LEVER

In Mechanics, an inflexible rod moveable about a fulcrum or prop, and having forces applied to two or more points in it. The lever is one of the mechanical powers; and, being the simplest of them all, was the first that was attempted to be explained. Its properties are treated of by Aristotle; but the first accurate explanation was given by Arch i me des, in his Treatise De Equipnnderantibus. In treating of the lever, it is convenient to distinguish the forces applied to it by different names. One is usually called the power, the other the weight or resistance. Levers are commonly divided into three kinds, accord(1-) (2.) ■SI ing to the relative positions of the power, the weight, (3.) w^and the fulcrum. In a lever of the first kind (fig. 1.), the fulcrum F is between the power P and the weight W. In a lever of the second kind (fig. 2.), the weight W is between the fulcrum F and the power P. In a lever of the third kind (fig. 3.), the power P is between the fulcrum F and the weight W. The general principle of the lever is, that when the power and weight are in equilibrio, they are to each other inversely as their distances from the fulcrum. This property is almost an obvious consequence from the principle of virtual velocities; but it may be deduced from more familiar considerations. Let A B be a cylinder or bar of homogeneous matter. If supported from the middle, O, the two ends would evidently balance each other, and the pre ure at O would be the same as if the whole matter of the bar were concentrated in that point. Suppose it to consist of two parts, A C and B C, these again would be separately supported at their middle points D and E; or the whole of the matter in A C may be conceived to be concentrated at D, and the whole of that in B C at E, and the equilibrium would not be disturbed. Hence the weight of A C attached at D, and the, weight of B C attached at E, would balance the inflexible line D E, if supported at O, the centre of the whole bar A B. But O D=s AO-AD = iAB-iAC = iBC;andOE = OBEB = 4AB — §CB= ^AC; consequently, O D is to O E asB C to A C; or O D is to O E as the weight concentrated at E to the weight concentrated at D. This demonstration is commonly ascribed to Arch i me des. {Maclaurin's Account of Newton's Principia.) This proposition shows the advantage obtained by using the lever as a mechanical engine. The arm PF (fig. 1.) is commonly longer than W F, and, consequently, when there is equilibrium the weight exceeds the power. The proportion in which the weight exceeds the power is called the mechanical advantage, or purchase. Suppose P F (figs. I. and 2.) = 4 feet, and W F = 1 foot; then a power of 1 lb. acting at P will overcome a resistance of 4 lbs. at W. Suppose the lever with the weights P and W to turn round the fulcrum, the two points to which P and W are attached will describe arcs proportional to the radii F P, F W; consequently the power P is to the weight W as the velocity of the weight to the velocity of the power. U u [s. 670]