

that

is

or 7-=a cos

shew that the path

its

is

a

|^^^—^

(9L according as fita^.

circle.

placed at rest in a rough tube at a distance a trom

Motion referred

to

polar coordinates

53

one end, and the tube starts rotating with a uniform angular velocity w about this end. Shew that the distance of the particle at time t is **'^*

ae~"^-

where tan

e is

One end

14.

velocity

w

[cosh [mt sec .

e)

+ sin e siuh

{oit

sec

e)],

the coefficient of friction.

^

of a rod

is

made

to revolve with uniform angular

in the circumference of a circle of radius a, whilst the rod itself

revolves in the opposite direction about that end with the same angular Initially the rod coincides with a diameter and a smooth ring velocity.

capable of sliding freely along the rod is placed at the centre of the Shew that the distance of the ring from A at time t is - [4 cosh

(o)^)

+ cos

circle.

1a>t\.

and P, where AP=r, is the position of OA and AP have revolved through an angle ^, ( = &)0, in opposite directions, the acceleration of A is aw^ along AG and the acceleration of P relative to is r - rff^, by Art. 49, i.e. r — rofi. Hence the total acceleration of P along AP is r - rw^ + aco^ cos 20^, and this is be the centre of the the ring at time t when both

circle

[If

zero since the ring

is

smooth.]

PQ is a tangent

15.

makes an angle accelerations of

at

§

to

a

6 with a fixed

P along and

circle of radius

a;

PQ\s

tangent to the circle

perpendicular to

QP

;

equal to p and shew that the

are respectively

Id

p-pd^ + ad, and -^(p^^h P<

[The accelerations of the accelerations of

Q

along and perpendicular to QP are ad and these same directions are

p-pe^ and

Two

16.

;

a,

masses

^^yh]

m and

m', connected by an elastic string are placed in a smooth tube of small bore which is length with angular velocity «.

particles, of

of natural length

aQ'^

P relative to Q in

made to rotate about a fixed point in its The coefficient of elasticity of the string is if

is

'imm'aa)^^{m-\-m'). Shew that, the particles are initially just at rest relative to the tube and the string just taut, their distance apart at time t is 2a — a cos mt.

An

elastic string is just stretched against

a rough wheel of radius uniform angular velocity co. Shew that the string will leave the wheel and expand to a maximum radius r given 17.

a,

which

is

set in rotation with

by

rHr-a)^Ma?^ ~ 2n\ r+a where 18.

'

M and X are the mass and modulus of elasticity of the A uniform

chain

AB is placed

in a straight tube

string.

GAB

which

re-

volves in a horizontal plane, about the fixed point 0, with uniform angular

Dynamics of a Particle

54 velocity

that the motion of the middle point of the chain

Shew

o).

is

the

same as would be the motion of a particle placed at this middle point, and that the tension of the chain at any point P is ^mai'.AP. PB, where m is the mass of a unit length of the chain. 53. is

A

particle moves in

a plane with an acceleration which in the plane

aliuays directed to a fixed point

differential equation

to obtain the

;

of its path.

as origin and a fixed straight line OX Referred to as initial line, let the polar coordinates of P be (r, 6). through If P be the acceleration of the particle directed towards 0, we

have, by Art. 49,

«

s-Kir=-^ Also, since there is

have,

by the same

no acceleration perpendicular to OP, we

article,

rdt\ (2) gives

r^

-—

.*.

^^"^

J

^^^

dr

_ d

= const. =

= — = hu-,

dt rvu

-^

__

d

/

J

du\

,

'^-''•

(say)

(3).

to -

r

_.du u'Td'dt" dd'

__1

^'It'

Tt~dt[uJ d^r

1 die

/i

u be equal ^

if

r-

/1\

"

dt)

du dd _

d (du\ dO

,

„

,

d'^w

w^ = dt[-^mr-^'dd[ddydt=-^'^'dd^'

Central Forces

Dynamics of a Particle

56

The constant h

is

thus equal to twice the sectorial area

described per unit of time.

Again, the sectorial area

= lA Now,

POQ =

1

its

As

in the limit

when Q

=

is

.*.

fi

v,

PQ

on

the perpendicular from

x per-

very close to P,

the velocity

and the perpendicular from

PQ

description

on PQ.

^. -r-x perpendicular from

-r-

=

in the limit h

on PQ, and the rate of

pendicular from

on the tangent at

= v.p,

I.e.

v

P =^.

= -. P

Hence, when a particle moves under a force to a fixed any point P of its path varies inversely as the perpendicular from the centre upon the tangent to the path at P. centre, its velocity at

Since v

=

-,

and in any curve

P

55. A 'particle moves in an ellipse under a force which is always directed towards its focus; to find the law of force, and the velocity at

any point of its path. to an ellipse referred

The equation

1

+ e cos ^ d'u

Hence equation

(4) of Art.

to its focus is

u=j + jCosd

I.e.

I

e

a

53 gives

P = .v[^ + .]4ÂŤ' The

(1).

I

(2).

acceleration therefore varies inversely as the square of

Central Forces

57

the distance of the moving particle from the focus and,

h

=

N iJbl

=

X semi-latus-rectum

w/j,

if it

be

(3).

Also

=

f

[1

+ 2. cos

^

+

= ;.

..]

1+^ - 1^]

[2

=4'-9'^>-«' where 2a

W'

the major axis of the

is

ellipse.

depends only on the distance r, that the velocity at any point of the path depends only on the distance from the focus and that it is independent of the direction of It follows, since (4)

the motion.

V

It also follows that the velocity

point whose distance from the focus

2a

—

,

of projection from any

is ro,

and that the a of the corresponding

Vn Since h

Periodic time. in a unit time,

is

less

than

given by

aj

equal to twice the area described

that if

it follows,

must be

ellipse is

T

be the time the particle

takes to describe the whole arc of the ellipse, then

^h X h

Also

= ^I

T = area

fjL

of the ellipse

X semi-latus-rectum

/

fi

—

.

7

fi

Ex.

=a

T = — — = -^a-

Hence 56.

= -rrab.

Find

r^=a"cosnd can

V/*

the laio of force toicards

the pole under lohich the curve

be described.

u"a"cos?i^ = l.

Here

Hence, taking the logarithmic

differential,

we obtain

du

y- = uta,nnd. da

dhi .'.

jg2

du

= Jg

tan nd + nu sec2 nd = ii [tan2 jid + n sec^

d?u •'

^

"^

"

^"

("

+

^) ^®*^^

ne = {n^-\)

nd].

a2''a2'H-i,

Dynamics of a

58 Hence

Particle

equation (4) of Art. 53 gives

P={n + l);i2a2''!(2«+3, the curve can be described under a force to the pole varying inversely as tho (2n + 3)rd power of the distance.

i.e.

Particular Gases.

I.

Let

n= -~,

so that the equation to the curve

i.e.

the curve

is

a parabola referred to

Let w = -

,

'

focus as pole.

its

so that the equation

= - (1 + cos 6),

is r

which

is

a cardioid.

