# Elasticity Dynamics

## weight, power, wheel, pulley, rope, axle, pullies, fixed, fig and wheels

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If teeth are cut in the circumference of a wheel, and if they work in the teeth of another wheel of the same size as fig. 9. it is evident that both the wheels will revolve in the same time ; and the weight appended to the axle of the wheel B, will be raised in the same time as if the axle had been fixed to the wheel A. But if the teeth of the second wheel be made to work in teeth made in the axle of the first, as at fig. 10. as every part of the cir cumference of the second wheel is ap plied successively to the circumference of the axle of the first, and as the former is much greater than the latter, it is evi dent, that the first wheel must go round as many times more than the second, as the circumference of the second wheel exceeds that of the first axle. In order to a balance here, the power must be to the weight, as the product of the circum ferences, or diameters of the two axles multiplied together, is to the circumfe rences or diameters of the two wheels. This will become sufficiently clear, if it be considered as a compound lever, which was explained above. Instead of a combination of two wheels, three or four wheels may work in each other, or • any number ; and by thus increasing the number of wheels, or by proportioning the wheels to the axis, any degree of pow er may be acquired. To this sort of en gine belong all cranes for raising great weights ; and in this case the wheel may have cogs ail round it, instead of handles ; and a small lanthorn, or trundle, may be made to work in the cogs, and be turned by a winch ; which will make the power of the engine to exceed the power of the man who works it, as much as the num ber of revolutions of the winch exceeds those of the axle, when multiplied by the excess of the length of the winch above the length of the semi-diameter of the axle, added to the semi-diameter or half thickness of the rope, by which the weight is drawn up. See CRANE.

The construction of the main-spring, box of the fusee of a watch, round which the chain is coiled, will illustrate the prin ciple of the wheel and axis. The box may be considered as the wheel, and the fusee the axle or pinion, to which the chain communicates the motion of the box. The power resides in the spring wound round an axis in the centre of the box, and the weight is applied to the low er circumference of the fusee. As the force of the spring is greatest when newly wound up, and gradually decreases as it unwinds itself, it is necessary that the fusee should have different radii, so that the chain may act upon the smallest part of the fusee when its force is greatest, and upon the largest part of the fusee when its force is least ; for the equable motion of the watch requires, that the inequality in the action of the spring should be counteracted so as to produce an uniform effect.

The pulley is a small wheel turning on an axis, with a drawing rope passing over it; the small wheel is usually called a sheeve, and is so fixed in a box, or block, as to be moveable round a pin passing through its centre. Follies are of two kinds ; fixed, which do not move out of their Places ; and moveable, which rise and fall with the weight.

When a pulley is fixed, as Plate a Mechanics; fig. 11. two equal weights

suspended to the ends of a rope passing over it will balance each other, for they stretch the rope equally, and if either of them be pulled down through any given space, the Other will rise through an equal space in the same time ; and con sequently, as the velocities of 'both are equal, they must balance each other. This kind of pulley, therefore, gives no mechanical advantage ; but its use con sists in changing the direction of the pow er, and sometimes enabling it to he ap .plied with more convenience. By it, a man may raise a weight to any point, as the top of a building, without moving front the place he is in ; whereas, other •wise, he would have been obliged to as cend with the weight ; it also enables se veral men together to strength to the weight by means of the rope. The moveable pulley represented at A • (fig. 12.) is fixed to the weight W, and rises and falls with it. In comparing this to a lever, the fulcrum must be consider ed as at A, the weight acts upon the cen tre c, and the power is applied at the ex tremity of the lever D. The power, therefore, being twice as far from the fulcrum as the weight is, the proportion between the power and weight, in order to balance each other, must be as 1 to 2. Whence it appears, that the use of this pulley doubles the power, and that a man may raise twice as much by it as by his strength alone. Again, every moveable pulley hangs by two ropes equally stretch ed, and which must, consequently, bear equal parts of the weight ; but the rope A B being made fast at _13,• half the weight is sustained by it, and the other part of the rope, to which the power is apfdied, has but half the weight to support ; conse quently, the advantage gained by this pul ley is as 2 to 1. When the upper and .fixed block contains two pullies, which only turn upon their axis, and the lower moveable block contains also two, which not only turn on their axis, but rise with the weight F (fig. 13.) the advantage gained is as 4 to 1. For each lower pulley will be acted upon by an equal part of the weight ; and because in each pulley that moves' with the weight a double in crease of power is gained, the force by which F may be sustained will be equal to half the weight divided by the number of lower pullies ; that is, as twice the Dumber of lower pullies is to 1, so is the weight suspended to the power. But if the extremity C (fig. 14.) be fixed to the lower block, it will sustain half as much as a pulley ; consequently, here the rule Will be, as twice the number of pullies adding unity is to 1, so is the weight to the power. These rules hold good, what ever may be the number of putties in the blocks. If, instead of one rope going round all the putties, the rope belonging to each pulley be made fast at top, as in fig. 15, a different proportion between the power and the weight will take place. Ilere it is evident, that each pulley dou bles the power; thus, if there are two pullies, the power will sustain four times the weight ; if three pullies, eight times the weight ; if four pullies, sixteen times; and so on : that is, the power P, of 11b. will sustain a weight W of 161b.

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