716 MECHANICS Newton s scholium. Conser vative system. Potential energy. statement, the parts of any machine) has its equivalent in 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 the resistance to acceleration, and the additional kinetic energy developed is equivalent to the work so spent." When part of the work is done against molecular forces, as in bending a spring, or against gravity, as in raising a weight, the recoil of the spring and the fall of the weight are capable, at any future time, of reproducing the work originally expended. But in Newton s day, and long after wards, it was supposed that work was absolutely lost by friction. 170. If a system of bodies, given either at rest or in motion, be influenced by no forces from without, the sum of the kinetic energies of all its parts is augmented in any time by an amount equal to the whole work done in that time by the stresses which we may imagine as taking place between its points. When the lines in which these stresses act remain all unchanged in length, the sum of the kinetic energies of the whole system remains constant. If, on the other hand, one of these lines varies in length during the motion, the stress in that line will do work or will consume work, according as the distance varies with or against it. 171. Experiment has shown that the mutual actions between the parts of any system of natural bodies always perform, or always consume, the same amount of work during any motion whatever, by which the system can pass from one particular configuration to another ; so that each configuration corresponds to a definite amount of kinetic energy. Hence no arrangement is possible in which a gain of kinetic energy can be obtained when the system is restored to its initial configuration. In other words, " the perpetual motion " is impossible. The "potential energy" ( 113) of such a system, in the configuration which it has at any instant, is the amount of work that its mutual forces perform during the passage of the system from any one chosen configuration to the configuration at the time referred to. It is generally convenient so to fix the particular configuration chosen for the zero of reckoning of potential energy that the potential energy in every other configuration practically considered shall be positive. As particular instances of this we may notice many of the results already given : for instance, the ordinary expres sion for the velocity acquired by a falling stone ( 28), |v 2 = gx ; for here ^mv z is the kinetic energy acquired, while mg.x is the work done by the weight (mg) during the fall. Similarly, we have in the motion of a planet, the ex- / 2 1 . v 2 - v* mu. pression v 2 = ,( ), which leads to m - 1 = (i - r}. r r a P 2 n ^ Here is the " mean value" of the force for distances from r to r v and therefore the right-hand side is the work done by the force, while the left-hand side is the increase of kinetic energy produced. To put this in an analytical form, we have merely to notice that, by what has just been said, the value of
- /(
~}ds ds . is independent of the paths pursued from the initial to the final positions, and therefore that is a complete differential. If, in accordance with what has just been said, this be called -dV, V is the potential energy, and dV -Ai= -j , . . . . dx l Also, by the second law of motion, if m : be the mass of a particle of the system whose coordinates are x l} y lt z 1 we have m-X, &c. =&c. and 2 dt dt 2 dt dt 2 The integral is , z = -dV. Conn vatic if that is, the sum of the kinetic and potential energies is con stant. This is called the "conservation of energy." In abstract dynamics, with which alone this article is concerned, there is loss of energy by friction, impact, &c. This we simply leave as loss, to be accounted for by Thermodynamics. 172. Hitherto, as we have been dealing with the motion of a single particle only, we have not required the assistance of even the third law. For, in those cases, already treated, in which one of the forces was not given, it was at all events due to a given constraint, and the geometrical circumstances of the constraint supplied the means of determining it. In fact we were not, in any case, concerned with reaction ; or, to use the more modern form of expression, we were engaged with one half, only, of a stress. When a stone s motion was investigated, no account was taken of the stone s attraction for the earth ; when we dealt with central forces, the centre was supposed to be fixed ; and, even in the cases in which variable constraint was supposed, the curve which produced it was assumed to move in a manner absolutely determined beforehand, and in no way affected by the reaction of the mass acted upon. But, in nature, circumstances are not so simple. Though, for all practical purposes, we may calculate the motion of an ordinary projectile as if its attraction had no influence upon the motion of the earth, we cannot do so in the case of the motion of the moon about the earth. The mass of the moon is about ^th of that of the earth, and its gravitation effects on the motion of the earth cannot be neglected. The moon, in fact, moves faster round the earth than would a projectile of less mass, though moving in precisely the same relative orbit ( 146). If the earth s motion were not accelerated by the reaction of the moon, the sole crest of the lunar tide-wave would be on the side of the earth next the moon, and there would be full-tide once only in a single rotation of the earth about its axis. We need not give further instances here ; they will pre sent themselves in almost every case we investigate. 173. To give a general notion of the applications of, and necessity for, the third law, we choose a few special cases, selected so as to give, in short compass, a sufficiently general glance at the whole subject. We take, first, the case of two stones or bullets con nected by an inextensible string passing over a smooth pulley. Let their masses be m and m. Our physical condition is that the tension of the string, whatever be its value, is the same throughout ; and this is accompanied by the geometrical condition that the length of the string is constant, or that the speeds of the two masses are equal but in opposite directions. Hence the amounts of increase of momentum in a given time are as the masses. But they are also as the forces, by the second law. Thus m : m : : T - mg : m g - T . This gives, at once, Fric- tiont dissi" tion. . Exam of tlrii law. At woe niachi m + m so that the whole downward force on m is m q - T = m , m + m" and the whole upward force on m is m in - m T - mg = m ; g. m + m
The motion of the system is therefore of precisely the same