William Kingdon Clifford. # Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) online

. **(page 5 of 9)**

Online Library → William Kingdon Clifford → Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) → online text (page 5 of 9)

Font size

disks, through the hole in the other, to a bullet on the

other side.

Hang up this spring in a horizontal position, by thin

strings fastened to the disks. Let the

bullet also be hung by a thin vertical

string. Now cut the string which fastens

the bullet to the further disk. The spring

will then open and the bullet will move

away. It will begin to move with a certain

acceleration, depending upon the compression of the spring.

Now suppose the spring to be hung up to the same

support as the bullet, so that the two may swing in con-

tact, in a direction perpendicular to the axis of the spring ;

the spring being compressed as before. At any instant of

the motion, let the string be again cut. Then the spring

will begin to open, and the bullet to move away ; but it

will always begin to move with the same acceleration in

the direction of the spring, provided that the compression

is always the same.

If we allow the spring and bullet to swing in the

direction of the spring's axis, the bullet will have an

acceleration in the direction of the axis except when it is

passing through the lowest point of its swing. If the

string be cut at that instant, the acceleration of the bullet

will be the same as before. But if the string be cut at

any other instant, the difference between the acceleration

before and after that instant will be precisely the accelera-

tion with which the bullet began to move away from rest.

58 DYNAMIC.

The greater the compression of the spring, the greater

will be this acceleration. The method of finding the

acceleration when the compression is given will be subse-

quently investigated.

We learn from these experiments that under certain

circumstances a body A (the bullet) in contact with a

strained body B (the spring) has an acceleration depending

upon the strain of B, but wholly independent of the

velocity of A or B. We have supposed the bullet and

string to be moving with the same velocity, in order to

make sure that the strain of the spring was always the

same. If however the same condition of strain could be

secured in any other way, the acceleration would be found

quite independent of the velocity of the spring.

MASS.

If we now cut the bullet in two, and use one half in

the same way, we shall find that for any given state of

strain of the spring the acceleration is double what it was

before. And generally, if we use any other piece of lead

we shall find that the accelerations in the two cases are

inversely proportional to the volumes of the pieces of lead;

or, we may say, to the quantities of lead.

The same thing is true if we take different pieces of

wood, or of any other substance, provided that the sub-

stance is homogeneous, that is, of the same nature all

through. But if we compare the accelerations of a piece

of wood and a piece of lead, we shall find that they are by

no means inversely proportional to the volumes of the two

bodies. A piece of wood, whose acceleration in the same

circumstances is the same as that of a piece of lead, will

be very much larger than the lead.

Let us now take an arbitrary body, say a certain piece

of platinum. Of every other sort of substance, as of lead,

wood, iron, etc., let us find a piece which under the same

circumstances (i.e. with the same strain of the spring) has

the same acceleration as the piece of platinum, and let

each of these be called the unit of quantity of the substance

MASS. 59

of which it is composed. Then if we consider a homo-

geneous body composed of one of these substances, the

number of units of quantity which it contains is called the

mass or measure of the body *. It is evidently the same as

the number of units of quantity of platinum in a lump

which has the same acceleration as the given body.

The piece of platinum actually -f- used is called the

" kilogramme des archives," and is preserved in Paris. For

convenience the unit of quantity is taken to be one-

thousandth part of this, and is called a gram. Thus we

may define:

The mass of a body is the number of grams of platinum

in a lump which under the same circumstances has the same

acceleration as the given body.

If the body is not homogeneous, but is made up of

parts of different kinds, the mass of the body is the sum of

the masses of the parts.

The mass of a cubic centimetre of any substance is

called the density of that substance.

We have supposed the densities of different substances

to be measured by means of a certain spiral spring. We

should, however, find exactly the same densities if we had

used any other strained body.

The product of the mass of a body by its acceleration

shall be called for shortness the mass-acceleration. Since

we have found that the accelerations of any two bodies, in

a given state of strain of the spring, are inversely propor-

tional to their masses, it follows that the mass-acceleration

of all bodies in contact with the spring in a given state of

strain, is the same. This mass-acceleration is called J the

stress belonging to that state of strain.

In all this we have supposed the motion of the body to

be pure translation.

The product of the mass of a body by its velocity is

* [contrast with definition on p. 1.]

t [? theoretically.]

J [rather ' taken as the measure of'.]

60 DYNAMIC.

called its momentum. Thus the mass-acceleration is the

rate of change of momentum. Both momentum and mass-

acceleration are directed quantities.

LAW OF COMBINATION.

If a body be in contact at the same moment with two-

strained bodies, its acceleration is the resultant of the

accelerations which it would have when placed in contact

with the two bodies separately. And in general, if a body

be in contact with any number of strained bodies, its actual

acceleration is the resultant of the accelerations due to the

several bodies. Here again we assume that the whole

motion is one of pure translation, or that the body may be

regarded as a particle.

LAW OF RECIPROCITY.