^ ^ Zi'

Here

Let n=l, so that the equation to the curve

III.

with a point on

its

is

r=a cos ^,

i.e.

a circle

circumference as pole.

Pen

Here IV.

+ cos ^

1

Pxi.

Here II.

is

2a

a ^e cos2-

—.. ,.8

Let n=2, so that the curve

r2

is

= a2 cos 25,

a lemniscate

i.e.

of

1

rT V.

Let

n=

- 2,

so that the curve is the rectangular hyperbola

the centre being pole, and

P oc

force is therefore repulsive

from the

- r,

since in this case (n

+ 1)

a'^

=

r'^

cos 26,

negative.

is

The

centre.

EXAMPLES

A particle shew that the

P

describes the following curves under a force force is as stated

:

1.

Equiangular spiral

Px

2.

Lemniscate of Bernouilli

P x-..

3.

Circle, pole

,.7

on

its

circumference

4.

- = e"«, n6, cosh n6, or sin n6

5.

r"cos«^ r^cosnd = a''; a^;

6.

r'^

= A cos, n6 B a\n nO -\-

;

Pa:—..

;

,.3-

Pccr^^'^. ;

Px

7.

r=asmnd:

P~

8.

aM = tanh f-^) or cothf-^j;

P oc

W'2^

1 y.2»

+3

2»2a2 ,

•

n2

._ 1 ,

to the pole,

Central Forces cosh ""

coshg + 2

cosh ^ + 1

cosh 6

cosh2^-l "'"'= cosh2^ + 2

"

„

_

2(9

59 1^ *

— \'

cosh

„

„

lO-

^-2

r*

+1

%osh2^-2

'

_ 1 ^^,-T-

Find the law of force to an internal point under which a body will Shew that the hodograph of such motion is an ellipse. describe a circle. [Use formula (5) of Art. 53. The hodograph of the path of a moving draw a straight line OQ point P is obtained thus From a fixed point the locus of the point parallel to, and proportional to, the velocity of P Q, for the diflerent positions of P, is the hodograph of the path of P.] 11.

:

;

12. a,

A particle of

unit mass describes an equiangular spiral, of angle

under a force which

always in a direction perpendicular to the

is

straight line joining the particle to the pole of the spiral force is /ir^*®°"*~^,

the pole

is

-

and that the rate of description of

Vm sin a

.

cos a

.

;

shew that the

sectorial area about

r^^^"^":

In an orbit described under a force to a centre the velocity at is inversely proportional to the distance of the point from the centre of force shew that the path is an equiangular spiral. 13.

any point

;

14.

The velocity at any point

of a central orbit

be for a circular orbit at the same distance varies as

— — and -,

;

is

- th of what

it

shew that the central

that the equation to the orbit

would force

is

X-i=a''*-icos{(?i2-i)^}.

57.

Apses.

An

apse

is

a point in a central orbit at which

the radius vector drawn from the centre of force to the moving

maximum or minimum value. By the principles of the Differential Calculus u is a maximum

particle has a

or a of

minimum

if -j^ is zero, ciu

u that does not vanish

is

and

if

the

first differential coefficient

of an even order.

If p be the perpendicular from the centre of force upon the tangent to the path at any point whose distance is r from the origin, then 1

When

-7^ is zero,

the case of the apse apse the particle

is

,

— = w^ = —

is

fdiC\

,

so that the perpendicular in

equal to the radius vector.

moving

Hence

at an

at right angles to the radius vector.

Dynamics of a

60 58.

When

Particle

the central acceleration is

when

a single-valued function

a function of the always the same at the same distance), every o.pse-line divides the orbit into two equal and similar portions and thus there can only he tivo apse-distances.

of the distance

and

distance only

Let

{i.e.

ABC be

apses A, B, and

the acceleration is

is

a portion of the path having three consecutive

C and

let

be the

c

centre of force.

Let

V

be the velocity of the particle at B. Then, if the velocity

/

of the particle were reversed at B,

would describe the path BPA. For, as the acceleration depends on the distance from only, the velocity, by equations (1) and (3) of Art. 53, would depend only on the distance from and not on the

/"^---.^^

/

/

/

^'^JXP\

/

/

it

V

^n^

./

.^^

/,/'

\

-^

/^^^'"'^

O

—

""

""--—

~^^

/

~~"^

—7 /

direction of the motion.

Again the

original particle starting from

from

B

and the reversed

B

with equal velocity F, must describe similar paths. For the equations (1) and (3) of Art. 53, which do not depend on the direction of motion, shew that the value of r and d at any time t for the first particle {i.e. OP' and Z BOP') are equal to the same quantities at the same time t particle, starting

for

the second particle

{i.e.

OP

and Z BOP).

BPA are exactly the same by being rotated about the line OB, would give the other. Hence, since A and G are the points where the radius vector is perpendicular to the tangent, we have OA = 00. Hence the curves BP'C and

either,

Similarly, if

OB

and

OD

D

equal,

were the next apse after

and

C,

we should have

so on.

Thus there are only two different apse-distances. The angle between any two consecutive apsidal distances

is

called the apsidal angle.

59.

When

power of the

the central acceleration varies as some integral /i^i'^ it is easily seen analytically that

distance, say

there are at most two apsidal distances.

A])ses

and Apsidal Distances

For the equation of motion

The

particle

is

61

is

when ;to=^> ^^d then

at an apse

this

equation gives

n

—

1 h^

2

^

„

^

fi

Whatever be the values of n or C this equation cannot have more than two changes of sign, and hence, by Descartes' Rule, it cannot have more than two positive roots.

A

60. to

particle moves luith

•^

,

—

,. ^

- ;

{distancef

find the path and to distinguish the cases. The equation (4) of Art. 53 becomes

d^

^+«= Case to

a central acceleration

Let

,

I.

A*

Ai

•

^,«,

< fi,

,.e.

so that

d^ii

^-g,

|^

.\ f/jb = (p-lj«

—

1

is

,_.

(1).

positive

and equal

n^, say.

d-u

The equation

(1) is -^^

= «^it, the general

solution of which

is,

as in Art. 29,

u=

+ 5e-"« = L cosh nd + M sinh nO, M are arbitrary constants.

Ae-"^

where ^, 5, or X,

This is a spiral curve with an infinite number of convolutions about the pole. In the particular case when A on B vanishes, it is an equiangular spiral. Case II.

Let

A^

=

/x,

so that the equation (1) "• dO"-

.*.

where

A

u=Ae^B = A{d-a\

and a are arbitrary constants.

becomes

Dynamics of a

62

This represents

Let

Case III. to

— n-,

a reciprocal

A

when

particular case

/i^

so that |^

-

a

general.

In the

circle.

negative and equal

1 is

say,

nnti'nn CW The equation (1)

which

is

where

A

u=

A

j^^

dd

+ B) = A

cos {rtO

is

P is

= a,

given hj 6

The equations

61.

f.ViprpfnrA therefore

is is

= — n^u,

cos

—

n {d

the solution of

a),

and a are arbitrary constants.

