The quantity on right-band side is the sum of the products of each value of F, by the corresponding space ds, through which the particle moved under its action. It is therefore the whole work done by the force. On the left hand, we find half the prod uct of the mass, and the square of the velocity it has acquired; in other words, the vis viva. Hence, in this case, the vis-viva acquired equals the amount of work (q.v.) expended by the force.
It appears from a general demonstration (founded on the experimental laws of motion, and therefore true, if they are), but which is not suited to the present work, that if, in any system of bodies, each be made up of particles or atoms, and if the forces these mutually exert be in the line joining each two, and depend merely on the distance between them, then we can express the required proposition in the following form: Any change of ris-viva in the system corresponds to an equal amount of work gained or lost by the attractions of the particles on each other.
What is spent, then, in work, is stored up in vis-viva; and conversely, the system. by losing some of its vis-viva, will recover so much work-producing power. If we call the former, as is now generally done, kinetic, and the latter potential, energy, we may express the above by saying, that in any system of bodies where the before-mentioned restrictions are complied with, the sum of the kinetic and potential energies cannot be altered by the mutual action of the bodies. The most simple and evident illustrations of this proposition are to be found in the case of the force know'. as gravitation. The potential energy of a mass on the earth's surface is zero, because, not being able to descend, it has, in common language, no work-producing power. If it be raised above the surface, and then dropped, it is easy to see that the work expended in raising it will be exactly recovered as vis-viva after its fall. For (see FALLING BODIES) a mass falling through a space, h, to the earth acquires a velocity a, such that vl = 2gh, or if in be the ' my mass, = mg.h. The left-hand side gives the vis-viva acquired by the fall—the right
is the product of the weight (mg) and the height fallen through—or is the work required to elevate the mass to its original altitude.
• Hence we may calculate the amount of work which can be obtained from a head of water in driving water-wheels. etc., remembering, however, that there is always a loss (as it is usually called) due to friction, etc., in the machinery. That there is a loss in useful power, is true, but we shall find presently that in energy there is none, as indeed our general result has already shown. Where the apparently lost energy goes, is another question.
Another good example of potential energy is that of the weights in an ordinary clock. It is the gradual conversion of potential into kinetic energy in the driving weight which maintains the motion of the clock, in spite of friction, resistance of the air, etc.; and we have in the kin'etic energy of sound (which depends on vibrations in the air) a considerable portion of the expended potential energy of the striking weight. A coiled Vateh-spring, it drawn bow, the charged receiver of an air-gun, are good examples of mores of potential energy, which can be directly used for mechanical purposes.
The chemical arrangement of the different components of gunpowder, or gun-cotton, is such as corresponds to enormous potential energy, which a single spark converts into the equivalent active amount. But here, heat has a considerable share in the effects produced; it may then be as well, before proceeding further, to consider how we can take account of it, and other (so-called) physical forces, as forms of energy.
Correlation of Physical Forces,—So far as we yet know, the physical forces may be thus classified: I. Gunvrr.ttioN (e:v.); II. MOLECULAR FORCES—0011ESION (including CAPILLARITY), ELASTICITY, CHEMICAL AFFLNITY; III. HEAT AND LIGHT; ELEr;