# LEVER

, a straight bar of iron or wood, &c, sup posed to be inflexible, supported on a fulcrum or prop by a single point, about which all the parts are moveable.

The Lever is the first of those simple machines called mechanical powers, as being the simplest of them all; and is chiefly used for raising great weights to small heights.

The Lever is of three kinds. First the common sort, where the weight intended to be raised is at one end of it, our strength or another weight called the power is at the other end, and the prop or fulcrum is between them both. In stirring up the fire with a poker, we make use of this Lever; the poker is the Lever, it rests upon one of the bars of the grate as a prop, the incumbent fire is the weight to be overcome, and the pressure of the hand on the other end is the force or power. In this, as in all the other machines, we have only to increase the distance between the force and the prop, or to decrease the distance between the weight and the prop, to give the operator the greater power or effect. To this kind of Lever may also be referred all scissais, pincers, snuffers, &c. The steel-yard and the common balance are also Levers of this kind.

In the Lever of the 2d kind the prop is at one end, the force or power at the other, and the weight to be raised is between them. Thus, in raising a waterplug in the streets, the workman puts his iron bar or Lever through the ring or hole of the plug, till the end of it reaches the ground on the other side; then making that the prop, he lifts the plug with his force or strength at the other end of the Lever. In this Lever too, the nearer the weight is to the prop, or the farther the power from the prop, the greater is the effect. To this 2d kind of Lever may also be referred the oars and rudder of a boat, the masts of a ship, cutting knives fixed at one end, and doors, whose hinges serve as a fulcrum.

In the Lever of the third kind, the power acts between the weight and the prop; such as a ladder raised by a man somewhere between the two ends, to rear it against a wall, or a pair of tongs, &c.

It is by this kind of Lever too that the muscular motions of animals are performed, the muscles being inserted much nearer to the centre of motion, than the point where is placed the centre of gravity of the weight to be raised; so that the power of the muscle is many times greater than the weight it is able to sustain. And in this third kind of Lever, to produce a balance between the power and weight, the power or force must exceed the weight, in the same proportion as it is nearer the prop than the weight is; whereas in the other two kinds, the power is less than the weight, in the same proportion as its distance is greater; that is, universally, the power and weight are each of them reciprocally as their distance from the prop; as is demonstrated below.

Some authors make a 4th sort of what is called a bended Lever; such as a hammer in drawing a nail, &c.

In all Levers, the universal property is, that the effect of either the weight or the power, to turn the Lever about the fulcrum, is directly as its intensity and its distance from the prop, that is as di, where d denotes the distance, and i the intensity, strength, or weight, &c, of the agent. For it is evident that at a double distance it will have a double effect, at a triple distance a triple effect, and so on; also that a double intensity produces a double effect, a triple a triple, and so on: therefore universally the effect is as di the product of the two. In like manner, if D be the distance of another power or agent, whose intensity is I, then is DI the effect of this also to move the Lever. And if these two agents act against each other on the Lever, and their effects be supposed equal, or the Lever kept in equilibrío by the equal and contrary effects of these two agents; th en is , which equation resolves into this analogy, viz, ; that is, the distances of the agents from the prop, are reciprocally| er inversely as their intensities, or the power is to the weight, as the distance of the latter is to the distance of the former.

Writers on mechanics commonly demonstr<*>te this proportion in a very absurd manner, viz, by supposing the Lever put into motion about the prop, and then inferring that, because the momenta of two bodies are equal, when placed upon the Lever at such distances, that these distances are reciprocally proportional to the weights of the bodies, that therefore this is also the proportion in case of an equilibrium; which is an attempt absurdly to demonstrate a thing supposing the contrary, that a body is at rest, by supposing it to be in motion. I shall therefore give here a new and universal demonstration of the property, on the pure prineiples of rest and pressure, or force only. Thus, let PW be a lever, C the prop, and P and W any two forces acting on the lever at the points P and W, in the directions PO, WO; then if CE and CD be the perpendicular distances of the directions of these forces from the prop C, it is to be demonstrated that . In order to which join CO, and draw CB parallel to WO, and CF parallel to PO. Then will CO be the direction of the pressure on the prop, otherwise there could not be an equilibrium, for the directions of three forces that keep each other in equilibrium, must necessarily meet in the same point. And because any three forces that keep each other in equilibrium, are proportional to the three sides of a triangle formed by drawing lines parallel to the directions of these forces; therefore the forces on P, C, and W, are as the three lines BO, CO, CB, which are in the same direction, or parallel to them; that is the force P is to the force W, as BO or its equal CF is to CB. But the two triangles CDF, CEB are similar, and have their like sides proportional, viz, ; and because it was ; theresore by equality ; that is, each force is reciprocally proportional to the distance of its direction from the fulcrum. And it will be found that this demonstration will serve also for the other kinds of Levers, by drawing the lines as directed. Hence if any given force P be applied to a Lever at A; its effect upon the Lever, to turn it about the centre of motion C, is as the length of the arm CA, and the sine of the angle of direction CAE. For the perp. CE is as CA × sin. [angle]A.

In any analogy, because the product of the extremes is equal to that of the means; therefore the product of the power by the distance of its direction is equal to the product of the weight by the distance of its direction. That is, .

If the Lever, with the two weights fixed to it, be made to move about the centre C; the momentum of the power will be equal to that of the weight; and the weights will be reciprocally proportional to their velocities.

When the two forces act perpendicularly on the Lever, as two weights &c; then, in case of an equilibrium, E coincides with P, and D with W; and the distances CP, CW, taken on the Lever, or the distances of the power and weight, from the fulcrum, are reciprocally proportional to the power and weight.

In a straight Lever, kept in equilibrio by a weight and power acting perpendicularly upon it; then, of these three, the power, weight, and pressure on the prop, any one is as the distance of the other two.

And hence too , and ; that is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.

Also, if several weights P, Q, R, S, &c, act on a straight Lever, and keep it in equilibrio; then the sum of the products on one side of the prop, will be equal to the sum on the other side, made by multiplying each weight by its distance from the prop; viz,

Hitherto the Lever has been considered as a mathematical line void of weight or gravity. But when its weight is considered, it is to be done thus: Find the weight and the centre of gravity of the Lever alone, and then consider it as a mathematical line, but having an equal weight suspended by that centre of gravity; and so combine its effect with those of the other weights, as above.

Upon the foregoing principles depends the nature of scales and beams for weighing all bodies. For, if the distances be equal, then will the weights be equal also; which gives the construction of the common scales. And the Roman statera, or steel-yard, is also a Lever, but of unequal arms or distances, so contrived that one weight only may serve to weigh a great many, by sliding it backwards and forwards to different distances upon the longer arm of the Lever. See Balance, &c.

Also upon the principle of the Lever depends almost all other mechanical powers and effects. See Wheel-and-axle, Pulley, Wedge, Screw, &c.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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