Suppose two ivory balls, A, B, to be hung up side by

side ; let A be pulled away and then

let go so as to impinge on B. The

velocity of A will appear to be suddenly

changed, and B will appear to suddenly

acquire a velocity. The change, how- "*" OCX

ever, is not really sudden. At the

moment of contact A has a certain velocity, but no

tangential acceleration ; B has neither velocity nor

acceleration. After the contact, both bullets become

compressed in the neighbourhood of contact ; and then

A has at every instant an acceleration opposite to its

velocity, depending on the strain of B, while B has an

acceleration in the direction of A's velocity, depending on

the strain of A. After the compression has attained a

certain magnitude, the compressed parts begin to expand

again ; and so long as any strain remains, each of them

has an acceleration depending on the strain of the other.

The two strains cease at the same moment, and then the

bullets separate. But the whole time during which they

are in contact is too short to be perceived by ordinary

means.

MOMENTUM. 61

During all this time, however, the mass-acceleration of

A, due to the strain of B, is equal and opposite to the mass-

acceleration of B, due to the strain of A. And the result is

that the change of momentum of A during the contact is

equal and opposite to the change of momentum of B. Or,

as we may otherwise state it, the sum of the momenta of

the two balls is the same before and after the impact.

The same thing holds good in the case of the bullet

and spiral spring which we previously considered. In that

case, however, the different parts of the spring are moving

with different velocities and accelerations. But if we

reckon the mass-acceleration of the whole spring as the

sum of the mass-accelerations of its parts, it will still be

true that at every instant the mass-acceleration of the

bullet, due to strain of the spring, is equal and opposite to

the mass-acceleration of the spring, due to strain of the

bullet. For there is a slight compression, both of the

bullet and of the disk with which it is in contact.

If we shorten the spring, by cutting off a part from the

end away from the bullet, we shall make no difference to

the acceleration of the bullet, provided that the remaining

part is kept in the same state of strain as before. The

only difference will be that this acceleration will diminish

more rapidly and last a much shorter time ; so that the

bullet will acquire on the whole a less velocity. And

finally if we remove the spring altogether, leaving only the

disk at the end ; and suppose the string stretched in any

other way so as to produce the same compression of the

disk as before*, and then to be suddenly cut; the accelera-

tion of the bullet will be the same. But in this case it

will exist only for an imperceptible time, during which the

bullet will acquire only a very small velocity.

In general, the mass-acceleration due to the strain of

two bodies hi contact depends only on the strain of each

at the surface of contact. No body can have a strain at its

surface unless it is in contact with another body.

If we draw an ideal surface separating a body into two

parts, each of these parts has a mass-acceleration due to

* [Is the disk compressed?]

62 DYNAMIC.

the strain at the surface of separation. For example, if we

divide the spiral screw by an imaginary plane of section,

the mass-acceleration of the portion to the left of the

plane is equal and opposite to that of the

portion to the right together with the bullet,

and each is due to the strain of the spring

at the point of section. This mass-accelera-

tion is called the stress across the section.

When it is away from the section on both sides, it is called

pressure; when it is towards the section, it is called tension.

Thus we see that what takes place at the common surface

of two bodies in contact is a particular case of what takes

place throughout the interior of any body.

GRAVITY.

When bodies are let go in the open air, they fall with

more or less rapidity to the ground. This difference of

velocity is found to depend on the presence of the air; and

in the exhausted receiver of an air-pump the most differ-

ent bodies fall through the same distance in the same

time ; having, as we remarked before, a constant accelera-

tion of 981 centimetres a second per second. Thus every

body left free in vacuo has a mass-acceleration vertically

downwards, proportional to its mass.

The acceleration of the Moon is found to be very

approximately the resultant of two accelerations, one

directed towards the Earth, and the other towards the

Sun. The acceleration towards the Earth is about one

3600th part of the acceleration of gravity, but varies with-

in certain limits, being inversely as the square of the

distance from the Earth's centre. Now the Moon is

distant from that centre on the average about 60 times

the Earth's radius, which is the distance from the centre

of bodies on the Earth's surface. Hence the acceleration

of the Moon towards the Earth is the same as that of any

terrestrial body would be at the distance of the Moon,

supposing the acceleration of the terrestrial body to vary

inversely as the distance from the Earth's centre. Thus

the Moon is to be regarded as a falling body.

GRAVITY. 63

Its acceleration towards the Sun is inversely as the

square of the distance from the Sun; but although that

distance is 400 times the distance from the Earth, this

acceleration is always greater than the other, so that the

orbit of the Moon is everywhere concave to the Sun.

The acceleration of the Earth is very approximately

the resultant of two accelerations, one towards the Moon

and one towards the Sun, both inversely as the square of

the distance. The accelerations of the Earth and the

Moon towards the Sun are equal at the same distance.

These descriptions of the acceleration of the Earth and

Moon are only approximate, because each of them has

other components, directed towards the planets, and

inversely as the squares of the distances from them.