The apse

when

in

spiral

zero, it is

is

> ^,

Particle

= A. 53

(4) or (5) of Art.

given and also the

A

u

will give the

path

initial conditions of projection.

which varies inversely ; if it be projected from an apse at a distance a from a velocity which is sJ2 times the velocity for a circle of radius a, shew that the equation to its path is Ex.

1.

particle moves with a central acceleration

as the cube of the distance the origin with

rcos-^ = a. Let the acceleration be

yuw^.

If Fi be the velocity in a circle of radius a with the

Fi2 — i- = normal

a

if

V be

.

^^

-a2-

the velocity of projection in the required path,

^ The

acceleration, then

a^ ••

Hence,

same

a

acceleration = —„

differential equation of the

path

dhi

a from equation

is,

fiu^

(4)

of Art. 53,

ij.

dT2+"=Pir2 = p"Hence, multiplying by

-j^

da

and

we have

integrating,

-'»•

r'=i"'[(sy"']=f"'-^ The

initial

conditions give that

when m = - then ,

a

Hence(l)gives

l'i

=

h^ =

from equation

(1)

l^^'^[_^^ 2ij.

and

= £. + 0. C--

2a2'

we have

m^"

2-

_

1 ,,

.

2a2*

-r-

dd

= 0,

and

v

— ^_:^. a

Examples

Central Forces.

6

63

%-^lif^) adn

(^>-

,

.

be measured from the initial radius vector, then ^ =

If

when u = -, and

7=-sin-i(l):

therefore

2

.Fir

e-\ cos

x/2j

Hence the path If

the curve r cos

is

hand

the negative sign on the right

we take

v/2

-^ = a. side of (2),

we obtain the same

result.

Ex.

A

2.

particle, subject

towards the origin,

from

velocity

equation

is

infinity at

to the

path

to

a force producing an acceleration

projected from the point

{a, 0)

an angle cot~^ 2 with

/j.

with a velocity equal

the initial line.

Shew

—g— to the

that the

is

r = a(l

+ 2sin

and find the apsidal angle and distances. The " velocity from infinity " means the

6),

would be acquired by the particle in falling with the given acceleration from infinity to the point under

consideration.

Hence

if

velocity that

V we

this velocity be

have, as in Art. 22,

F2= 1^

SO that

The equation

of

motion of the d^u

(1).

particle is

n

•••r'=2rH^^)j='^L3+2«"'j+^ If

jection,

we havepo = *

Hence,

initially,

sin a,

where cot a = 2,

(duy

„

Hence

(2) gives, initially,

5^

so that

C=

From

(2)

and

h^

i.e.

Pq=—-.

we have

'"nw

i.e.

(2)-

Pq be the perpendicular from the origin upon the initial direction of pro-

from ;i2

5

(1)

5

1

-^7==^

and

r

1

'''•

(3)

1

"1

^

= ~. da

we then have

(^)

=w2[2au + 3a2u2-l] = M2[au+l][3a«-l].

Dynamics of a Fartide

64 putting u = -

On

this equation gives

,

^^)=(a + .)(3a-r), and hence

^

0=

'

J sj{a+r ){Ba-r)

Putting r = a + y, we have

^

6=

I

-

,

=pin-i -^-

I: '2a

we measure

If

from the

6

hence 7 = 0. Therefore the path dr Clearly

37^ at/

= 0,

i.e.

initial

is

apse,

TT

.

2a r = a,

when

and

= a{l + 2 sin 6).

r

we have an

'

radius vector, then ^ =

^ = 2'

when

Sir

5ir

T'

T'

,

^*''-

Hence the apsidal angle is w and the apsidal distances are equal to 3a and and the apses are both on the positive directions of the axis of y at distances 3a and a from the origin. The path is a lima9on and can be easily traced from its equation. a,

EXAMPLES

A particle moves

1.

and

under a central repulsive force

the equation to the path t

—

"^^

=

projected from an apse at a distance a with velocity V.

is

time

]

is

Shew

t

that

rcos^^ = a, and that the angle 6 described

in

is

-tan-iK^

A

2.

;;

,

where

^0^

moves with a central

particle

=^

.

acceleration,

= -rTT—^^^

r-j,

and

is

projected from an apse at a distance a with a velocity equal to n times

that which would be acquired in falling from infinity apsidal distance ^ If

path 3.

72

is

= 1,

is

.

si if-

shew that the other

-I

particle be projected in

and the

;

.

any

direction,

shew that the

a circle passing through the centre of force.

A

moving with a central acceleration

particle,

projected from an apse at a distance a with a velocity

the path

1^

V shew ;

is

—

rcosh according as

F

^-

is

$

—

^

6

\

= a,

or t-cos

the velocity from infinity.

j^

;o

(distance)'

- 6 =a,

that

Examples

Central Forces,

(55

A particle moving under a constant force from a centre is projected a direction perpendicular to the radius vector with the velocityacquired in falling to the point of projection from the centre. Shew that its path is 3 , ($)' :C0S-2. 4.

in

move in a straight line through the path had always been this line. If the velocity of projection be double that in the previous case, shew that the path is

and that the origin in the

particle will ultimately

same way

^

5.

as

if its

/r-a

_i

,

1

,

/r-a

_,

/

A

particle

moves with a central

acceleration

/^

'/*

(

+

2a'^\ —_.)

being

,

projected from an apse at a distance a with twice the velocity for a circle at that distance find the other apsidal distance, and shew that the ;

equation to the path -^

6.

A

is

= tan-i(W3)-^tan-(x/g^). particle

where

^^

moves with a central acceleration

= 3^, /x (

?'

+

projected from an apse at distance a with a velocity 2'\/na; it

describes the curve r^ [2 7.

A

particle

+ cos

>^3d]

path

is

A

moves with a central acceleration

the curve

A-*

+2/*

shew that

= 3a^.

projected from an apse at distance c with a velocity its

being

-3 j

being

iji{r'-c*r),

./ -^ c^

',

shew that

= c*.

moves under a central force /nX I3a^u*+8au^] it is projected from an apse at a distance a from the centre of force with shew that the second apsidal distance is half the first, velocity ^^lOA and that the equation to the path is 8.

particle

;

;

2r=a\

l

+ sech-y^

.

A particle describes an orbit with a central acceleration fiu^—Xu^, 9. being projected from an apse at distance a with a velocity equal to that from infinity ; shew that its path is r

Prove also that

= a cosh

it will

n

,

where

^i^

+ 1 = Tlr^^

_

X

be at distance r at the end of time

Dynamics of a Particle

6G

lu a central orbit the force

10.

is fxv? {Z

projected at a distance a with a velocity

tan"^ - with the radius,

A

11. is

projected from an apse at a

shew that

its orbit is r

A

12. it

shew that the equation

the particle be

to the path is r

making

= a tan 6.

force

= a + b cosd.

moves with a central

particle

if

in a direction

m^ {3«?t* - 2 (a^-b^) u^}, a>b, and distance a + b with velocity '\/fi-7-{a + b)

moves under a

particle

+ ^i'^h'^);

/—^

acceleration \^ (8aii^ + ahc^)

;

projected with velocity 9X from an apse at a distance f from the

is

origin

;

shew that the equation to

its

J3\/ au-3 13.

