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the circumference of the wheel, while
180. Corol. 1. If the wheel be put in motion; then, the spaces moved being as the circumferences, or as the radii, the velocity of w will be to the velocity of P, as CA to CB; that is, the weight is moved as much slower, as it is heavier than the power; so that what is gained in power, is lost in time. And this is the universal property of all machines and engines.
181. Corol. 2. If the power do not act at right angles to the radius CB, but obliquely; draw CD perpendicular to the direction of the power; then, by the nature of the lever,
182. To this power be-
183. And the same for all cranes, capstans, windlasses, and such like; the power being to the weight, always as the radius or lever at which the weight acts, to that at which the power acts; so that they are always in the reciprocal ratio of their velocities. And to the same principle may be referred the gimblet and auger for boring holes.
184. But all this, however, is on supposition that the ropes or cords, sustaining the weights, are of no sensible thickness. For, if the thickness be considerable, or if there be several folds of them, over one another, on the roller or barrel; then we must measure to the middle of the outermost rope,
the radius of the roller; or, to the radius of the roller we
185. The wheel-and-axle has a great advantage over the
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 e, and this turn the small wheel or axle R,
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.
188. If a Power sustain a Weight by means of a Fixed Pulley;
FOR, through the centre c of the pulley
189. Corol. Hence, if the pulley be put
190. If a Power sustain a Weight by means of One Moveable
FOR, here AB may be considered as a lever of the second
kind, the power acting at a,
191. Corol. 1. Hence it is evident, that when the pulley is put in motion, the velocity 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 r 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.
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 the powers that ustain them by laying them on the inclined planes.
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 gravity, acting perp. to Ac, or parallel to BA; 2d,
the weight of the body w is as the line
AB, BE, AE; that is,
and the pressure on the plane as the line AE.
But the two triangles ABE, ABC are équiangular, 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,
BC, AE, CE,
which therefore are as the three forces w, p, p,
See more on the Inclined Plane, at p. 144, &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.