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the radius of the roller ; or, to the radius of the roller we must add half the thickness of the cord, when there is but one fold.
185. The wheel-and-axle has a great advantage over the simple lever, in point of convenience. For a weight can be raised but a little way by the lever; whereas, by the continual turning of the wheel and roller, the weight may be raised to any height, or from any depth.
186. By increasing the number of wheels too, the power may be multiplied to any extent, making always the less wheels to turn greater ones, as far as we please : and this is commonly called Tooth and Pinion Work, the teeth of one circumference working in the rounds or pinions of another, to turn the wheel. And then, in case of an equilibrium, the power is to the weight, as the continual product of the radii of all the axles, to that of all the wheels. So, if the power ?
turn the wheel Q, and this turn the small wheel or axle R, and this turn the wheel s, and this turn the axle T, and this turn the wheel v; and this turn the axle x, which raises the weight w; then p:W::CB.DE . FG: AC. BD . EF. And in the same proportion is the velocity of w slower than that of P. Thus, if each wheel be to its axle, as 10 to 1; then P:w:: 13 : 103 or as 1 to 1000. So that a power of one pound will balance a weight of 1000 pounds; but then, when put in motion, the power will move 1000 times faster than the weight:
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
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 as 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 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.
190. If a Power sustain a Weight by means of One Moveable
Pulley; the Power is but Half the Weighton For, here as may be considered as a lever of the second
kind, the power acting at A,
= LAB, therefore
191. Corol. 1. Hence it is
WO 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 à 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 imity.
Here P = w.
OF THE INCLINED PLANE. 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.
PROPOST ION 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 Weigłt 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 De the plane Bc, or to dw. Then we are to consider
W that the body w is sustained by three forces, viz. Ist, its own weight or the force of 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
A B, BE, AE ; that is, the weight of the body w is as the line AB, the power p is as the lin
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
AB, BE, AE, 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 p 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. Tlie Inclined plane comes into use in same situations in which the other niechanical 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 pipon.
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 wn, 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.
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 aş much. But then if they be set a-moving, the perp. descent of the power P, will be equal to 3 tiines as much as the perp. 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 Jength 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.