A particle,

path

is

/au + 5

1

'^°^J6'

subject to a central force per unit of mass equal to fi{2{a^ + b'^)ii^-3a^'^vJ],

is

projected at the distance

angles to the initial distance 7*2

14.

it is

A

particle

a with a

velocity

—-

shew that the path

;

=a2 cos^

d+b"^ sin^

moves with a

in a direction at right is

the curve

6.

central acceleration

projected at a distance a with a velocity

/25

*/ -rr

fi

(u^--—ii~\

;

times the velocity for

4 a circle at that distance and at an inclination tan~' ^ to the radius vector.

Shew

that

its

path

is

the curve

A

particle is acted on by a central repulsive force which varies as 15. the nth power of the distance if the velocity at any point of the path be equal to that which would be acquired in falling from the centre to the point, shew that the equation to the path is of the form ;

71+3

r

An

^

„

—^ 6 = const. ,

cos

I, is tied to a particle at one end to a point in a smooth horizontal table. The particle can move on the table and initially is at rest with the string A blow (which, if directed along the string straight and unstretched. would make the particle oscillate to a maximum distance 2l from the fixed end) is given to the particle in a direction inclined at an angle a to the Prove that the maximum length of the string during the ensuing string. motion is given by the greatest root of the equation

16.

end and

elastic string, of natural length

is fixed

at its other

Examples

Central Forces.

A

17.

particle of

m

mass

is

67

attached to a fixed point by an elastic

nmg

string of natural length a, the coefficient of elasticity being

projected from an apse at a distance a with velocity

the other apsidal distance

A

18.

particle acted

on by a repulsive central force

projected from an apse at a distance c with velocity

it

will describe a three-cusped

Also

To

(5)

put

r^

\/^

-=-

(r^

- Qc^)-

shew that

J

and we have Sp'-=Qc^-r'^.

Vr^ - p2

,

ht=

giving

I

7

c

-^-a/ d V

-^ r-

5-.

- c-

= c- + 8c^cos^(p.]

Find the path described about a

19.

when the

/xr

hypocycloid and that the time to the cusp

of Art. 53,

hdt=p .ds=pdr

integrate,

it is

(r—a) — 2pha {r + a) = 0.

is

[Use equation

;

shew that

;

given by the equation

is

nr'^

^2pgh

fixed centre of force

acceleration toward the centre is of the form

by a

particle,

in terms of ^ + ^, ^3'

J.2

V

the velocity

Shew

20.

at an apse

whose distance

is

a from the centre of

force.

that the only law for a central attraction, for which the any distance is equal to the velocity acquired in

velocity in a circle at falling

from

infinity to that distance, is that of the inverse cube.

A particle

moves in a curve under a central attraction so that its any point is equal to that in a circle at the same distance and under the same attraction shew that the law of force is that of the inverse cube, and that the path is an equiangular spiral. 21.

velocity at

;

A

moves under a

central force m/x-r (distance)" (where be projected at a distance ^ in a direction making an angle ^ with the initial radius vector with a velocity equal to that due to a fall from infinity, shew that the equation to the path is 22.

n>l

particle

but not = 3).

If

it

w-3

If

n>3

n-3

/

_.)

X

shew that the maximum distance from the centre

is

R cosed"' ~^,3, and

if

23.

?i<3 then the particle goes to

A

particle

moves with

central

velocity of projection at distance

ultimately go

ofi"

to infinity if

infinity.

R

is

^^> -/} +

acceleration fiu^ + vu^ and the V; shew that the particle will

^i-

5—2

Dynamics of a

68

A

24.

—^

^(?i

per unit of mass

central attraction equal to

— l)a"-3r-" + Xr~3j where

shew that

;

In a central orbit 25. one of the conies (cm 26.

A

from an apse at a distance a with a velocity

particle is projected

and moves with a

particle, of

pole equal to

-~

mass m, moves under an

find the orbit

move with a

and interpret the

a/ -^

from an apse

to the orbit is r(l-4-cos''6) is (3a)

-

and

sin d in succession h^ {u

central acceleration

^

(1

+F

sin^ 6)

and

h^ {u sin

EUminating

u,

~

^,

result geometrically. "^f

by

and integrating, we have

cosd + u sin

^- mcos ^)=

= 2a,

x —^-

[Multiplying the equation of motion, h^ {u + u)=iji{l+k^sm^ 6)~ cos 6

is

attractive force to the

with velocity

It is projected

Shew that the equation

If a particle

time

P=^iu'^{cu + Q,os,d)~'^, shew that the path

if

and that the time of a complete revolution

27.

?i>3,

will arrive at the centre of force in

it

+ cos 6Y = a-\-b cos {±6 + a),

sin^ 6.

at a distance a.

Particle

6)

=fi sin

-ja cos ^ (1

(1 +F siu^ 6) ' ^+A, +F sin2 e)~^^i\ +k'^) + B.

(9

we have

A2tt=/i(H-Fsin2^)2^(l + P) + ^sin^-5cos^.] 28.

and

A particle

moves

in a field of force

whose potential

is fxr ~

-

cos 6

projected at distance a perpendicular to the initial line with

it is

2 velocity -

-Jy.

;

shew that the =

a sec

orbit described is

J 2 log tan -——

.

A

particle is describing a circle of radius a under the action of a 29. constant force X to the centre when suddenly the force is altered to \+fiii\n7it, where /x is small compared with X and t is reckoned from the Shew that at any subsequent time t the distance of instant of change.

the particle from the centre of force

3X

What

is

is

- an^(^Vi:^^"(V?)-""^' I

the character of the motion

if 2>\

= an'^

\

Stability [Use equations (1) and and the first then becomes

Xa3

r

r

=a+^

A

62.

where ^

small,

is

;

^.]

nearly a circle

is

find the condition that this

may

is

d^u

This will give a

r^ = sj\a?,

^ 7it.

.

A — /I sin

a path which

particle describes

of motion

.

the second gives

;

and neglect squares of

about a centre of force (= fxu^) be a stable motion.

The equation

69

orbits

of Art. 53

—^ = -

..

Put

(2)

of

„

/"-

circle of radius

-

if

/-IN

„

h'^

= fic^''^

(2).

Suppose the particle to be slightly displaced from the circular path in such a way that h remains unaltered (for example, suppose it is given a small additional velocity in a direction away from the centre of force by means of a blow, the perpendicular velocity being unaltered).

