WORK ENERGY. Two general formula were developed in the discussion of translation and rotation, Yr= LO= The first formula may he expressed in words as follows: if a particle whose mass is m is moving with a speed in any direction, this will he changed to s in that same direction under the action of a constant force F in that direction. provided time distance traversed in that time i x as given by the relation Fx = An illustration is afforded by an arrow shot from a bow: = 0, then F.r = MR. Fx is called the 'work' done by the how, and the quantity is called the kirmriir cs,rgy of trmislation. Any body. not itself in motion, which has the power of producing kinetic. energy in another body is said to have potcntial cnrrgy. Thus a bent how, a compressed spring. a stretched elastic cord, etc., have potential energy. To bend the how, compress the spring, stretch the cord, etc., a force must he overcome; that is, motion is pr..dneed in a direction contrary to the elastic force of the body. The numerical value of the potential energy is defined as equal to the prod uct of the force overcome and the distance thromdi which this has been done. i.e. to the 'work done on' the bow. spring, or string.. If the spring is compressed by a body falling upon it, the spring gains potential enemy since work is done on it and the body loses kinetic energy. (The spring and body together would naturally continue to vibrate up and down, but it may be supposed here that the spring is caught and held \viten it is compressed to its greatest extent.) if F is the force of opposition due to the spring; the distance required to change the speed of the body of mass m from s to ti": the gain of potential energy of the spring in that distance is nr, and the loss of kinetic energy is 2 where Et Sint ilarly, if the spring expels the hotly, the spring Joe, work on the body and loses potential energy, and the body gains kinetic energy; the loss in po tential energy being Kr and the gain in kinetic energy being if in the distance x the speed is increased from s, to s; and as before Fx= 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 equals the gain of kinetic energy of the particle accelerated: or. the gain of tential of a system producing retardation the loss of kinetic of the retarded particle. Kinetic energy may also be produced by the impact of another body; and all ments are in accord with the idea that the kinetic energy gained by a body in this case equals tInft lost by the impinging particle pro robd no other effects are Produced. This is illustrated by the impact. of perfectly elastic bodies. (In general. when there is impact. such as rise of temperatnre are produced. in which case the kinetic energy gained by the particle does not equal that lost.) In general. then, in one body loser; energy :Mother body gains an equal amount, work be ing simply the transfer of the energy. Work is done in two ways: producing a change in spend and overcoming some opposing elastic force. Unless there is motion in the direction of the force. no work is dune.
It is evident that the kinetic energy of a 1.:11ving hotly involves the idea of speed, not refoc lty, because the amount of work it can du is independent of the direction of the motion. (Also if there is no charge in the speed of a body, the force is at right angles to the motion and so 110 is done, whatever the chanr„e in direction may ite.1 Illustrations of the second formula, , are given by the turning of a sindstone, and by a being set in motion or -.topped.
There are other ways of doing work than in ,,‘,.eopining elastic forces and producing speed, raising a up from the earth, separating a of iron front a magnet, separating two bodies elect ri tied oppositely, ov'reoming the force of etc. In all these eases, the body the work loses energy and the system nn which work is done gains energy. The 'prin ciple of the eonservation of is that in every ease the energy lost by the former equals that gained by the latter: so that on the whole is no change. Every phenomenon in nature is in accord with this principle so far as is know n.