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William Kingdon Clifford.

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

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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


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