LEVER. In Mechanics, an inflexible rod movable about a fulcrum or prop, and forces applied to two or more points in it. The lever is one of the me chanical powers ; and, being the simplest of them all, was the first that was attempt ed to be explained. Its properties are treated of by Aristotle; but the first ac curate explanation was given by Archi medes, in his Treatise De Equipanderanti bus.
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, according to the relative positions of the power, the weight, 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 crum F and the power P. In a lever of the third kind (fig. 3), the power P is be tween the fulcrum F and the weight W. The general principle of the lever is, that when the power end weight are in equilibria they are to each other inverse ly as their distances from the fulcrum. This property is almost an obvious conse quence from the principle of virtual ve locities ; 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 0, the two ends would evidently balance each other, and the pressure at 0 would be the same as if the whole matter of the bar were con centrated in that point. Suppose it to consist of two parts, A C and B 0, these again would be separately supported at their middle points D and E; or the whole of the matter in A C may be con ceived to be concentrated at D, and the whole of that in B Cat E, and the equili brium would not be disturbed. Hence the weight of A C attached at D, and the weight of B C attached at E, would bal ance the inflexible line D E, if supported at 0, the centre of the whole bar A B.
But 0 D=A 0—A A B-1 A C=1 BC• and 0 E=0 B--E B=4 A B-1 B =I- A C; consequently, 0 D is to 0 E as B C to A C; or 0 D is to 0 E as the weight concentrated at E to the weight concentrated at D. This demonstration is commonly ascribed to Archimedes. This proposition shows the advantage obtained by using the lever as a mechan ical engine. The arm P F (fig. 1), is com monly longer than W F, and, consequent ly, when there is equilibrimn the weight exceeds the power. The proportion in which the weight exceeds the power is called the mechanical advantage, or pur chase. Suppose P F (figs. 1 and 2)=4 feet, and W F=1 foot ; then a power of 11b. 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. Therefore in this, as in all mechanical en gines, 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 perpendicu lar 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 effect.