Lastly, we have the potential energy of a body occupying a position of visible advantage with respect to some force. If the force be that of gravity, we have the energy of a stone at the top of a cliff, of a head of water, of a clock wound up, and so on; or again, if the force be that due to elasticity, we have the energy of position of a cross-bow bent, or of a spring stretched, with many other similar instances.

Now, under certain conditions, these various forms of visible energy are transmuted into one another, while under other conditions they are transmuted into the various forms of molecular energy but as these last will form the subject of future chapters, we shall at present mainly confine ourselves to a description of the various forms of visible energy and their transmutations into one another.

a

118. Linear Velocity.-Let us begin with the energy of rifle-ball. In its rapid flight through the air the ball imparts some of its motion to the particles of air with which it comes in contact; but neglecting this in the meantime, let us suppose that it ultimately strikes a heavy mass of wood hung by a string, and so forming a pendulum, in the centre of which it lodges.

Let us suppose that the weight of the ball is 20 grammes and its velocity 200 metres per second, and that the weight of the heavy block of wood in which it lodges is 20 kilogrammes. Before impact the momentum of the ball was 20 X 200 = 4,000, representing a mass equal to 20 moving with a velocity equal to 200. After the impact, we have, of course, the same momentum of 4,000, but it will now represent a mass equal to 20,020, moving with a velocity equal to o‘2 nearly.

Now, according to the method of estimating energy (Art.

20 (200)2
19.6

100), that of the ball before impact will be X

1000

=

= 40.8 nearly, whereas after impact the energy of the united mass

(ball plus pendulum) will be

20,020 (0°2)2

X
1000 19.6

= = 0.0408. We

thus see that although in conformity with the third law of motion the momentum is preserved, yet the energy after

impact is a thousand times less than the energy before, so that most of this energy has disappeared from the category of visible motion. Into what form, therefore, has it been transmuted? We answer, the ball has worked its way into the heart of the log of wood. In doing so, its energy has been spent in accomplishing the disintegration of the log of wood; it has, in fact, been spent against a species of friction or resistance opposing its passage, and it will be found that the production of heat has been the result. So that in this case the result of the transference of a quantity of momentum from a small to a large mass has been the conversion of visible energy into heat.

119. Resistance of Air.—So in like manner the momentum originally communicated to the air by the passage of the ball gradually becomes distributed over larger and larger masses of air, and in this process the forward momentum in the direction of motion of the ball is strictly preserved, but the energy represented by this momentum becomes less according as the moving mass of air becomes greater. As we know there is no loss of energy, we conclude that it has passed into heat; and could we only perform the experiment, we should find that when the disturbance produced in the air by the ball had become so spent as to be insensible, there would be a certain increase of temperature, representing the energy derived from the ball.

We are thus prepared to recognise an extension of the first law of motion; for, in the first place, when the moving body is not acted upon by any external force it will continue moving for ever with a uniform velocity, neither losing momentum nor energy; while, again, if it be acted upon by some external force, such as the resistance of the air, it loses both momentum and energy; and while the momentum which it loses is being communicated to larger and larger masses of air, and is thus preserved, the energy lost by it ultimately takes the shape of heat, and is thus preserved likewise.

120. Impact of Inelastic Bodies. Let us now vary the case by considering two inelastic solids striking against each

other. Let the one weigh 20 grammes, and have a velocity equal to 20, and let the other weigh 10 grammes, and have a velocity of 16 in an opposite direction ; we have thus a momentum equal to 400 in one direction, and one equal to 160 in the opposite, giving an excess in the direction of motion of the larger solid equal to 400 - 160, or 240. Now this residual momentum must be preserved after impact, by the third law of motion, and hence the united mass, or 30 (for the balls, being inelastic, will move together), will, after impact, move with a velocity equal to 8. But the united energy before impact 20 (20)2 ΙΟ (16)2

was

X
X
+
1000 19.6 1000 19.6
30 (8)2

impact is only X

1000

=

=

0'098.

