This force of 100 pounds, being applied to the power end of the third lever, will act upon the same principles as the others, and raise the weight end with a force of 1,000 pounds.

Fig. 156.

[In calculating the action of any compound system of levers, it does not affect the principles of calculation if some of the levers are of the first kind, and some of any other. The rule is to "multiply the weight on any lever by its distance from the fulcrum, and multiply the power by its distance from the same point; if the products are equal, then the weight and power will balance each other." If we wish to calculate the effect of the system given in Fig. 155, we must multiply the length of the long arm by the power, and multiply the short arm by the weight or resistance offered.]

For many of the purposes of science, levers are used to magnify small motions. The power causing

the motion is applied by a short arm near to the fulcrum of the lever; and the other arm, which may be ten, twenty, or more times longer, moves over a graduated scale. The pyrometer is an example of this application.

The most accurate means of determining the weight of bodies is by the lever. When arranged for this purpose, it passes under the name of "The Balance." It is a lever of the first kind with equal arms. Various forms are given to it, and various contrivances annexed for the purpose of insuring its lightness, its inflexibility, and the absolute equality of the lengths of its arms. Fig. 156 represents one of the best kinds: a a is the beam; c is the fulcrum, or centre of motion; d dare the scale-pans, in which the weights and objects to be weighed are applied; their points of suspension are at a a. With a view of reducing friction, the axis of motion, e, and both the points of suspension, are knife-edges of hard steel, working on planes of agate; and to preserve them uninjured, the beam and the scale-pans are supported upon props, except at the time a substance is to be weighed. Then, by moving the handle, f, the axis of motion is deposited slowly on its agate plane, and the scale-pans on their points of suspension, and the beam thrown into action.

In balances it is essential that the centre of gravity should have a particular position. The cause of this will be appreciated from what has been said in Chapter XXIV. Thus, if the centre of gravity coincided with the centre of motion, the balance beam would not vibrate, but would stand in a position of different equilibrium, whatever angular position might be given to its arms.

If the centre of gravity was above the axis of motion, the balance would be in a condition of unstable equilibrium, and would overset by the slightest increase of weight on either side, the centre of gravity coming down to the lowest point. But when it is beneath the axis of motion, the balance vibrates like a pendulum, and neither sets nor oversets. It is essential,

therefore, that in all these instruments the centre of gravity should be below

D B

Н

E

the centre of motion. And it might be shown that the sensibility of the balance, or, in other words, the smallness of the weight it will detect, becomes greater as these two centres approach K each other.

The different kinds of weighing-machines are either modified levers or combinations of levers. Examples occur in the machine for weighing loaded carts, in the steelyard, which is a lever of unequal arms, and in the bent lever balance. The latter is represented in Fig. 157. It consists of a bent lever, A B C, the end of which, C, is loaded with a fixed weight. This lever works on a fulcrum, B, supported on a pillar, H, J. From the arm, A, is suspended a scale-pan, E, and to Fig. 157. the pillar there is affixed a divided scale, F G, over which the lever moves. Through B draw the horizontal line, G K, and let fall from it the perpendiculars, A K, D C. Then if B K and B D are inversely proportional to the weight in the scale, E, and the fixed weight, C, the balance will be in equilibrio; but if they are not, then the lever moves, C going farther from the fulcrum, and stopping when equilibrium is attained. The scale, F G, is graduated by previously putting known weights in E.

CHAPTER XXVIII.

THE PULLEY-THE WHEEL AND AXLE.

Description of the Pulley-Laws of the Lever apply to it-Use of the Fixed Pulley-The Movable Pulley-Runners-Systems of Pulleys-White's Pulley-Law of Equilibrium-Advantages of the Wheel and Axle over the Lever-Windlass-Capstan-Wheelwork-Different kinds of Toothed Wheels.

THE pulley is a wheel, round the rim of which a groove is cut, in which a cord can work, and the centre of which moves on pivots in a block. The wheel sometimes passes under the name of a sheave.

b

By a fixed pulley, we mean one which merely revolves on its axis, but does not change its place. The power is applied to one end of the cord, and the weight to the other.

The action of the pulley may be readily understood from that of the lever. Let c, Fig. 158, be the axis of the pulley; b, the point to which the weight is attached; a, the point of application of the power; draw the lines, c b, c a-they represent the arms of a lever and the law of the equilibrium of a lever, therefore, applies in this case also; and as these Fig. 158. arms are necessarily equal to each other, the pulley will be in equilibrio when the weight and power are equal.

If the direction in which the power is applied, instead of being P a, is P'a', the same reasoning holds good. For, on drawing C a', as before, it is obvious that b c a represents a bent lever of equal arms.

of equilibrium is, therefore, the same.

The condition

The fixed pulley does not increase the power, but it renders it more available, by permitting us to apply it in any desired direction.

To prove the properties of the pulley experimentally, hang to the ends of its cord equal weights; they will remain in equilibrio. Or, if the power be increased, so as to make the weight ascend, the vertical distances passed over are equal.

P

W

The movable pulley is represented at Fig. 159. Its peculiarity is that, besides the motion on its own axis, it also has a progressive one. Let b be the axis of the pulley, and to it the weight, W, is attached; the power is арplied at a. Draw the diameter, a c, then c is the fulcrum of a c, which is in reality a lever of the third order, in which the distance, a c, of the power is twice that, bc, of the weight. Consequently "the movable pulley doubles the effect of the power," and the distance traversed by the power is twice that traversed by the weight.

