The unseen universe

by Balfour Stewart; Peter Guthrie Tait

(2.) Conservation of Moment of Momentum.—Here we deal with quantities of the order of the moments of forces about an axis, i.e. couples in Poinsot’s sense. These also are directed magnitudes depending for their conservation upon the first interpretation of Newton’s third law, and therefore the same remarks apply to them as to the preceding.

(3.) Conservation of Vis Viva.—Vis viva is the old name for energy of motion or the consequent power of doing work. We now deal with quantities which cannot possess direction, because they are essentially products of pairs of quantities similarly directed, and are therefore all to be treated as of the same algebraic sign, or rather (to adopt the language of Sir W. R. Hamilton) as signless quantities. With such there can of course be no cancelling.

To make our meaning clear, let us consider upon what vis viva depends. It depends upon and is proportional to the product of the mass into the square of the velocity. Compare, or rather contrast, this with the definition of momentum given above, and it will be seen that vis viva is the product of the momentum and the velocity. Now mass is of course a signless quantity; evidently we cannot have negative mass. Then with regard to the square of the velocity, this will be positive whether the velocity be positive or negative, whether it be in one direction or the opposite. Vis viva, therefore, or energy of motion, is something which is not affected with the sign of direction, or, as we have already said, it is a signless quantity. It is found to be convenient to measure it as half the product of the moving mass into the square of its velocity. So measured, it is now called (see [§ 99]) kinetic energy.

Now to our cannon again. Before firing there is no vis viva of either cannon or ball. After firing each has vis viva, but that of the ball is greater than that of the cannon in the proportion in which the cannon’s mass exceeds that of the ball. And the system as a whole has vis viva though it has no momentum. If, as before, we could reverse the motions of cannon and ball, then, even when they impinged, the vis viva would not be lost. As will presently be seen, it would be employed in heating both the impinging bodies.

98. We have said that the energy which a body contains—its vis viva—its power of doing work, is independent of the direction in which it is moving; and, further, that while the mass is the same, it is proportional to the square of the velocity. For instance, we may measure the energy of a cannon-ball or of an arrow by the distance it will carry itself up against the force of gravity, represented by its own weight, when shot vertically upwards, and we find that with a double velocity it will go four times as high. Or we may point the cannon horizontally, and measure the energy of the same ball by the number of planks of oak wood which it can penetrate, and we shall find that a ball with double the velocity will penetrate nearly four times as many as one with the single velocity. All such experiments concur together in convincing us that the energy of the ball is independent of the direction in which the cannon is pointed, and is proportional to the square of the velocity, so that a double velocity will give a fourfold energy.

99. We have just now spoken about a cannon-ball fired into the air against the force of gravity. Such a ball, as it mounts, will each moment lose part of its velocity, until it finally comes to a standstill, after which it will begin to descend. When it is just turning it is perfectly harmless, and if we were standing on the top of a cliff to which it had just reached, we might without danger catch it in our arms and lodge it on the cliff. Its energy has apparently disappeared. Let us, however, see whether this is really true or not. It was fired up at us, let us say, by a foe at the bottom of the cliff, and the thought occurs to us to drop it down upon him again, which we do with great success, for he is smashed to pieces by the ball.

In truth, dynamics informs us that such a ball will again strike the ground with a velocity, and therefore with an energy precisely equal to that with which it was originally projected upwards. Now, when at the top of the cliff, if it had not the energy due to actual motion, it had nevertheless some sort of energy due to its elevated position, for it had obviously the power of doing work. A pond of still water, unless it can fall, i.e. unless it has what is technically called a ‘head,’ is of no use in driving a water-wheel. The head, or the power of descending, gives it a store of dormant energy, which becomes active as the water gradually descends. And the same amount of work may be obtained (by means of a turbine for instance) from a small quantity of water, provided it has a great ‘head,’ as can be obtained (by means of an ordinary overshot or breast wheel) from water with far less head, provided it be supplied in proportionally greater quantity. We thus recognise two forms of energy which change into one another, the one due to actual motion and the other to position; the former of these is generally called kinetic, and the latter potential energy.

All this appears to have been clearly perceived by Newton, who gave it as a second interpretation of his Third Law of Motion. His statement is equivalent, in modern language, to the following:[36]—Work done on any system of bodies has its equivalent in the form of work done against friction, molecular forces, or gravity, if there be no acceleration; but if there be acceleration, part of the work is expended in overcoming resistance to acceleration, and the additional kinetic energy developed is equivalent to the work so spent.

100. Thus Newton expressly tells us (though not in these words) that we are to include in the same category work done by or against a force—whether that force be due to gravity, friction, or molecular action (such as elasticity, for instance), or even to acceleration.

(a.) When work is done against gravity, as in lifting a mass from the ground, we have just seen that it is (as it were) stored up in the raised mass; we can recover it at any time by letting the mass descend. Thus it is that we furnish a clock with motive power sufficient to keep it going for a week in spite of friction and other resistance, by simply winding up its weights.