By the experiments of Cavendish it is shewn that bodies

on the Earth's surface have accelerations towards each

other which vary inversely as the squares of their distances;

but these accelerations are very small and difficult to

observe. The accelerations of all bodies towards a body A

are equal at the same distance, but the accelerations at

the same distance towards two bodies A and B are directly

proportional to their masses.

Since then the acceleration of B towards A is to the

acceleration of A towards B as the mass of A to the mass

of B, it follows that the mass-acceleration of A is equal

and opposite to the mass-acceleration of B, and each of

them bears a fixed ratio to the product of the masses

divided by the square of the distance.

In this case the mass-acceleration of a body depends,

not upon the strain of an adjacent body as before, but

upon the position of a distant body. The mass-acceleration

in this case is called attraction, namely, the attraction of

gravity. And we may now state the proposition enun-

ciated by Newton, that every particle of matter attracts

every other particle, with an attraction proportional to the

product of their masses divided by the square of the distance.

Newton assumed that the Law of Reciprocity was true

in the case of attraction, because he had proved it true by

64 DYNAMIC.

experiment in the case of the pressure of adjacent bodies.

When the Law of Reciprocity is assumed, it is sufficient

to shew that the mass-acceleration of all bodies, due to

any one (say the Earth), is the same at the same distance.

This Newton did by his experiments on pendulums made

of different substances, and by comparing the acceleration

of the Moon with that of a falling body.

(B.)

ELECTRICITY.

If we rub a rod of glass with a piece of silk, and then

touch with the glass rod two pith balls hung near one

another by silk threads, they will move away from one

another. The same thing happens if we touch both of

them with the silk. But if we touch one with the silk

and the other with the rod, they will move together until

they come into contact, after which they will hang down

as before.

This is commonly described by saying that both glass

and silk acquire a certain charge of electricity, one positive

and the other negative, which is partially communicated

to the pith balls. Two bodies having like charges (both

positive or both negative) move away from one another

when free ; two bodies having unlike charges (one positive

and the other negative) move towards one another.

In either case the mass-accelerations of the two bodies

are equal and opposite, and each is proportional to the

product of the charges divided by the square of the dis-

tance. This mass-acceleration is called attraction or

repulsion, accordingly as the bodies approach, or recede

from, one another.

When a charge is communicated to a piece of metal

supported on a glass rod, it distributes itself all over the

surface of the metal, so that it must have gone from one

part of it to another. A body which admits of this travel-

MAGNETISM. 65

ling is called a conductor. Other bodies [which do not

admit of this travelling], such as glass, are called insulators.

When two conductors are placed near one another, and

one or both of them charged, there is found a certain dis-

tribution of charge over the surface of both, which is the

same as it would be if each element of charge had an

acceleration compounded of accelerations from all other

elements of the same kind and to all elements of different

kinds, proportional to their magnitudes divided by the

square of the distance, while those elements which are on

the surface of the conductor have also a normal accelera-

ition inwards, equal and opposite to the normal component

of the resultant of all the other accelerations. Thus we see

that any two elements of charge have charge-accelerations

which are equal and opposite, and proportional to the pro-

duct of the charges divided by the square of the distance.

We do not call these charge-accelerations attraction or

repulsion, because there is reason to think that a charge is

not a body, but a strain or displacement of something

which freely pervades the interstices of bodies.

MAGNETISM.

A kind of iron ore, called loadstone, is found to attract

pieces of iron placed near it. This property may be com-

municated to iron by rubbing it with loadstone; the piece

of iron is then said to be magnetized and is called a

magnet. If a long thin bar be uniformly magnetized and

hung by its centre, it will point nearly north and south ;

the end towards the north is called the north pole of the

magnet, the other end the south pole. Two such bars

placed in the same straight line, north and south, in their

natural positions, will have accelerations towards each

other inversely proportional to their masses; the mass-

accelerations are equal and oppo-

g -7 -, site, and each is the resultant of

four. Namely the mass-accelera-

tion of SN is compounded of mass-

accelerations towards S'N', inversely as the squares of the

distances NS', SN' : and of mass-accelerations away from

C. 5

66 DYNAMIC.

S'N', inversely as the squares of the distances SS', NN'.

Thus we may say that there is attraction between a north

and a south pole, and repulsion between two north or

two south poles, proportional in each case to the product

of their strengths divided by the square of the distance.

The two poles of any one magnet are always of the same

strength.

In any other position of the two magnets, each of

them has an angular acceleration, or tends to turn round.

ELECTEIC CURRENTS.

A copper wire connecting two plates, of copper and

zinc respectively, placed in dilute sulphuric acid, is found

to be in a peculiar state, which is described by saying

that it carries an electric current. If the wire be divided,

the two ends are found to be oppositely electrified ; and

when they are brought together again, there is a con-

tinuous passage of electric charge from one to the other.

There are other substances besides copper which will

carry an electric current, and other modes of producing

it besides the arrangement just described, which is called

a battery.