In (1) put u

= c + X, ,

,

where x

_(c +

d^x

is

small

then

;

it

gives

^)"--

Neglecting squares and higher powers of is always small, we have

x,

i.e.

assuming

that X

d'^x

._ __ = _(3-n)^.

If n be

<

3, so

that 3 a? = ^

If

n be >

3,

so that

x=

?i

—

.

n

is positive, this

cos [\/3

—

3

is

-nd

-Y

gives

B].

positive, the solution is

A leV'^^s 6 + Bye--J~>^^ ^

X continually increases as 6 increases hence x is not always small and the orbit does not continue to be nearly

so that

;

circular.

If

n<

3,

the approximation to the path

u=

c

is

+ Acosl'^S-nd + B]

(4).

Dynamics of a

70 The i.e.

;77i

= 0,

= sin[\/3-?i^ + 5]. of this

solutions

difference

This

c^n^tCl^.^ c\r\ by the equation

apsidal distances are given

by

The

Particle

is

If

w

between

equation are a series of angles, the

their

successive

values

beinef

.

,

therefore the apsidal angle of the path.

=

3,

this apsidal

angle

is

infinite.

In this case

it

would be found that the motion is unstable, the particle departing from the circular path altogether and describing a spiral curve.

The maximum and minimum values

c+A

are

and

c

— A,

of

so that the motion

it,

is

in the case n

<

3,

included between

these values. case may be considered in the Let the central acceleration be ^ (u). The equations (1) and (2) then become

The general

63.

same

manner.

and

h'c'

Also (3)

is

=

fx,i>

(o)

(6).

now

de'^^'^''''J{cy {c±xy

= c-1x + X -^tM

neglecting squares of x.

,

(p{c)

••

and the motion

In

is

dd'

{

4>o

stable only if

this case the apsidal angle is

Central

and

transversal accelerations

1

If, in addition to the central acceleration P, we have 64 an acceleration T perpendicular to F, the equations of motion

are

= -P dt'

dt

dd

and Let r

Dynamics of a

72

If the nearly circular orbit of

2.

shew that the apsidal angle [Using equation

A

3.

particle

apsidal angle

a particle be

53 we see that

(5) of Art.

(a'"

""*-?"»-

= Jni^

2)

P

*/ 1 +T2>

varies as r'""^; the

62.]

moves with a central acceleration

is 7r-=-

^''^

-j- nearly.

is

from Art.

result then follows

Particle

where -

is

,-(^

-

-jj

;

shew that the

the constant areal velocity.

Find the apsidal angle in a nearly circular orbit under the central

4.

force ar»'

+ 6>'".

Assuming that the moon

5.

is

acted on by a force

.

,.

J^

r-j

to the

earth and that the effect of the sun's disturbing force is to cause a force m? X distance from the earth to the moon, shew that, the orbit being nearly circular, the apsidal angle

mean lunar month, and cubes

A

6.

particle

is

shew

that, if the

mean

iri\+^~A

nearly,

where

—

is

a

m are neglected.

in

an approximately circular orbit under the

^ and

a small constant tangential retardation/;

moving

action of a central force

of

is

3 f distance be a, then 6 = nt + ^ - t\ squares of

/

being neglected.

M and

ends of an a smooth fixed ring, the whole The particle m being projected at right shew that its path is

T\uo particles of masses

7.

in are attached to the

inextensible string which passes through

resting on

angles

a

horizontal table.

to the string,

\_VnVVJj^\The

tension of the string being T, the equations of motion are

fdd\-^_T ^_ m df ^\dt)

and

^^^'

-rdtV'^tr^

^^)'

^^^^-^'^^-J

^•^-

Examples

Central Forces. r^e=h

(2) gives

and

and then

(1)

since r

zero initially,

is

73

fH

(3) give

This equation and

when

(4)

(4),

= -^.

\r

r=^a.

give

r^-d^

•

and

C vanishes

^^ \

'•

if

/

m

f

rr=

—adr

.

/-i

a

a

= C03-1 - +

~ (7,

6 be measured from the initial radius vector. .-.

a = rcos[^^^^^]isthepath.

Two masses i/, m are connected by a string which passes through 8. a hole in a smooth horizontal plane, the mass m hanging vertically. Shew that describes on the plane a curve whose diflferential equation

M

is

/ m\ d'^u _ mg 1 \-^Jl)'d&^^^~~M A2^-

Prove also that the tension of the string

In the previous question

9.

the plane with velocity

if

m=M,

a/ —^ from

is

and the

latter be projected

an apse at a distance

the former will rise through a distance

a,

on

shew that

a.

M

Two

particles, of masses and m, are connected by a light the string passes through a small hole in the table, in hangs vertically, and describes a curve on the table which is very nearly a circle whose centre is the hole ; shew that the apsidal angle of the 10.

string

;

M

orbit of

M

/

,.-

m can move on a smooth horizontal table. which passes through a smooth hole in the and is attached to table, goes under a small smooth pulley of mass a point in the under side of the table so that the parts of the string hang vertically. If the motion be slightly disturbed, when the mass m 11.

A

is it *

particle of

mass

It is attached to a string

is

M

describing a circle uniformly, so that the angular

changed, shew that the apsidal angle

is it

a/

—

-t^

momentum •

is

un-

Dynamics of a Particle

74 12.

Two

particles

on a smooth horizontal table are attached by an and are initially at rest at a distance a

elastic string, of natural length a,

One particle is projected at right angles to the string. Shew that the greatest length of the string during the subsequent motion be 2a,

apart. if

then the velocity of projection

*/ -^—

is

,

between the masses of the particles and X

where is

m

is

the harmonic

mean

the modulus of elasticity of

the string.

is

[Let the two particles be the one that is projected.

and therefore

T=\

of tension T, such that

T

is

A and B of masses M and M', of which B When the connecting string is of length r

Yf along AB, and that of

B

.

is

T

—

along

,

,

a

the acceleration of

BA.

To

.4

get the relative

B

and A an acceleration equal and opposite to that of A. The latter is then "reduced to rest" and the acceleration of B relative to A is along BA and

motion we give to both

_T ~M

The equation

T^_2 M'

m

to the relative path of

(Pu

_

r-a _

2X l-ait

~ ma

a

u

B is now 2\

\

— au

Integrate and introduce the conditions that the particle

an apse at a distance a with velocity apse at a distance 2a determines F.]

V,

The

is

projected trom

fact that there is another

CHAPTER V UNIPLANAR MOTION WHEN THE ACCELERATION IS CENTRAL AND VARYING AS THE INVERSE SQUARE OF THE DISTANCE 65.

In the present chapter we shall consider the motion when

the central acceleration follows the Newtonian

Law

of Attrac-

tion.

This law particles, of

may be

masses

mutual attraction

between every two and m^ placed at a distance r apart, the

expressed as follows

wij

;

is

7

—~

is a constant, depending on the units of mass and length employed, and known as the constant of

units of force, where 7

gravitation.

