An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery.
When a machine undergoes alternate acceleration and retardation, so that at certain instants of time, occurring at the end of intervals called periods or cycles, it returns to its original speed, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other; and at the end of each cycle the principle of the equality of energy and work stated in § 87, with all its consequences, is verified exactly as in the case of machines of uniform speed.
At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, as applied to direct deviation, or acceleration and retardation.
§ 116. Energy of Acceleration and Work of Retardation for a Shifting Body.—Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time Δt let its velocity be increased at a uniform rate of acceleration from v1 to v2. The rate of acceleration will be
dv/dt = const. = (v2 − v1) Δt;
and to produce this acceleration a uniform effort will be required, expressed by
P = w (v2 − v1) gΔt
(71)
(The product wv/g of the mass of a body by its velocity is called its momentum; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)
To find the energy which has to be exerted to produce the acceleration from v1 to v2, it is to be observed that the distance through which the effort P acts during the acceleration is