When a small magnet hung by its centre is brought

near a wire carrying a current, it places itself at right

angles to the direction of the wire. When we come to

consider the motion of a rigid body having different ac-

celerations in its different parts, we shall be able to shew

that the flux of the velocity-system of the magnet is

equivalent to two mass-accelerations of

its two poles in different directions.

These follow the following law. Let oc

be a line in the direction of a small

piece of wire at o carrying a current,

and of length representing the pro-

duct of the length of the wire by the

strength of the current. Then the

mass-acceleration of a magnetic north pole at p will be per-

pendicular to the plane poc, proportional to oc sin 6 : op 2 ,

or, which is the same thing, proportional to the vector pro-

duct of oc and op divided by the cube of the distance op.

LAW OF FORCE. 67

The mass-acceleration of each of two small pieces of

wire carrying currents due to the position of the other is

not certainly known, as it is only possible to experiment

upon closed circuits. The law of dependence on the

position is too complicated in this case to be explained at

present ; but it agrees with the other cases which we have

examined in these important respects : The mass-accelera-

tion of each conductor depends on the position of the other

and the strength of the two currents, and the two mass-

accelerations are equal and opposite.

LAW OF FORCE.

We have now briefly examined various cases of mass-

acceleration or rate of change in the momentum of a body.

We have found that it depends upon one of two things :

the strain of an adjacent body, or the position and state

of a distant body. But it does not depend, in any case

which we have examined, on the velocity either of the

body itself or of other bodies.

This quantity, then, the rate of change in the mo-

mentum of a body, may be calculated in two ways. First,

by observing the motion of the body; in this case the

quantity is called mass-acceleration, or flux of momentum.

Secondly, by observing the strain of adjacent bodies and

the position and state of distant bodies; when so calculated,

it is called force. Force is a name given to the flux of

momentum of a body, which is intended always to remind

us that it depends partly on the strain of adjacent bodies

and partly on the position and state of distant bodies.

In all cases the actual flux of momentum is the re-

sultant of those which are severally due to the strains of

different adjacent bodies and the position and state of

different distant bodies.

When, for example, a book rests on a table, it has a

mass-acceleration downwards equal to its mass multiplied

by 981 centimetres a second per second. This is due to

the position of the Earth, and depends on the mass of the

Earth and the distance of the book from its centre. The

book has also an equal mass-acceleration vertically up-

52

68 DYNAMIC.

wards, due to the strain of the table, which is slightly

compressed under it. If the Earth, excepting this table

and book, could be suddenly annihilated, the book would

begin to move upwards from the table with an acceleration

of 981 centimetres a second per second. But this accelera-

tion would diminish so very rapidly and disappear in so

minute a time (the strains being very small) that the

book would acquire on the whole only a very small

velocity.

An electrified magnet suspended by an elastic string

at its middle, in the presence of electrified and magnetic

bodies, will have at every instant a mass-acceleration com-

pounded of those due to the position of the Earth, the

position and state of the electrified and magnetic bodies,

and the strain of the elastic string. The composition

takes place according to the already known laws of com-

position or addition of vectors and of rotors passing

through the same point. It is understood that for the

present the rotation of the magnet is neglected in this

computation.

There are certain cases of apparent exception to the

Law of Force which shall be here briefly mentioned. A

bullet travelling through the air has a mass-acceleration

opposite to its velocity, which varies according to a com-

plicated law so long as the velocity is below the velocity

of sound, but afterwards is nearly proportional to the

square of the velocity. In this case, however, the mass-

acceleration depends directly on the strain of the air,

which itself depends on the velocity of the body during a

short time previous to the moment considered ; so that

indirectly the mass-acceleration depends on the velocity

of the body a little before. This mass-acceleration, depend-

ing on the strain of a fluid, is called resistance, or fluid

friction.

Two solids in contact experience equal and opposite

mass-accelerations, ca,Ued friction, parallel to the surface of

contact, which are independent of the velocity when they

are once moving, but different from their values when the

solids are relatively at rest. This kind of friction, like

that of fluids, is really due to a shearing strain of the

LAW OF FORCE. 69

surfaces in contact, and the difference between friction in

rest and in motion is to be accounted for by a change in

the nature of the surfaces.

The reader must be very careful to distinguish be-

tween the technical meaning of the word force, explained

in this section, and the various meanings which the word

has in conversation or in literature. He must especially

learn to dissociate the dynamical meaning from the idea

of muscular exertion and the feelings accompanying it.

When I press any object with my hand, a very complex

event takes place. As a consequence of a certain mole-

cular disturbance in my brain, nervous discharges go to

the muscles of my arm and hand. The effect upon the

muscles is to produce an internal strain, in virtue of

which my hand receives a certain mass-acceleration. The

part of it in immediate contact with the object, and the

object itself, are slightly compressed. The object has

then a mass-acceleration due to this strain of an adjacent

body. The compression of my hand, and the continued

strain of the muscles, are followed by nerve-discharges

which travel back to my brain, there to result in a further

disturbance. Besides these physical facts, there coexist

other side.

