WORK. To do work is to overcome resistance. If we try to lift a ton-weight. how ever we may fatigue ourselves, we cannot move it, and therefore we do no work., But we can lift with ease a hundred-weight, and then we do more work in proportion as we raise it higher, In lifting coals from a pit, the work done is evidently in proportion to the depth of the pit, and to the weight of the coals raised. This and numberless other instances are too well known to need further description. Wo may therefore at once define the done by a-force as the product of the force into the space through which it mores its point of application in its own direction, and it is u7ually measured by engineers and others who do not require absolute accuracy, in the work required to raise ayonad one foot high. If the motion of the point of application be in the opposite direction to that of the force, the work is done against the force. If the motion be per pendicular to the direction of the force, no work is clone by or against the force. Thus, the work spent in projecting a curling-stone, in opening a massive gate, or in turning a large fly-wheel of grind-stone, has nothing whatever to do with the force of gravity—the body moved, in all these cases, is, as a whole, neither raised nor lowered as regards its distance above the earth's surface. If the direction of the force be oblique to the direc tion in which the point of application moves, we must resolve the force, by the-law of the parallelogram of forces (see COMPOSITION OF FORCES), into two components, one in the direction of motion, the other perpendicular to it. The former is the working com ponent; the latter, as we have just seen, does no work. A good illustration of this is found in the case of raising stones from a quarry by carting them up a series of inclined planes, as contrasted with hauling them up vertically. The work done in either case is measured by the product of the weight of the stones, and the height through which they have been raised ; and thus, for the same load of stones, it will be the same whichever process is adopted. This is evident from the property of the inclined plane—viz., that the force required to support a body resting on the plane (which is the force that has to be overcome when we haul it up the plane) is to the weight of the body as the height of the plane to its length. Hence, this force, multiplied into the length of the plane, gives the same product as the whole weight into the height of the plane; and these are the two quantities of work wo are comparing.
When work is done upon a body, there is always an increase of velocity unless other forces act on the body, so that it does an equal amount of work against them. Thus, if we push a movable body, such as a cart, along a road, the velocity gradually increases, and would increase indefinitely were there no friction and no resistance of the air (forces against which work has to be done), and could we move fast enough to keep continu ally pushing it, however great its velocity may become. If, on the other band, by means of a rope and pulley, we raise a stone, if once started, it will ascend uniformly, so long as we pull with a force just equal to its weight, because, then, as much work is done on the stone by the hand as it does against gravity. If we pull with a force
greater than its weight, we do more work on the stone than it does against gravity, and the upward velocity increases; if with a force less than the weight, the stone has to do more work against gravity than is done on it by the rope, and its velocity upward becomes less. The measure of the excess of work done on a body over that which it does against resistance is the increase of the product of half the mass into the square of the of what was formerly called the of the body, what is now called its actual, or preferably, its kinetic energy. See FORCE. Hence, as it is evident that if a body, or system, be acted on by a set of forces which are in equilibrium, it will have no tendency to lose or to acquire velocity, its kinetic energy will remain unchanged, and therefore as much work must be done upon it by some of the applied forces, as it does against the rest, in any displacement so slight as not to change the circumstances of the particular arrangement. That is, when forces are in equilibrium on a body, if the body be slightly displaced, the sum of the products of each force by the effective component of the dis placement of its point of application is zero—the product being positive when the force does work, negative when work is done against it. This is the celebrated principle of vir tual velocities, the term virtual velocity having been, very inconveniently, applied to what we have called above the effective component of the displacement of the point of appli cation of a force. It was often employed as the basis of the whole of statics, and very curious attempts have been made to give proofs of it (independent of the laws of composi tion of forces), especially by Lagrange. But the principle of work, or energy, of which that of virtual velocities is a mere particular case; and which is at once applicable to the whole range of dynamical science, is distinctly enunciated by Newton in a scholium to his third law of motion. See MOTION, LAWS OF. His words are memorable, and should be universally cestimetur agentis actio ex dus vi et velocitate conjunctim; et sim -iliter resistentis reactia cestinietur conjunctim ex ejus partium singularum velocitatzbus et vizi bus resistendi ab earum attrition, cohasione, pondere, et accelerations oriundis; erupt actio et reactio, in omni instrumentorum usu, sibi invicem serum. cequales. Newton has defined what he means by the velocity of an agent—viz. the component of the velocity of its point of application which is in the direction agent—viz., the agent. He has also shown what is the measure of resistance arising from acceleration see VELOCITY); so that, merely using modern terms instead of those employed by Newton, but in nowise altering the scope of the above remarkable passage, we have the following version of it: Work done upon any system of bodies (literally, 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.