If the masses be

the value of 7 is

is

measured in pounds, and the length in feet, and the attraction

1*05 x 10~' approximately,

expressed in poundals. If the masses be

measured in grammes, and the length in 7 is 6'66 x 10~^ approximately, and

centimetres, the value of

the attraction 66.

A

is

expressed in dynes.

particle moves in a -path so that

its

a fixed point and

to

always directed sheiu that its

to

path

is

a conic

the three cases that arise.

section

is

and

equal to

acceleration is

jj^.

x; ;

to

distinguish between

Dynamics of a Particle

76

When

P=

-, the equation (5) of Art. 53 becomes

p'dr Integrating

we

^'

have, by Art. 54,

= ^ = 2^ +

.^

:....(2).

(7

r

jf-

Now

^

r'

the (p, r) equation of an ellipse and hyperbola, re-

ferred to a focus, are respectively

-= — -1,

- = — +1

and

r

p-

(3),

r

p>

where 2a and 26 are the transverse and conjugate axes. Hence, when G is negative, (2) is an ellipse; when

G

is

positive, it is a hyperbola.

Also (p, r)

when (7=0,

— = constant,

becomes

(2)

Hence

and

its focus.

(2) always represents a conic section,

at the centre of force,

this is the

r

equation of a parabola referred to

and which

is

whose focus

is

an negative]

ellipse "I

parabola

>

according as

G is

or hyperbola] i.e.

according as

velocity at

v^

| -^

any point

P

zero or positive

,

according as the square of the

i.e.

^

2.-1

=

is

,

where S

Again, comparing equations (2) and of the ellipse,

(3),

the focus.

is

we

have, in the case

h^^/^^G_ b'

.'.

h

= Kf

/J,

- =

*

1

"J/xx semi-latus-rectum,

Hence, in the case of the So, for the hyperbola,

and, for the parabola,

-

a

v-

v-

ellipse, v^

= =

(2 yu,

(

2a -

-

.

= fi

\\

+-

1

( ~^

\v

and G

—

-] aj

=

.

(4).

Law

Central Forces.

of

the Inverse

Square 11

be noted that in each case the velocity at any point

It will

does net depend on the direction of the velocity.

Since h

is

twice the area described in the unit of time

(Art. 54), therefore, if

T

be the time of describing the

ellipse,

we

have

^

area of the ellipse

irah

%

time varies as the cube of the

so that the square of the periodic

major

Itr

axis.

Cor.

in

any direction the path

hyperbola, according as F^

Now falling

R

If a particle be projected at a distance

1.

V

velocity

<=>

is

an

ellipse,

with

parabola or

^

the square of the velocity that would be acquired in

from infinity to the distance R, by Art. 81.

-/:(-r^)-=HMHence the path

is

an

ellipse,

ing as the velocity at any point

is

parabola or hyperbola accord-

< = > that acquired

in falling

from infinity to the point. Cor. radius

2.

R

is

The

velocity Vi for the description of a circle of

given by

-^ = normal

acceleration

li _.

= -^,

,

so that Vi^

JrC'

•

V — '^^'^ocity from

=

^ K

.

infinity

In the previous article the branch of the hyperbola is the one nearest the centre of force. If the central acceleration be from the centre and if it vary as the inverse square of the distance, the further branch is described. For in this case the equation of motion is 67.

described

^^£ = -ii.

.-X^-^ + C

(1).

Dynamics of a

78

Now bola

the (p, r) equation of the further branch of a hyper•

is

^-

f~ and

Particle

= v/i

_

T'

X semi-latus-rectum, and

a -

h— =

with (1) provided that

this always agrees

that h

1

^^

=

(7,

so

/I

h'~

v'

P

Construction of the orbit given the point of projection and

68.

and magnitude of the velocity of projection. Let 8 be the centre of attraction, P the point of projection, TPT the direction of projection, and V the velocity of pro-

the direction

jection.

Case

Let F^ <

I.

o^

whose major axis 2a

ellipse

is

a)

by Art.

'

'

QQ, the path is

so that

^^'^

2a = 2ix

Draw PS', so that PS' and PS are on the same making Z T'PS' = Z TPS, and take PS'

Then

=2a-SP = 2a-R =

S' is the

an

given by the equation

R = SP,

V^ = ,j,(%-^, where

M

then,

;

- V'U

side of

TPT',

2fi-V'R'

second focus and the elliptic path

is

therefore

known. Case II.

Draw

Let V^

- ^^

,

so

path

that the

a parabola.

is

the direction PS' as in Case I; in this case this

direction of the axis of the parabola.

meet

TPT in

to

YA

perpendicular to SU. is S,

Then

A

the perpendicular from

Let V^

S

the

to

TPT

and

the vertex of the required

SY2

.

-^^^

=

2«o" -^

where

,

«o is

on the direction of projection.

> ^^

transverse axis 2a given

is

is

parallel to

and the curve can be constructed.

The semi-latus-rectum = 2SA =

Case III.

SU

U; draw >SF perpendicular

PS'

parabola whose focus

Draw

,

so that the path

is

a hyperbola of

by the equation

and hence 2a = (l + a)'

— V K» —

-,r.,

s~ Zfjb

•

Kepler's Laivs In this case PS'

such that Z

TPS =

79

on the opposite side of TPT' from PS, z TPS', and S'P - SP = 2a, so that

lies

The path can then be

constructed, since S'

is

the second focus.

Kepler's Laws. The astronomer Kepler, after many 69. years of patient labour, discovered three laws connecting the motions of the various planets about the sun. They are

Each planet

1.

describes

an

ellipse

having the sun in one of

its foci. 2.

to the

The areas described by the radii draum from the planet are, in the same orbit, proportional to the times of

sun

describing them. 3.

are

The squares of

pt'^'opoi'tional to the

the periodic times

cubes of the

of the various planets major axes of their orbits.

From the second law we conclude, by Art. 54, that 70. the acceleration of each planet, and therefore the force on it, is directed towards the Sun. From the first law it follows, by Art. 55 or Art. 66, that the acceleration of each planet varies inversely as the square of its distance from the Sun.

From

the third law

it follows,

T^

since from Art. 66

we have

= ^-^.a^ H-

that the absolute acceleration distance from the Sun)

Laws

is

/x

{i.e.

the acceleration at unit

the same for

all planets.

similar to those of Kepler have been found to hold

for the planets

and their

satellites.

It follows from the foregoing considerations that

assume Newton's Law of Gravitation

to

we may

be true throughout the

Solar System. Kepler's

71.

Laws were obtained by him, by a

process of

he found one that was suitable he started with the observations made and recorded for many years by Tycho Brahe, a Dane, Avho lived from A.P. 1546 to 1601. continually trying hypotheses until ;

80

Dynamics of a Particle The

first and second laws were enunciated by Kepler 1609 in his book on the motion of the planet Mars. The third law was announced ten years later in a book

in

entitled

On

the

Harmonies of

these laws was given by

The explanation of in his Principia published

the World.

Newton

in the year 1687.