Hang up this spring in a horizontal position, by thin

strings fastened to the disks. Let the

bullet also be hung by a thin vertical

string. Now cut the string which fastens

the bullet to the further disk. The spring

will then open and the bullet will move

away. It will begin to move with a certain

acceleration, depending upon the compression of the spring.

Now suppose the spring to be hung up to the same

support as the bullet, so that the two may swing in con-

tact, in a direction perpendicular to the axis of the spring ;

the spring being compressed as before. At any instant of

the motion, let the string be again cut. Then the spring

will begin to open, and the bullet to move away ; but it

will always begin to move with the same acceleration in

the direction of the spring, provided that the compression

is always the same.

If we allow the spring and bullet to swing in the

direction of the spring's axis, the bullet will have an

acceleration in the direction of the axis except when it is

passing through the lowest point of its swing. If the

string be cut at that instant, the acceleration of the bullet

will be the same as before. But if the string be cut at

any other instant, the difference between the acceleration

before and after that instant will be precisely the accelera-

tion with which the bullet began to move away from rest.

58 DYNAMIC.

The greater the compression of the spring, the greater

will be this acceleration. The method of finding the

acceleration when the compression is given will be subse-

quently investigated.

We learn from these experiments that under certain

circumstances a body A (the bullet) in contact with a

strained body B (the spring) has an acceleration depending

upon the strain of B, but wholly independent of the

velocity of A or B. We have supposed the bullet and

string to be moving with the same velocity, in order to

make sure that the strain of the spring was always the

same. If however the same condition of strain could be

secured in any other way, the acceleration would be found

quite independent of the velocity of the spring.

MASS.

If we now cut the bullet in two, and use one half in

the same way, we shall find that for any given state of

strain of the spring the acceleration is double what it was

before. And generally, if we use any other piece of lead

we shall find that the accelerations in the two cases are

inversely proportional to the volumes of the pieces of lead;

or, we may say, to the quantities of lead.

The same thing is true if we take different pieces of

wood, or of any other substance, provided that the sub-

stance is homogeneous, that is, of the same nature all

through. But if we compare the accelerations of a piece

of wood and a piece of lead, we shall find that they are by

no means inversely proportional to the volumes of the two

bodies. A piece of wood, whose acceleration in the same

circumstances is the same as that of a piece of lead, will

be very much larger than the lead.

Let us now take an arbitrary body, say a certain piece

of platinum. Of every other sort of substance, as of lead,

wood, iron, etc., let us find a piece which under the same

circumstances (i.e. with the same strain of the spring) has

the same acceleration as the piece of platinum, and let

each of these be called the unit of quantity of the substance

MASS. 59

of which it is composed. Then if we consider a homo-

geneous body composed of one of these substances, the

number of units of quantity which it contains is called the

mass or measure of the body *. It is evidently the same as

the number of units of quantity of platinum in a lump

which has the same acceleration as the given body.

The piece of platinum actually -f- used is called the

" kilogramme des archives," and is preserved in Paris. For

convenience the unit of quantity is taken to be one-

thousandth part of this, and is called a gram. Thus we

may define:

The mass of a body is the number of grams of platinum

in a lump which under the same circumstances has the same

acceleration as the given body.

If the body is not homogeneous, but is made up of

parts of different kinds, the mass of the body is the sum of

the masses of the parts.

The mass of a cubic centimetre of any substance is

called the density of that substance.

We have supposed the densities of different substances

to be measured by means of a certain spiral spring. We

should, however, find exactly the same densities if we had

used any other strained body.

The product of the mass of a body by its acceleration

shall be called for shortness the mass-acceleration. Since

we have found that the accelerations of any two bodies, in

a given state of strain of the spring, are inversely propor-

tional to their masses, it follows that the mass-acceleration

of all bodies in contact with the spring in a given state of

strain, is the same. This mass-acceleration is called J the

stress belonging to that state of strain.

In all this we have supposed the motion of the body to

be pure translation.

The product of the mass of a body by its velocity is

* [contrast with definition on p. 1.]

t [? theoretically.]

J [rather ' taken as the measure of'.]

60 DYNAMIC.

called its momentum. Thus the mass-acceleration is the

rate of change of momentum. Both momentum and mass-

acceleration are directed quantities.

LAW OF COMBINATION.

If a body be in contact at the same moment with two-

strained bodies, its acceleration is the resultant of the

accelerations which it would have when placed in contact

with the two bodies separately. And in general, if a body

be in contact with any number of strained bodies, its actual

acceleration is the resultant of the accelerations due to the

several bodies. Here again we assume that the whole

motion is one of pure translation, or that the body may be

regarded as a particle.

LAW OF RECIPROCITY.

