LEVER ((erare, to lift up), the name of a common mechanical instrument, consisting of a simple bar of wood or metal, by fixing one point of which, called the fulcrum, a pressure at the end more distant from the fulcrum is made to counterbalance a larger pressure at the nearer end; or if both ends beequallv distant from the fulcrum, equal pressures are made to balance each other. [ lataiNcE.] The lover, considered as a machine, Would require no further notice than a reference to the article POWER for the correction of a mistake incident to the conception of this and other machines. But as one of the fundamental principles of mechanics receives its most simple form in its application to the common lever, this instrument assumes a degree of theoretical importance which will justify some discussion of the subject : and the principle of the lever, which is often confounded with the lever itself, must be explained. Thus when it is said in popular writings on mechanics that all machines are reducible to the lever and the inclined plane (an assumption of a startling character if we consider, for instance, the works of a Common watch) it is meant that every mode of communicating or relieving pressure is explicable upon the principle of one or other of those machines.
The first explanation of the lever was given by Archimedes, and that in so simple a manner, that while his method has always been the best for a popular view of the subject, it has never been surpassed, or even equalled, in rigor or purity, considered as a foundation for the science of STATICS.
It assumes two principles; firstly, that when a system is in equi librium, the state of rest will not be disturbed if additional pressures, such as compensate each other, and would by themselves produce no motion, be introduced or removed ; secondly, that when a weight is made to rest by being attached to an immoveable point (say it is sus pended by a string) the point or pivot of suspension undergoes a pressure equal to the weight of the system, whatever may be the form of that system, or the disposition of its parts. Every science must be founded upon some axiomatic assumption; and perhaps there is none which is better entitled to preference than the fact that a given weight, say a pound, suspended by a string, exerts the same pressure on the string whatever its shape may be ; namely, a pressure equal to the weight of the body. This being premised, a cylindrical or prismatic bar of uniform material will necessarily rest if a pivot be passed through its middle section at A : since there is no reason why it should pre ponderate on either side. Divide the her into two parts, n c and c v, of which K and a are the middle points. At ic and L suspend weights equal to the weights of s c and c D; but at the same time apply counterpoises of equal weight acting over fixed pulleys c and 11: so that the new forces being such that each pair would be in equilibrium, they would not affect the equilibrium already established by means of the equality of tho parts or the bar on each side of A. Now, equi librium existing, we are at liberty to remove any forces which equi librate each other, such as are the upper v and the weight of n 0 ; such also as are the upper w and the weight of c D. For n c, if detached,
would exert on the string which goes over the pulley o a pressure neither more nor less than its own weight (which is v); and o o, if detached from the pivot and from a o, would exert on the string of the pulley H a pressure equal to its own weight, or to w. But when these pairs are removed, there remain only the lower weights v and w ; the substance and rigidity of the lever being retained to connect thcm,though its weight is removed or counterpoised. And K being the sum of the halves of the parts, is equal to half of the whole length, or to II A : task away the common part A K, and there remains n e, equal to A a, or x o equal to A L ; also A x is equal to o is Also v is to w in the proportion of n o to c v, or of K o to C L, or of A 1. to A K ; that is, the weights v and w balance each other when they are inversely as their distances from the fulcrum a.
It only remains to show that no other weight except v, proportioned to w as above, will counterbalance w. If possible, let another weight V, produce this effect when applied at x ; and upwards, by means of the pulleys n and a, apply pressures equal to w and v', the old weights v and w remaining as before. Then there are two systems which being separately in equilibrium are so when existing in connection. But the under and upper w balance each other ; remove them, and there remain two unequal pressures, v and v', which acting in opposite directions at the same points, balance each other : a manifest absur dity. Consequently, no other weight except v can balance w when placed at K.
The most simple way in which the preceding result can be stated is ae follows : when v placed at K balances w placed at L, about the pivot A, the number of pounds in v multiplied by the number of feet in lc A gives the same product as the number of pounds in w multiplied by the number of feet in A a ; (any other units of weight and length will do equally well, if only the same be used in both). The product of a pressure and the perpendicular let fall upon its direction from a fixed pivot or fulcrum is sometimes called the moment, sometimes the leverage, of the weight.
As the pressure on the pivot A is the sum of the weights v and w, if the lever were suspended at A, by a string passing over a pulley, a counterpoise might be applied in the shape of a weight equal to the sum of the weights v and w. But when a system is at rest, the equili brium is not disturbed by making any point an immoveable pivot, and taking away any weight which may be there, leaving its place to be supplied by the re-action of the pivot. If then we were to makes a pivot, weights equal to w and v +w, acting downwards and upwards at a and A, would counterbalance one another, and since vxse=wxa A, add wxxe to both sides, and we have (v + w)s A=wxK L.
In English treatises on mechanics, it is customary to call oue of the pressures which balance on a lever, the power, and the other the weight. Levers are thus distinguished as of the, first, second, or third kind, according as the fulcrum, the weight, or the power, is in the middle. •