Kepler's third law, in the form given in Art. 69, is 72. only true on the supposition that the Sun is fixed, or that the mass of the planet is neglected in comparison with that of the Sun.

A more accurate form is obtained in the following manner. Let S be the mass of the Sun, P that of any of its planets, and 7 the constant of gravitation. between the two

is

thus 7

—

V-

.

,

The

force

where r

is

of attraction

the distance

r-

between the Sun and planet at any instant.

The

acceleration of the planet

the Sun, and that of the

Sun

is /3

is

(

then a

= -^ j

(

=

^

j

towards

towards the planet.

To obtain the acceleration of the planet relative to the Sun we must give to both an acceleration ^ along the line PS. The acceleration of the Sun is then zero and that of the planet is a + /3 along PS. If, in addition, we give to each a velocity equal and opposite to that of the Sun we have the motion of P relative to the Sun supposed to be at rest. The relative acceleration of the planet with respect to the Sun then=a + ^ = :yl^\ Hence

the

/x.

we then have T

of Art. 66

is

JliS + P)

7 {S + P), and, as in that

article

Laws

Kepler's

If 7\ be the time of revolution

and

a^

of the relative path of another planet Pj, rp

27r

^

I

T-

Law,

Kepler's ^

that

^,

as

a/

1\^

true,

follows

it

similarly

a?

.

varies

777-

Si

approximately

the semi-major axis

we have

>S+P T^_^ "S + P^T,^ a,^'

JjiS + Pr)"^' Since

81

that

4-

very ^

is

P

—^

-^

very nearly

is

and hence that P and Pj are either very nearly equal or very small compared with S. But it is known that the masses of the planets are very different hence they must be very small compared with that of the Sun. unity,

;

73.

to give

The

corrected formula of the last article

an approximate value

Sun

planet to that of the

to

may be used

the ratio of the mass of a

in the case

small satellite, whose periodic time and

where the planet has a

mean

distance from the

planet are known.

In the case of the the force which for

all

the attraction of the planet

satellite

is

practical purposes determines its path.

If P be the mass of the planet and D its mean distance from the Sun, then, as in the previous article,

T= p

Similarly, if

^"^ ,

^y{S + P)

be the mass of the

distance from the planet, and 27r

3

~\/j(P + p) The

quantities T,

value for As

all

^ -^p P

a numerical

t,

D

t

its ^

'

d

satellite,

its

mean

periodic time, then

S + P T'_D'

" P+p

f-

~d'-

and d being known,

this

gives

a

.

example take the case of the Earth

Now T = 365i days, ÂŤ = 27^ days, the values being approximate.

L. D.

D\

E

D= 93,000,000 miles,

and the Moon m.

and d = 240,000 miles,

6

Dynamics of a Particle

82 Therefore

Moon. 5f

.-.

.5

+

= 325900

7?

sum

times the

the Earth

of the masses of

and

m=^E nearly.

Also

= 330000 E

nearly.

a fairly close approximation to the accurate result. If the Sun be assumed to be a sphere of radius 440,000 miles and mean density n times that of the Earth, assumed to be a sphere of radius 4000 miles,

This

is

this gives

n X (440000)3 = 330000 x 71=

.'.

330000

=

1103

Hence the mean density

of the

mean

1331

(4000)3.

,1

= about ''"""" 4*

Sun

= - that of the Earth =- x 5 so that the

330

density of the

.

527 = about 1*4 grammes per cub. cm.,

Sun

is

much

again as that of

mean

distance and

nearly half as

water.

74.

It

not necessary to

is

periodic time of the planet

rather the

sum

of its mass

P

know

the

in order to determine its mass, or

and that of its

satelHte.

be the masses of the Earth and Moon, R the distance of the Earth from the Sun, r that of the Moon from the Earth, if Y denote a year and y the mean lunar

For

if

E

and

m

month, then we have

F= ,-^

^ R^

9r

y= ^

,^

^ry

(1),

3

-- .r^

(2).

(E+ 711)

Also, as in the last article,

t= From

(1)

and

(3),

From

(2)

and

(3),

,

^„

-

-d'

(3).

(P+^)| = (^+m).^ Equation (4) gives the ratio oi P Equation (5) gives the ratio of P

(5).

+p

to

S+

+ _p

to

^ + m.

E.

Law

Examples

of the Inverse Square.

83

EXAMPLES Shew

1.

that the velocity of a particle moving in an ellipse about

a centre of force in the focus

is

perpendicular to the radius and

^

of

two constant

that, at

any point of

its

velocities,

^

perpendicular to the major axis.

A particle describes an ellipse about a centre

2.

shew

compounded

of force at the focus

;

path, the angular velocity about the other

focus varies inversely as the square of the normal at the point.

A particle

3.

moves with a central acceleration

projected with velocity

V

rectangular hyperbola

the angle of projection

if

at a distance R.

.

A

particle describes

,.

that

—^

f"

its

it is

;

path

is

a

is

fi

_i

VR 4.

Shew

=

an

(--1/

ellipse

under a force

,.

/^

-^

towards

the focus if it was projected with velocity Ffrom a point distant r from the centre of force, shew that its periodic time is ;

27rr2_ r2-|-f

If the velocity of the Earth at any point of its orbit, assumed to 5. be circular, were increased by about one-half, prove that it would describe a parabola about the Sun as focus.

Shew

also that, if a

body were projected from the Earth with a velocity it will not return to the Earth and may even

exceeding 7 miles per second, leave the Solar System.

A

from the Earth's surface with velocity v diminution of gravity be taken into account, but the resistance of the air neglected, the path is an ellipse of major axis 6.

shew

— —

—-^ "zga

particle is projected

;

that, if the

?,

where a

is

the Earth's radius.

v''

Shew that an

7.

unresisted particle falling to the Earth's surface from

a great distance would acquire a velocity

'^'iga,

where a

is

the Earth's

radius.

Prove that the velocity acquired by a particle similarly

Sun

is

falling into the

to the Earth's velocity in the square root of the ratio of the

diameter of the Earth's orbit to the radius of the Sun.

6—2

Dynamics of a

84

Farticle

If a planet were suddenly stopped in its orbit, supposed circular,

8.

shew that

it

would

fall

into the

Sun

in a time

which

is

^ 8

times the

period of the planet's revolution.

The

9.

eccentricity of the Earth's orbit round the

Sun

is

—

;

60

shew

Sun exceeds the length of the semiduring about 2 days more than half the year.

that the Earth's distance from the

major axis of the orbit 10.