Suppose two ivory balls, A, B, to be hung up side by

side ; let A be pulled away and then

let go so as to impinge on B. The

velocity of A will appear to be suddenly

changed, and B will appear to suddenly

acquire a velocity. The change, how- "*" OCX

ever, is not really sudden. At the

moment of contact A has a certain velocity, but no

tangential acceleration ; B has neither velocity nor

acceleration. After the contact, both bullets become

compressed in the neighbourhood of contact ; and then

A has at every instant an acceleration opposite to its

velocity, depending on the strain of B, while B has an

acceleration in the direction of A's velocity, depending on

the strain of A. After the compression has attained a

certain magnitude, the compressed parts begin to expand

again ; and so long as any strain remains, each of them

has an acceleration depending on the strain of the other.

The two strains cease at the same moment, and then the

bullets separate. But the whole time during which they

are in contact is too short to be perceived by ordinary

means.

MOMENTUM. 61

During all this time, however, the mass-acceleration of

A, due to the strain of B, is equal and opposite to the mass-

acceleration of B, due to the strain of A. And the result is

that the change of momentum of A during the contact is

equal and opposite to the change of momentum of B. Or,

as we may otherwise state it, the sum of the momenta of

the two balls is the same before and after the impact.

The same thing holds good in the case of the bullet

and spiral spring which we previously considered. In that

case, however, the different parts of the spring are moving

with different velocities and accelerations. But if we

reckon the mass-acceleration of the whole spring as the

sum of the mass-accelerations of its parts, it will still be

true that at every instant the mass-acceleration of the

bullet, due to strain of the spring, is equal and opposite to

the mass-acceleration of the spring, due to strain of the

bullet. For there is a slight compression, both of the

bullet and of the disk with which it is in contact.

If we shorten the spring, by cutting off a part from the

end away from the bullet, we shall make no difference to

the acceleration of the bullet, provided that the remaining

part is kept in the same state of strain as before. The

only difference will be that this acceleration will diminish

more rapidly and last a much shorter time ; so that the

bullet will acquire on the whole a less velocity. And

finally if we remove the spring altogether, leaving only the

disk at the end ; and suppose the string stretched in any

other way so as to produce the same compression of the

disk as before*, and then to be suddenly cut; the accelera-

tion of the bullet will be the same. But in this case it

will exist only for an imperceptible time, during which the

bullet will acquire only a very small velocity.

In general, the mass-acceleration due to the strain of

two bodies hi contact depends only on the strain of each

at the surface of contact. No body can have a strain at its

surface unless it is in contact with another body.

If we draw an ideal surface separating a body into two

parts, each of these parts has a mass-acceleration due to

* [Is the disk compressed?]

62 DYNAMIC.

the strain at the surface of separation. For example, if we

divide the spiral screw by an imaginary plane of section,

the mass-acceleration of the portion to the left of the

plane is equal and opposite to that of the

portion to the right together with the bullet,

and each is due to the strain of the spring

at the point of section. This mass-accelera-

tion is called the stress across the section.

When it is away from the section on both sides, it is called

pressure; when it is towards the section, it is called tension.

Thus we see that what takes place at the common surface

of two bodies in contact is a particular case of what takes

place throughout the interior of any body.

GRAVITY.

When bodies are let go in the open air, they fall with

more or less rapidity to the ground. This difference of

velocity is found to depend on the presence of the air; and

in the exhausted receiver of an air-pump the most differ-

ent bodies fall through the same distance in the same

time ; having, as we remarked before, a constant accelera-

tion of 981 centimetres a second per second. Thus every

body left free in vacuo has a mass-acceleration vertically

downwards, proportional to its mass.

The acceleration of the Moon is found to be very

approximately the resultant of two accelerations, one

directed towards the Earth, and the other towards the

Sun. The acceleration towards the Earth is about one

3600th part of the acceleration of gravity, but varies with-

in certain limits, being inversely as the square of the

distance from the Earth's centre. Now the Moon is

distant from that centre on the average about 60 times

the Earth's radius, which is the distance from the centre

of bodies on the Earth's surface. Hence the acceleration

of the Moon towards the Earth is the same as that of any

terrestrial body would be at the distance of the Moon,

supposing the acceleration of the terrestrial body to vary

inversely as the distance from the Earth's centre. Thus

the Moon is to be regarded as a falling body.

GRAVITY. 63

Its acceleration towards the Sun is inversely as the

square of the distance from the Sun; but although that

distance is 400 times the distance from the Earth, this

acceleration is always greater than the other, so that the

orbit of the Moon is everywhere concave to the Sun.

The acceleration of the Earth is very approximately

the resultant of two accelerations, one towards the Moon

and one towards the Sun, both inversely as the square of

the distance. The accelerations of the Earth and the

Moon towards the Sun are equal at the same distance.

These descriptions of the acceleration of the Earth and

Moon are only approximate, because each of them has

other components, directed towards the planets, and

inversely as the squares of the distances from them.

