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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. The condition of equilibrium is, therefore, the same.

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

Fig. 160.

ascend, the vertical distances
passed over are equal.

P

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 applied 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.

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.

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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, cd, 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,

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 ou can be brought into action again; but the wheel and axle continue their operation constantly in the same direction.

[graphic]

Fig. 163.

[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 b that of the weight, W. Draw the line a cb; it evidently represents 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.

Fig. 165.

W

[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;

α

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 66 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.

The paddle-wheel of a steamboat acts on the same principle; the water, which offers a resistance to the motion of the paddle-boards, is the power.] Wheels and axles are often made to act upon one another by the aid of eogs, as in clockwork and mill machinery. In these cases the cogs on the periphery of the wheel take the name of teeth, those on the axle the name of leaves, and the axle itself is called a pinion.

The law of equilibrium of such machines may be easily demonstrated to be, that the power multiplied by the product of the number of teeth in all

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A

W

Fig. 167.

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the wheels, is equal to the weight multiplied by the product of the number of leaves in all the pinions.

A system of wheel and pinion work is represented at Fig. 169. It is scarcely necessary to observe, that in it, as in all other cases, the law of virtual velocities holds good-the power multiplied by the velocity of the power is equal to the weight multiplied by the velocity of the weight.

In the construction of such machinery attention has to be paid to the form of the teeth, so that they may not scrape or jolt upon one another. Several of them should be in contact at once, to diminish the risk of fracture and the wear.

If the teeth of a wheel be in the direction of radii from its centre it is called a spur-wheel.

If the teeth are parallel to the axis of the wheel it is called a crown-wheel. If the teeth are oblique to the axis of the wheel it is called a bevelledwheel.

By combining these different forms of wheel suitably together, the resulting motion can be transferred to any required plane. Thus by a pair of bevelled-wheels motion round a vertical axis may be transferred to a horizontal one, or, indeed, one in any other direction.

When a pinion is made to work on a toothed bar, it constitutes a rack. This contrivance is under the same law as the wheel and axle.

CHAPTER XXIX.

THE INCLINED PLANE-THE WEDGE-THE SCREW.

Description of the Inclined Plane-Modes of applying the Power-Conditions of Equilibrium when the Power is Parallel to the Plane or Parallel to the Base-Position of Greatest Advantage-Description and Mode of using the Wedge-Formation of the Screw.

By the inclined plane we mean the unyielding plane surface inclined obliquely to the resistance to be overcome.

In Fig. 170, A C represents the inclined plane; the angle at A is the elevation of the plane; the line A C is the length, CB is the height, A B the base.

B

In the inclined plane the power may be applied in the following directions:-1. Parallel to the plane; 2. Parallel to its base; 3. Parallel to neither of these lines.

As in the former cases, so in this-the conditions of the equilibrium may be deduced from those of

the lever.

Fig. 170. Let us take the first instance, when the power is applied parallel to the inclined plane. Let Q, Fig. 170, be a body placed upon the plane, A C, the height of which is B C, and the base AB. The weight of this body acts in the vertical direction, a W; the body rests on the point, c, as on a fulcrum; and the power, P, under the supposition, acts on Q, in the direction a P. From the fulcrum, c, draw the perpendicular, cb, to the line of direction of the weight, a W; draw also ca. Then does be a repre

sent a bent lever, the power being applied to the point, a, and the weight at the point, b; and, therefore, the power is to the weight as b c is to ac; but the triangles, a bc, A B C, are similar to each other. Therefore we arrive at the following law :

When the power acts in a direction parallel to the inclined plane, it will be in equilibrio with the weight when it is to the weight as the perpendicular of the plane is to its length.

In a similar manner it may be shown that when the power acts parallel to the base it will be in equilibrio with the weight, if it be to the weight as the perpendicular of the plane is to its base.

In different inclined planes the power increases as the height of the plane, compared with its length, diminishes, and the best direction of action is parallel to the inclined plane. This is very evident from the consideration that, if the power be directed above the plane, a portion of it is expended in lifting the weight off the plane, while the diminished residue draws it up. If it be directed downward a part is expended in pressing the weight upon the plane, and the diminished residue draws it up. Therefore, if the power acts parallel to the plane, it operates under the most advantageous condition. The laws of the inclined plane may be illustrated by an instrument such as is represented in Fig. 171, in which A c Ac' is the plane, which may be set at any angle. It works upon an axis, A A'. Upon the plane a roller, e, moves. It has a string passing over a pulley, d, and terminating in a scale-pan, f, in which weights may be placed. The direction of the string may be varied, so as to be parallel to the plane, or the base, or any other direction.

A

Fig. 171.

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The inclined plane is used for a variety of purposes-very frequently for facilitating the movements of heavy loads

THE WEDGE.

The wedge may be regarded as two inclined planes laid base to base

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