LEVER

In Mechanics, an inflexible rod movable 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 Aristolle; but the first accurate explanation was given by Arch i me des, in his Treatise Dc Equiponderant ibus. 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. Lovers are commonly divided into three kinds, according (1.) (2.) \ V to the relative i sitions of the power, the weight, and the (3.) fulcrum. In a lever of the first kind, (fig. 1), the fulcrum F is between the 3. v.'.i j, power P and the weight W. In a Ie- 3" ver of the second kind (fig. 2), the weight W is between the fulcrum F and the power «'. 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 = AO-AD = J AB-UC = £BC; and OE=OB-EB = | AB-JCB=fAC; consequently, O D Is to O E as B C to A C; or O D is to O E as the weight concentrated at E to the weight concentrated atD. This demonstration is commonly ascribed to Arch i me des. (Maclaurin's Account of Newton's Principia.) This proposition shows the advantage obtained bv using the levet as a mechanical engine. The arm P F (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. 1 and acting2) at—4P will feet, overcome and W F a= resist an cel foot; then of 4 albs.power at W.of "I lb. 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. Therefore in this, as in all mechanical engines, when a small power raises a great weight, the velocity of the power is much greater than the velocity of the weight; and what is gained in force is therefore said to be lost in time. When the power and the weight do not act on the lever in directions perpendicular to its length, or when the arms of the lever are not in the same straight line, or are bent, then the power and the weight are not to each other reciprocally as the arms of the lever, but as the straight lines drawn from the fulcrum perpendicular to the respective directions in which the power and the weight take efTect 662 [s. 676]