By the experiments of Cavendish it is shewn that bodies

on the Earth's surface have accelerations towards each

other which vary inversely as the squares of their distances;

but these accelerations are very small and difficult to

observe. The accelerations of all bodies towards a body A

are equal at the same distance, but the accelerations at

the same distance towards two bodies A and B are directly

proportional to their masses.

Since then the acceleration of B towards A is to the

acceleration of A towards B as the mass of A to the mass

of B, it follows that the mass-acceleration of A is equal

and opposite to the mass-acceleration of B, and each of

them bears a fixed ratio to the product of the masses

divided by the square of the distance.

In this case the mass-acceleration of a body depends,

not upon the strain of an adjacent body as before, but

upon the position of a distant body. The mass-acceleration

in this case is called attraction, namely, the attraction of

gravity. And we may now state the proposition enun-

ciated by Newton, that every particle of matter attracts

every other particle, with an attraction proportional to the

product of their masses divided by the square of the distance.

Newton assumed that the Law of Reciprocity was true

in the case of attraction, because he had proved it true by

64 DYNAMIC.

experiment in the case of the pressure of adjacent bodies.

When the Law of Reciprocity is assumed, it is sufficient

to shew that the mass-acceleration of all bodies, due to

any one (say the Earth), is the same at the same distance.

This Newton did by his experiments on pendulums made

of different substances, and by comparing the acceleration

of the Moon with that of a falling body.

(B.)

ELECTRICITY.

If we rub a rod of glass with a piece of silk, and then

touch with the glass rod two pith balls hung near one

another by silk threads, they will move away from one

another. The same thing happens if we touch both of

them with the silk. But if we touch one with the silk

and the other with the rod, they will move together until

they come into contact, after which they will hang down

as before.

This is commonly described by saying that both glass

and silk acquire a certain charge of electricity, one positive

and the other negative, which is partially communicated

to the pith balls. Two bodies having like charges (both

positive or both negative) move away from one another

when free ; two bodies having unlike charges (one positive

and the other negative) move towards one another.

In either case the mass-accelerations of the two bodies

are equal and opposite, and each is proportional to the

product of the charges divided by the square of the dis-

tance. This mass-acceleration is called attraction or

repulsion, accordingly as the bodies approach, or recede

from, one another.

When a charge is communicated to a piece of metal

supported on a glass rod, it distributes itself all over the

surface of the metal, so that it must have gone from one

part of it to another. A body which admits of this travel-

MAGNETISM. 65

ling is called a conductor. Other bodies [which do not

admit of this travelling], such as glass, are called insulators.

When two conductors are placed near one another, and

one or both of them charged, there is found a certain dis-

tribution of charge over the surface of both, which is the

same as it would be if each element of charge had an

acceleration compounded of accelerations from all other

elements of the same kind and to all elements of different

kinds, proportional to their magnitudes divided by the

square of the distance, while those elements which are on

the surface of the conductor have also a normal accelera-

ition inwards, equal and opposite to the normal component

of the resultant of all the other accelerations. Thus we see

that any two elements of charge have charge-accelerations

which are equal and opposite, and proportional to the pro-

duct of the charges divided by the square of the distance.

We do not call these charge-accelerations attraction or

repulsion, because there is reason to think that a charge is

not a body, but a strain or displacement of something

which freely pervades the interstices of bodies.

MAGNETISM.

A kind of iron ore, called loadstone, is found to attract

pieces of iron placed near it. This property may be com-

municated to iron by rubbing it with loadstone; the piece

of iron is then said to be magnetized and is called a

magnet. If a long thin bar be uniformly magnetized and

hung by its centre, it will point nearly north and south ;

the end towards the north is called the north pole of the

magnet, the other end the south pole. Two such bars

placed in the same straight line, north and south, in their

natural positions, will have accelerations towards each

other inversely proportional to their masses; the mass-

accelerations are equal and oppo-

g -7 -, site, and each is the resultant of

four. Namely the mass-accelera-

tion of SN is compounded of mass-

accelerations towards S'N', inversely as the squares of the

distances NS', SN' : and of mass-accelerations away from

C. 5

66 DYNAMIC.

S'N', inversely as the squares of the distances SS', NN'.

Thus we may say that there is attraction between a north

and a south pole, and repulsion between two north or

two south poles, proportional in each case to the product

of their strengths divided by the square of the distance.

The two poles of any one magnet are always of the same

strength.

In any other position of the two magnets, each of

them has an angular acceleration, or tends to turn round.

ELECTEIC CURRENTS.

A copper wire connecting two plates, of copper and

zinc respectively, placed in dilute sulphuric acid, is found

to be in a peculiar state, which is described by saying

that it carries an electric current. If the wire be divided,

the two ends are found to be oppositely electrified ; and

when they are brought together again, there is a con-

tinuous passage of electric charge from one to the other.

There are other substances besides copper which will

carry an electric current, and other modes of producing

it besides the arrangement just described, which is called

a battery.