19

o'539, and that after

What, therefore, has become of the remainder of the energy ? We reply, as before, it has been transmuted into heat. It would thus appear that the collision of inelastic balls results in a transfer of visible motion into heat.

121. Impact of Elastic Bodies.-But the case is altered if the balls be perfectly elastic,' for there is then no transmutation into heat, but the energy of visible motion is preserved, as well as the momentum, and is the same both before and after impact.2

For example, let two perfectly elastic balls weighing respectively 4 and 3 kilogrammes, moving in the same direction, with velocities 5 and 4, impinge against each other, then we know, by the laws of elastic bodies, that after impact the

velocity of the first or largest ball will e

36 7

29

יך

and that of the

second or smallest ball- Now, in the first place, the mo

mentum before impact was 4,000 × 5 +3,000 X 4

= 32,000,

while after impact it is 4,000 X +3,000 X

29 7

36

= 32,000,

7

I Two bodies are perfectly elastic when the momentum impressed during restitution is equal to that spent in producing compression.

2 Part of the energy is probably changed into vibrations of the elastic bodies, but for our present purpose this may be neglected.

which is the same as before. Also the energy before impact

fore both the momentum and the energy are unaltered by impact.

The most interesting case of impact is when one elastic ball strikes centrically another of the same size at rest, in which case the first ball entirely loses its motion, which is transferred to the second. Therefore if we have a row of such balls placed near each other, and if an impulse be communicated to the first of them in the direction of the row, it will in time be transmitted along the whole series until it reaches the last ball, which will then (being the last) start off and leave the series.

122. Energy of Rotation.-Let us next consider very shortly the case of a disc in rapid rotation. We have already (Art. 17) explained that such a motion implies great force of cohesion, for the particles at the circumference of the disc have, by the first law of motion, a tendency to move in a straight line with a uniform velocity; as, however, they move in a circle, they must be continually acted upon by some force. The tendency to rectilinear motion is in fact continually resisted by the force of cohesion, tending to prevent the separation of the particles of the disc, and the resulting motion is a compromise between the centrifugal tendency and this force. But while there is thus a continual change in the direction of motion of a particle, the velocity will remain the same, for if the velocity of the various particles composing the disc were to change, it would imply that the energy of motion of the whole disc had changed also; but this cannot be, for the disc will retain its energy unchanged by the law of the conservation of energy, unless it be acted upon by friction or resistance, in which case the energy of the disc will be gradually transferred to the bodies rubbing against it.

And generally speaking, wherever we have in nature a strictly circular orbit of particles round a central force we have a uniform velocity, and the energy of visible motion of the mass remains constan*

123. Energy of a Body moving in an Ellipse.— Not so, however, if the orbit be elliptical. Let us consider, for instance, the motion of a comet-a body which describes a very elongated elliptical orbit, having the sun in one of its foci.

Let the comet be farthest from the sun at B, and nearest him at A. Now the comet, while moving from B to A, has gradually been approaching the sun, and the same thing will happen to it as when a stone falls to the earth.

In this case we know that the energy of position of the stone is gradually changed into the energy of actual motion, and so in like manner the energy of position which the comet has at B (being there very far

FIG. 41.

from the centre of gravitating force) becomes changed as it approaches the sun into the energy of actual motion, until at A it is moving with a very great velocity; it has in fact fallen towards the sun,

from the distance BS to the distance A S, and its increase of energy will be that which would be acquired by a body of its own weight falling direct towards the sun from a distance BS to one A S, without any regard to the path by which it has passed from the one position to the other. The same thing takes place in the case of the earth and the other planets. Thus, when the earth is nearest the sun it is moving fastest. If we assume the greatest distance of the earth from the sun to be 92,965,oco miles, and the least distance 89,895,000 miles, the difference, or 3,070,000 miles, represents the distance through which the earth has fallen towards the sun; the energy of actual motion of the earth will therefore be greater at perihelion than at aphelion, by that due to the mass of the earth falling through 3,070,000 miles under the attraction of the sun's gravitating force.