Page:The New International Encyclopædia 1st ed. v. 13.djvu/280

MECHANICS. 252 MECHANICS. ilarly. if the spring expels the body, the spring dop.s work on tlu' liody and loses potential energy', and the body gains kinetic energy ; the loss in po- tential encrgj' being Fa; and the gain in kinetic energy being ms- — itits^- if in the distance x the speed is increased from s, to s; and as before Fic = ims- — AmSo". The kinetic energy of the spring itself is neglected. In words, this formula means that the loss of potential energy of the system producing the acceleration cquaJs the gain of kiiu'tie energy of the particle accelerated; or, the gain of po- tential energj' of a system ])ro<lueing retardati(m c(pials the loss of kinetic energy- of the retarded imrticle. Kinetic energy may also be produced by the impact of another body; and all experi- ments are in accord with the idea that the kinetic energy gained by a body in this case equals tluft lost by the impinging particle pro- rilled nu other effects are produced. This is illustrated by the impact of perfectly elastic bodies. (In general, when there is impact, heat- effects such as rise of temperature are produced, in which case the kinetic energy gained by the |)article docs not equal that lost.) In general, then, in mechanics, whenever one body Iosch energy another body gains an equal amount, work be- ing simply the transfer of the energy. Work is done in two ways: pnjdueing a change in s|)eed and overcoming some opposing elastic force. Unless ihere is motion in the direction of the force, no work i> done. It is evident that the kinetic energj- of a moving body involves the idea of speed, not veloc- it;/, because the amount of work it can do is independent of the direction of the motion. (Also if there is no change in the speed of a body, the force is at right angles to the motion and so no work is done, whatever the change in direction may be.) Illustrations of the second formula, Lfi= 5 IW — iI«o-, ire given by the turning of a /TPindstone, and by a fly-wheel being set in motion or stopped. There are other ways of doing work than in overcoming elastic forces and producing speed, e.g. raising a body up from the earth, separating a ])iece of iron from a magnet, separating two bodies electrified oppositely, overcoming the force of friction, etc. In all these cases, the body doing the work loses energj' and the system on which work is done gains energy. The 'prin- ciple of the conservation of energy' is that in every case the energy lost by the former equals that gained by the latter; so that on the whole there is no change. Kvery phenomenon in nature is in accord with this principle so far as is know n. When a body is raised from the earth, work is done eipial to the product of the weight of the body and the rertical height it is raised, mf/h. This amiumt of energy is gained by the system consisting of the earth and the body whose mass is ni : but mitil gravitation is understood it will be impossible to locate the energy' in any definite place in- |)laees. If n body falls through a height h. it ami the earth lose potential energy, mr/ft, which is gained in the form of kinetic energy by the falling body ami the earth, principally by the former, since the change in the speed of the earth occasioned by the body as it falls toward it is so infinitesimal. If, after the body falls a dis. lance, h. its speed is s, its kinetic energj- is Vims', and therefore mgh = "4 ms' or s' = 2gh. This formula .shows that the speed of a falling body depends upon the vertical height traversed, not on the slope or length of the |)atli itself; it may fall vertically, or down an inclined plane, or down a spiral, etc. The cases of work being done against electri- cal and magnetic forces are discussed under Electricity and JIaq.etism. Whenever work is done in overcoming friction, it is observed that heat-elfects are produced, which can be traced to the fact that the minute portions of the body on which the work is done gain energy. This question is filly discussed under Heat. Since, when any inelastic body is de- formed in any way, there is internal friction, part of the energy gained by such a body when it strikes another body goes into producing heat- effects. It is a general property of motion, which fol- lows at once from the delinition of ]iotential energj', that all motions take place of themselves in such a manner as to make the potential en- ergj- of the svstem decrease, and that eiiuilibriuni is not reached until the potential energj' liaa reached a value such that it is a mininuim — that is. is as small as is possible under existing condi- . tions. ' ( The unit of work or energj' is that correspond- ing to a unit force acting through a distance of a unit length. On the C. G. S. system this unit is, then, tlut corresponding to a force of 1 dvne acting through 1 cm.: it is called an 'erg.' An erg is, however, such a small imit that 10' ergs — a 'joule,' as it is called — is ordinarily used as the practical unit. The amount of work done in a unit interval of time by any agency is called its 'activity' or 'power' (q.v.). On the C. 0. S. sys- tem the unit is, then. 1 erg per second. The jiractical unit is. however. 1 joule per second; this is called a 'watt.' M.vciii.NEs are mechanical aiqiliances by means of which a force apiilied at one point and in a definite direction is made to produce a different fence at another point and generally in a dilferent direction; the work done bv means of the latter force can never be greater than that done by the former — it is in practice always less, owing to friction and other causes. The 'mechanical ad- vantage' of the machine is the ratio of the two forces described above. There are many forms of machim^s: levers, ]>ulleys, inclined jdane, w-edge, screw. winiUass. etc. (See the separate articles.) The problem in any one case is to determine the theoretical mechanical advantage of a machine; that is, on the assumption that there is no fric- tion when the forces are w-orking. There arc two general methods of solving this: one is to imagine a certain foree acting on the madiine and to determine by the ordinarj principles of eipii- librium what second force will just balance the action of the first: the second is to consider the machine in eipiilibrium under the action of these two forces, then to imagine a small displacenu'nt, and to express the fact that the work done by one force equals that done against the other. Kor the application of these prineiidcs to the various machines reference should bi' made to the separate articles in which they are de- scribed. Rllil.liHiR.MMlY. . brief useful treatise for the gcni-ral reader, which gives a clear conception of the elementarv principles of mechanics, is Max- well, .l/((/(cr and Motion (New York, 1892). The