When a small magnet hung by its centre is brought

near a wire carrying a current, it places itself at right

angles to the direction of the wire. When we come to

consider the motion of a rigid body having different ac-

celerations in its different parts, we shall be able to shew

that the flux of the velocity-system of the magnet is

equivalent to two mass-accelerations of

its two poles in different directions.

These follow the following law. Let oc

be a line in the direction of a small

piece of wire at o carrying a current,

and of length representing the pro-

duct of the length of the wire by the

strength of the current. Then the

mass-acceleration of a magnetic north pole at p will be per-

pendicular to the plane poc, proportional to oc sin 6 : op 2 ,

or, which is the same thing, proportional to the vector pro-

duct of oc and op divided by the cube of the distance op.

LAW OF FORCE. 67

The mass-acceleration of each of two small pieces of

wire carrying currents due to the position of the other is

not certainly known, as it is only possible to experiment

upon closed circuits. The law of dependence on the

position is too complicated in this case to be explained at

present ; but it agrees with the other cases which we have

examined in these important respects : The mass-accelera-

tion of each conductor depends on the position of the other

and the strength of the two currents, and the two mass-

accelerations are equal and opposite.

LAW OF FORCE.

We have now briefly examined various cases of mass-

acceleration or rate of change in the momentum of a body.

We have found that it depends upon one of two things :

the strain of an adjacent body, or the position and state

of a distant body. But it does not depend, in any case

which we have examined, on the velocity either of the

body itself or of other bodies.

This quantity, then, the rate of change in the mo-

mentum of a body, may be calculated in two ways. First,

by observing the motion of the body; in this case the

quantity is called mass-acceleration, or flux of momentum.

Secondly, by observing the strain of adjacent bodies and

the position and state of distant bodies; when so calculated,

it is called force. Force is a name given to the flux of

momentum of a body, which is intended always to remind

us that it depends partly on the strain of adjacent bodies

and partly on the position and state of distant bodies.

In all cases the actual flux of momentum is the re-

sultant of those which are severally due to the strains of

different adjacent bodies and the position and state of

different distant bodies.

When, for example, a book rests on a table, it has a

mass-acceleration downwards equal to its mass multiplied

by 981 centimetres a second per second. This is due to

the position of the Earth, and depends on the mass of the

Earth and the distance of the book from its centre. The

book has also an equal mass-acceleration vertically up-

52

68 DYNAMIC.

wards, due to the strain of the table, which is slightly

compressed under it. If the Earth, excepting this table

and book, could be suddenly annihilated, the book would

begin to move upwards from the table with an acceleration

of 981 centimetres a second per second. But this accelera-

tion would diminish so very rapidly and disappear in so

minute a time (the strains being very small) that the

book would acquire on the whole only a very small

velocity.

An electrified magnet suspended by an elastic string

at its middle, in the presence of electrified and magnetic

bodies, will have at every instant a mass-acceleration com-

pounded of those due to the position of the Earth, the

position and state of the electrified and magnetic bodies,

and the strain of the elastic string. The composition

takes place according to the already known laws of com-

position or addition of vectors and of rotors passing

through the same point. It is understood that for the

present the rotation of the magnet is neglected in this

computation.

There are certain cases of apparent exception to the

Law of Force which shall be here briefly mentioned. A

bullet travelling through the air has a mass-acceleration

opposite to its velocity, which varies according to a com-

plicated law so long as the velocity is below the velocity

of sound, but afterwards is nearly proportional to the

square of the velocity. In this case, however, the mass-

acceleration depends directly on the strain of the air,

which itself depends on the velocity of the body during a

short time previous to the moment considered ; so that

indirectly the mass-acceleration depends on the velocity

of the body a little before. This mass-acceleration, depend-

ing on the strain of a fluid, is called resistance, or fluid

friction.

Two solids in contact experience equal and opposite

mass-accelerations, ca,Ued friction, parallel to the surface of

contact, which are independent of the velocity when they

are once moving, but different from their values when the

solids are relatively at rest. This kind of friction, like

that of fluids, is really due to a shearing strain of the

LAW OF FORCE. 69

surfaces in contact, and the difference between friction in

rest and in motion is to be accounted for by a change in

the nature of the surfaces.

The reader must be very careful to distinguish be-

tween the technical meaning of the word force, explained

in this section, and the various meanings which the word

has in conversation or in literature. He must especially

learn to dissociate the dynamical meaning from the idea

of muscular exertion and the feelings accompanying it.

When I press any object with my hand, a very complex

event takes place. As a consequence of a certain mole-

cular disturbance in my brain, nervous discharges go to

the muscles of my arm and hand. The effect upon the

muscles is to produce an internal strain, in virtue of

which my hand receives a certain mass-acceleration. The

part of it in immediate contact with the object, and the

object itself, are slightly compressed. The object has

then a mass-acceleration due to this strain of an adjacent

body. The compression of my hand, and the continued

strain of the muscles, are followed by nerve-discharges

which travel back to my brain, there to result in a further

disturbance. Besides these physical facts, there coexist