OF THE PULLEY.

187. A PULLEY is a small wheel, commonly made of wood or brass, which turns about an iron axis passing through the centre, and fixed in a block, by means of a cord passed round its circumference, which serves to draw up any weight. The pulley is either single, or combined together, to increase the power. It is also either fixed or moveable, according as it is fixed to one place, or moves up and down with the weight and power.

PROPOSITION XXXIII.

188. If a Power sustain a Weight by means of a Fixed Pulley : the Power and Weight are Equal.

FOR through the centre c of the pulley draw the horizontal diameter AB: then will AB represent a lever of the first kind, its prop being the fixed centre c; from which the points A and B, where the power and weight act, being equally distant, the power P is consequently equal to the weight w.

189. Corol. Hence, if the pulley be put in motion, the power p will descend as fast as the weight w ascends. So that the power is not increased by the use of

W

the fixed pulley, even though the rope go over several of them. It is, however, of great service in the raising of weights, both by changing the direction of the force, for the convenience of acting, and by enabling a person to raise a weight to any height without moving from his place, and also by permitting a great many persons at once to exert their force on the rope at P, which they could not do to the weight itself; as is evident in raising the hammer or weight of a pile-driver, as well as on many other occasions.

PROPOSITION XXXIV.

190. If a Power sustain a Weight by means of One Moveable Pulley; the Power is but Half the Weight.

FOR, here AB may be considered as a lever of the second

3

kind, the power acting at A, the weight at C, and the prop. or fixed point at B ; and be

cause P: W :: CB : AB,

and CB =AB, therefore

P = w, or w = 2p.

191. Corol. 1. Hence it is evident, that, when the pul'ley is put in motion, the ve locity of the power will be double the velocity of the weight, as the point p moves

twice as fast as the point c and weight w rises. It is also evident, that the fixed pulley F makes no difference in the power P, but is only used to change the direction of it, from upwards to downwards.

192. Corol. 2. Hence we may estimate the effect of a combination of any number of fixed and moveable pulleys; by which we shall find that every cord going over a moveable pulley always adds 2 to the powers; since each moveable pulley's rope bears an equal share of the weight; while each rope that is fixed to a pulley, only increases the power by unity.

XXWW

av

Here Pw.

Here p = w

OF THE INCLINED PLANE.

·w + w + w

193. THE INCLINED PLANE, is a plane inclined to the horizon, or making an angle with it. It is often reckoned one of the simple mechanic powers; and the double inclined plane makes the wedge. It is employed to advantage in raising heavy bodies in certain situations, diminishing their weights by laying them on the inclined planes.

PROPOSITION

PROPOSITIOŃ XXXV.

194. The Power gained by the Inclined Plane, is in Proportion as the Length of the Plane is to its Height. That is, when a Weight w is sustained on an Inclined Plane BC, by a Power P, acting in the Direction DW, parallel to the Plane; then the Weight w, is in proportion to the Power P, as the Length of the Plane is to its Height; that is, w: P :: BC: AB.

FOR, draw AE perp. to the plane BC, or to Dw. Then we are to consider that the body w is sustained by three forces, viz. 1st, its own weight or the force of

DO

W

gravity, acting perp. to Ac, or parallel to BA; 2d, by the power P, acting in the direction wD, parallel to BC, or BE; and 3dly, by the re-action of the plane, perp. to its face, or parallel to the line EA. But when a body is kept in equilibrio by the action of three forces, it has been proved, that the intensities of these forces are proportional to the sides of the triangle, ABE, made by lines drawn in the directions of their actions; therefore those forces are to one another as the three lines AB, BE, AE; that is, the weight of the body w is as the line AB, the power P is as the line

BE,

and the pressure on the plane as the line AE.

But the two triangles ABE, ABC are equiangular, and have therefore their like sides proportional; that is,

the three lines

are to each other respectively as the three BC, AB, AC, or also as the three

BC, AE, CE,

which therefore are as the three forces W, P, p, where denotes the pressure on the plane. That is, W: P:: BC: AB, or the weight is to the power, as the length of the plane is to its height..

See more on the Inclined Plane, at p. 164, &c.

195. Scholium. The inclined plane comes into use in some situations in which the other mechanical powers cannot be conveniently applied, or in combination with them. As, in sliding heavy weights either up or down a plank or other plane laid sloping: or letting large casks down into a cellar, or drawing them out of it. Also, in removing earth from a lower situation to a higher by means of wheel-barrows, or otherwise, as in making fortifications, &c; inclined planes, made of boards, laid aslope, serve for the barrows to run upon.

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Of all the various directions of drawing bodies up an inclined plane, or sustaining them on it, the most favourable is where it is parallel to the plane BC, and passing through the centre of the weight; a direction which is easily given to it, by fixing a pulley at D, so that a cord passing over it, and fixed to the weight, may act or draw parallel to the plane. In every other position, it would require a greater power to support the body on the plane, or to draw it up. For if one end of the line be fixed at w, and the other end inclined down towards B, below the direction WD, the body would be drawn down against the plane, and the power must be increased in proportion to the greater difficulty of the traction. And, on the other hand, if the line were carried above the direction of the plane, the power must be also increased; but here only in proportion as it endeavours to lift the body off the plane.

perp.

If the length BC of the plane be equal to any number of times its perp. height AB, as suppose 3 times; then a power P of 1 pound, hanging freely, will balance a weight w of 3 pounds, laid on the plane; and a power p of 2 pounds, will balance a weight w of 6 pounds; and so on, always 3 times as much. But then if they be set a-moving, the perp. descent of the power P, will be equal to 3 times as much as the ascent of the weight w. For, though the weight w ascends up the direction of the oblique plane BC, just as fast as the power p descends perpendicularly, yet the weight rises only the perp. height AB, while it ascends up the whole length of the plane BC, which is 3 times as much; that is, for every foot of the perp. rise of the weight, it ascends 3 feet up in the direction of the plane, and the power p descends just as much, or 3 feet.

PROPOSITION XXXVI.

197. When a Wedge is in Equilibrio; the Power acting against the Back, is to the Force acting Perpendicularly against either Side, as the Breadth of the Back AB, is to the Length of the Side AC or BC.

FOR, any three forces, which sustain one another in equilibrio, are as the corresponding sides of a triangle drawn perpendicular to the directions in which they act. But AB is perp. to the force acting on the back, to urge the wedge forward; and the sides AC, BC are perp. to the forces acting on them; therefore the three forces are as AB, AC, BC.

which are per. to them.

And therefore the thinner a wedge is, the greater is its effect, in splitting any body, or in overcoming any resistance against the sides of the wedge.

SCHOLIUM.

199. But it must be observed, that the resistance, or the forces above-mentioned, respect one side of the wedge only. For if those against both sides be taken in, then, in the foregoing proportions, we must take only half the back AD, or else we must take double the line AC or DC.

In the wedge, the friction against the sides is very great, at least equal to the force to be overcome, because the wedge retains any position to which it is driven; and therefore the resistance is doubled by the friction. But then the wedge has a great advantage over all the other powers, arising from the force of percussion or blow with which the back is struck, which is a force incomparably greater than any dead weight or pressure, such as is employed in other machines. And accordingly we find it produces effects vastly superior to those of any other power; such as the splitting and raising the largest and hardest rocks, the raising and lifting the largest ship, by driving a wedge below it, which a man can do by the blow of a mallet: and thus it appears that the small blow of a hammer, on the back of a wedge, is incomparably greater than any mere pressure, and will overcome it,

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