W

Fig. 159.

A movable pulley is sometimes called "a runner;" and as it would be often inconvenient to apply the power in the upward direction, as at a P, there is commonly associated with the runner a fixed pulley, which, without changing the value of the power, enables us to vary the direction of its action.

Systems of pulleys are arrangements of sheaves, movable and fixed.

Fig. 160. When one fixed pulley acts on a number of movable ones, equilibrium is maintained when the power and weight are to each other as 1 to that power of 2 which equals the number of the movable pulleys. Thus, if there be, as in Fig. 160, three movable pulleys, the power is to the weight as 1: 23 that is, 1: 8; consequently, on such a system, a given power will support an eightfold weight.

When several movable and fixed pulleys are employed, as in Fig. 161, equilibrium is obtained when the power equals the weight divided by twice the number of movable pulleys. The weight being equally divided between the six lines, it follows that each is drawn by th of the weight, W. Consequently, if sixty pounds weight is suspended to the bottom, each line would be drawn upon by a force of ten pounds. If we wish to keep this machine in a state of equilibrium, we must attach a weight, P, of ten pounds to the end of the line.

In such systems of pulleys there is a great loss of power arising from the friction of the sheaves against the sides of the blocks, and on their axles.

H

In White's pulley this is, to a considerable extent, avoided. This contrivance is represented in Fig. 162. It consists of several sheaves of unequal diameters, all turned on one common mass, and working on one common axis. The diameters of these, in the upper blocks, are as the numbers 2, 4, 6, &c.; and in the lower, 1, 3, 5, &c. ; consequently they all revolve in equal times, and the rope passes without sliding or scraping upon the grooves.

WHEELS AND AXLES.

The wheel and axle consist of a cylinder revolving upon an axis, and having a wheel of larger diameter immovably affixed to it. The power is applied to the circumference of the wheel, the weight to that of the axle.

[Let a b be a wheel, c d, Fig. 163, its axle, and suppose the circumference of the wheel to be eight times as great as the circumference of the axle; then a power, P, equal to one pound, hanging by the cord, I, which goes round the wheel, will balance a weight, W, of eight pounds, hanging by the rope, K, which goes round the axle; and as the friction on the pivots, E F, or gudgeons of the axle, is but small, a small addition to the power will cause it to descend, and raise the weight; but the weight will rise with only an eighth part of the velocity wherewith the power descends, and consequently through no more than an eighth part of an equal space in the same time. If the wheel be pulled round by the handles, S S, the power will be increased in proportion to their length. G is a ratchet-wheel on with a catch, H, to fall in its teeth.-Ferguson's Lectures, 10th edition, page 55.]

W

Fig. 161.

one end of the axle,

Fig. 163.

W

Fig. 162.

The law of equilibrium is, that "the power must be to the weight as the radius of the axle is to that of the wheel."

This instrument is, evidently, nothing but a modification of the lever; it may be regarded as a continuously acting lever; in fact, it is sometimes called "the perpetual lever." In its mode of action the common lever operates in an intermittent way, and, as it were, by small steps at a time. A mass which is forced up by a lever a short distance must be temporarily propped, and the lever re-adjusted before it can be brought into action again; but the wheel and saxle continue their operation constantly in the same direction.

[The inconvenience of having a large wheel and very slender axle may be avoided, without lessening the mechanical advantage, by employing a

Fig. 164.

machine called the "Chinese wheel and axle," which consists of two cylinders, one larger than the other, turning about the same axis. The weight is attached to a pulley, which plays on a long cord, which is coiled round both axles in contrary directions. When the winch is turned, one end of the cord uncoils from the smaller cylinder, and is wound round the larger; thus the weight is elevated at each turn, through a space equal to half of the difference between the circumference of the two cylinders. Therefore the advantage of this machine, with its pulley, is in the ratio of the diameter of the larger cylinder to half its excess above that of the lesser one.] (Fig. 164.)

That this is its mode of action may be understood from considering Fig. 165, in which let c be the common centre of the axle, cb, and of the wheel, ca, a the point of application of the power, P, and 6 that of the weight, W. Draw the line a cb; it evidently represents al a lever of the first order, of which the fulcrum is c, and from the principles of the lever it is easy to demonstrate the law of equilibrium of this machine, as just given. Further, it is immaterial in what direction the power be applied, as P' at p the point, a'; for a'c b still forms a bent lever, and the same principle still holds good.

P

Fig. 165.

[The effect of the wheel depends upon the superiority of the radius, or diameter of the wheel, to that of the axle. In Fig. 166 we see that the weight, W, corresponds with the counteracting force, P, in an inverse ratio to the arms of the lever;

that is, inversely to the radii, a b and dc, of the wheel. Let us suppose that the radius, a b, of the axle is four times less than the radius, dc, of the wheel, we may equipoise a weight of eighty pounds by a force of twenty pounds.]

66

Sometimes the wheel is replaced by a winch, as in Fig. 167; it is then called a windlass," if the motion is vertical; but if it be horizontal, as in Fig. 168, the machine is called a "capstan," which differs from a windlass in having its revolving axis placed vertically. The circumference is pierced with holes, which receive long levers, called capstan-bars, by which it is worked by men, who walk round the capstan, and make it revolve by pressing the ends of the levers forward.

Fig. 166.

[The treadmill is another variety. In this case the weight of several people treading on the circumference of a long wheel causes it to revolve.