The mean distance

of

Mars from the Sun being 1-524 times that Mars about the Sun.

of the Earth, find the time of revolution of

The time

Mars about the Sun

is 687 days and his the distance of the Satellite Deimos from Mars is 14,600 miles and his time of revolution 30 hrs. 18 mins. shew that the mass of the Sun is a little more than three million times

11.

mean

of revolution of

distance 141| millions of miles

;

that of Mars. 12.

and

his

The time of revolution of Jupiter about the Sun is mean distance 483 millions of miles the distance ;

satellite is

11-86 years of his first

261,000 miles, and his time of revolution 1 day 18^ hrs. shew is a little less than one-thousandth of that of the ;

that the mass of Jupiter

Sun. 13.

and

its

The outer

16| days approximately, 26j radii of the latter. The revolves in 12 hours nearly find its distance

satellite of Jvipiter revolves in

distance from the planet's centre

last discovered satellite

is

;

from the planet's centre. Find also the approximate ratio of Jupiter's mean density to that of the Earth, assuming that the Moon's distance is 60 times the Earth's radius and that her siderial period is 27J days nearly. [Use equations (2) and (3) of Art. 74, and neglect m in comparison with E, and p in comparison with P.] 14.

A

planet

its velocity

that

is

Sun as focus shew when the radius vector to

describing an ellipse about the

away from the Sun

is

greatest

the planet is at right angles to the major axis of the path,

then

is

,

,

where 2a

is

the major axis,

e

;

and that

it

the eccentricity, and

T

the periodic time.

75.

To find

of a given arc of an end of the major aocis.

the time of description

elliptic orhit starting

from

The equation

=h

r^-j-

ht=

the nearer

of Art. 53 gives

j\"-dd Jo

=

\\.^—a^^.de

Jo(l +ecos^)2

(1) ' ^

Time of describing any arc If

\ be >

85

we have

1,

e

dO

\{\ +tan2-j + (l-tan2 /X +

=

^

tan-^ Itan ^

,

.

1

/^^^

, 6/

—

^

I)

,

.

I

Hence, by differentiation with respect to the constant have j(A,

+

. ^

,

tan~^

(\--l)*

r

.

-}-

['""IVx-Tl]

2!

A -11 +

tan ^ a /

2V

L

A.

r

^

X + lJ

= - we ,

sin d

— 1

?i

\ + cos^ ;^

x

r^;

X^-l

have

dd

+ eccos ey

2

=

5

(1

i

^

In this equation, putting

j(l

we

tan-

A.

=—

tan"

(V-l)S

2

2>"

2\

=-

dd cos^)2

\,

-

T.tan tz\/ ^ /l^^l

^ tan~^

q

—

sin ^ ^

« z

, :;

e cos

e^)

Hence substituting since Y

in equation (1)

we

^

have.

= -r= = -;= =

}.

a^(l-e^)^

L(i-e^)2

= —=

I

2 tan~^

I

tan

fvrr

vr

~|^

sin B

e

1

— e-1 + 1

ecos ^Jo

+ e cos

f^

Dynamics of a Particle

86 [An

alternative

method

of obtaining

this result

will

be

given in Art. 82.]

76.

To find

Here

e is

>

a hyperbolic

the time similarly for

1,

so that

we put

e

= - where X <

orbit.

1.

Hence sec^

2 ^^

l+\-{l-\) setf

1 1

2 ^^

+X

,

V1 =-.

tan"-

Vi-x + tan

/i-x,

1

J

/1

+ A,

tan

+ X + V 1 - A. tan

log

^1-^' "vrTx-Vl-Xtan| Differentiating with respect

to

X,

we

have,

after

simplification,

Vl + X +

_[_dO__ (1-v)^

\/l

- X tan

Vl + X - Vl - X

;

tan

iin^ 1

Replacing X by -

,

- X^ X + cos

we have

d0 •'(1

+

;

cos

ey

I

{e''

+—

-

1)^

Ve +

1

+ Ve- 1

tan

Ve +

1

- Ve - 1

tan

sin 6 1

1

+ e cos ^

01 t;

some

Time of describing

aiiy arc

87

Hence, since in this case ,a

h

'Jim

*

^fj,

the equation (1) of the last article gives ^

.

TZl ^15i_ _ log. 1+ecos^

'\/fi

In

77.

the case

of a paraholic

Ve +

+

1

-

A/e

1

tan

_

?

°

.

.

ve +

— ve— 1 tan-

1

orbit to find the corresponding

time.

The equation ^

to the parabola is r ^

the latus-rectum and

is

=

=

1

+ cos

measured from the

-^ ^

,

axis.

where 2d

is

Hence the

equation (3) of Art. 53 gives

h.t=(rW=[j^-^ '(l+cos6')^^^" coi J

.

ht

[6

J

d?

a be the

d'

projectile,

is

d«

of the air being

attraction of the Earth at a point outside

r irom the centre

vacuo

j.a

vacations of gravity being taken

into consideration but the resistance

The

J

apsidal distance.

Motion of a

78.

+

h^^^^

But

if

(1

ir

de

is •^.

Hence the path

neglected.

at a distance

of a projectile in

one of the cases of Art. QQ, one of the

described being at the centre of the Earth.

it

foci of

the path

Dynamics of a

88 If

R

Particle

be the radius of the Earth, then

gravity at the surface of the Earth

The path

= g,

~ R"

so that

yu,

the value of

= gR-.

of a projectile which starts from a point on the is therefore an ellipse, parabola, or hyperbola

Earth's surface

according as V^ =

~

,

i.e.

according as V^ = 2gR.

79. The maximum range of a particle starting from the Earth's surface with a given velocity may be obtained as follows Let S be the centre of the Earth and P the point of projection. Let be :

K

the point vertically above that,

by Art.

31,

P

to

which the

velocity, V, of projection is due, so

we have (1),

where

R

is

the radius of the Earth and

Hence, by equation

By comparing the second focus is

P

and radius

(4)

PK is h.

of Art. 66,

this with equation (1) is,

h.

we have

PH=h,

so that the locus of

for a constant velocity of projection, a circle

It follows

that the major axis of the path

is

whose centre

SP + PH

or

SK.

Planetary Motion The

ellipse,

whose

foci are

89

S and H, meets a plane LPM, passing through

the point of projection, in a point Q, such that SQ + QH—SK. Hence, if SQ meet in T the circle whose centre is S and radius SK, we have QT=QH. Since

H

is, in general, another point, H', on the circle of foci equidistant with from Q, we have, in general, two paths for a given range. The greatest range on the plane LPM is clearly Pq where qt equals qO. Hence Sq + qP=Sq + qO + OP = Sq + qt + PK^SK+ PK. Therefore q lies on an ellipse, whose foci are the centre of the Earth and the point of projection, and which passes through K. Hence we obtain the maximiun range.

there

Suppose that the path described by a planet

80.

the

Sun S

is

the ellipse of the figure.

major axis and produce Let G be the centre.

to the Q.

it

to

Draw

P about

PN perpendicular

meet the auxiliary

circle in

The points A and A' are called respectively the Perihelion and Aphelion of the path of the planet. The angle ASP is called the True Anomaly and the angle ACQ the Eccentric Anomaly. In the case of any of the planets the eccentricity of the path

is

small, being never as large as

1

except in the case of Mercury when it is "2 the foci of the path are therefore very near C, the ellipse differs little in actual ;

shape from the auxiliary the ,True and Eccentric