tioned as present in the universe, these letters representing variable quantities, then we shall always have s+t+u+v +w+x+y+z = a constant quantity; that is to say, while u may change into v or into w, and, in fact, while the various forms of energy may change into each other, according to the laws which regulate such changes, nevertheless the sum of all the energies present in the universe will always remain constant in amount; and this is the doctrine known as the Conservation of Energy. 116. Function of a Machine.-To realize the truth of this doctrine, ïet us take one of the ordinary mechanical combinations, such as a system of pulleys (Fig. 40), and see what we gain by its employment. In this system there are two blocks, the lower one moveable and the upper one fixed; while the same string goes round all the pulleys. The power P is applied to the extremity of this string, so that the tension of all parts of this string is equal to the weight of P. Now w is supported by six strings: hence we see that w must be 6 times as great as P in order that there may be equilibrium. = P W FIG. 40. Suppose, now, that P is equal to 1, and w to 6 kilogrammes, and that P is pulled down 6 metres, we have thus spent a quantity of energy upon the machine represented by 1×6 6. Our gain is that we have caused the weight w of 6 kilogrammes to rise, but in order that the law of the conservation of energy should hold good, w ought not to rise higher than one metre; for if it did, we should get back more energy than we had spent upon the machine. Now it will readily be seen that the rise of w will be 6 times less than that of P, because w is supported by six strings, while P is only supported by one; therefore by lowering P through six metres, each of these strings of w, and hence w itself, will be raised one metre; and hence the gain of energy by the raising of w into a position of advantage will be 6 x 1 = 6, or precisely what was spent in lowering P. We thus see that in such a machine what we gain in force we lose in space. The same law will hold for the hydraulic press (Fig. 23). In this case, if the area of the two pistons be as I: 100 and a weight of 10 kilogrammes be applied to the small piston, it will raise 1000 kilogrammes if put on the large one; but since the volume of water remains constant, the rise of the larger piston will only be th of the fall of the smaller. Now if the smaller piston falls one metre with a weight of 10 kilogrammes, we have spent 10 units of energy on our machine. But the large piston containing 1000 kilogrammes will have risen both of a metre, so that we shall have recovered by its rise, an amount of energy equal to 1000 X TOO IO; that is to say, neither more nor less than the 10 units of energy which we spent. = Here, too, therefore, and indeed in all machines, we do not create energy, but simply transform it into a kind more convenient for us, and the law holds universally that what we gain in force we lose in space, so that the power multiplied by its space of descent is always equal to the weight multiplied by its space of ascent CHAPTER IV. VISIBLE ENERGY AND ITS TRANSMUTATIONS. LESSON XVI.-VARIETIES OF VISIBLE ENERGY. 117.-By visible energy we mean the energy of visible motions and arrangements. Thus, for instance, a cannonball during its flight, or a flowing river, form examples of visible motion, and a stone on the top of a cliff is an example of a visible advantageous arrangement as far as energy is concerned. To begin with the first description of energy, or that due to visible motion; there are many varieties of this. Thus we have, first of all, the energy of a body in actual visible linear velocity, such as a railway train, a cannon-ball, a gale of wind, a stream of water, a meteor. But there is also the energy due to rotatory motion; as for instance, that of a top in rapid rotation, or that of the earth in its rotation round its axis. In the next place, there is the energy of oscillating and vibratory motion; in the former category we may place the motion of a pendulum, while the string of a musical instrument is a very good illustration of the latter. The whole phenomena of sound are to be included under this last head, for although the vibrations of sounding bodies are sometimes so rapid as to be invisible, yet they result from an arrangement and motion of particles on the large scale, and not from strictly molecular motions and arrangements, as is the case with that species of vibration which forms light. I 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. 118. Linear Velocity.-Let us begin with the energy of a 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 1000 196 100), that of the ball before impact will be = 40*8 nearly, whereas after impact the energy of the united mass (ball plus pendulum) will be 20,020 (0.2)2 = 0.0408. We 1000 19.6 = 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 |