130

Examples of the Wheel and Axle.

dicated, and are usually expressed, by comparison of the diameters of the wheel and the axle. Instead of the wheel, d, here shown, the handle or winch, c, may be substituted as explained in Art. 241. Fig. 40 is an end view of the same object. It explains why the wheel with its axle has been called a perpetual lever; for the two weights hanging in opposition, on the wheel at a, and on the axle at b, are always as if they were connected by a horizontal lever, a cò, whose arms are the half-diameters of the wheel and the axle, and fulcrum their centre, c; and while a simple lever could only lift through a small space, it is evident that this construction will lift as long as there is rope to be wound up.

11

Fig. 40. A common crane for raising weights, consists of an axle to wind up or receive the rope which lifts the weight, and of a winch or a large wheel, at the circumference of which the power is applied. The power may be animal effort exerted on the rim or outside of the wheel, or the Energy supplied by a steam engine. 241. The capstan, used on board of ships, is merely a large upright axle or spindle, b, which by turning pulls the cable or rope, abc (fig. 41). It is moved by the men pushing at the capstan-bars, d, e, f, which for the time are placed in holes in the broader part, i, or drum of the capstan, usually appearing above the deck, at the top of the spindle. These bars may be considered as the spokes of a large wheel, and the effect produced by a man working at one of them, is in proportion to his distance from the centre. The capstan is chiefly used on board ships for lifting the anchor, and for doing any other very heavy work. It is applied also to various purposes on shore.

Fig. 41.

The common winch with which a grindstone is turned, or a crane worked, or a watch wound up, is really in principle a wheel: for the hand of the worker describes a circle, and there is no difference in the result whether an entire wheel be turning with the hand or only a single bent spoke of a wheel.

242. The fusee in a watch is a beautiful illustration of the principle of the wheel and axle. The spring of a watch, immediately after winding up, being more strained, is acting more powerfully to

The Fusee.-Wheels and Pinions.

a

131

drive on the wheels than afterwards when slacker, and it would destroy the wished-for uniformity in the motion of the time-piece if there were no means of equalising its action. The fusee (fig. 42) is this means. It is a barrel or spindle, tapering from its large end, b, to its smallend, a, with a spiral groove cut in the surface to receive the chain, by pulling at which, the spring in the box,

, moves the watch.

Fig. 42.

Now when the watch has been wound up, by a key applied on the axle of the fusee near a, the fusee is covered with the chain up to the small end, and the newly bent and strong spring begins to pull by this small end or short lever; and afterwards, exactly as the spring becomes relaxed and weaker, it is pulling at a larger and larger part of the fusee-barrel, and so keeps up an equal or uniform effect on the general movement.

In place of a common cylindrical axle, a large fusee is often used with a winch, for drawing water by bucket and rope from very deep wells. When the bucket is near the bottom of the well, and the labourer has to overcome the weight of the long rope, in addition to that of the bucket and water, he does so more easily by beginning to wind the rope on a small axle, that is to say, on the small end of the fusee; and in proportion as the length of rope diminishes, he lifts by a larger axle.

The same thing happens, in principle, when the rope, by coiling on itself, increases gradually the diameter of the axle

243. By means of a wheel, which is very large in proportion to its axle, forces of very unequal intensities may be balanced, but the machine becomes of inconvenient proportions. It is found preferable, therefore, when such an end is desired, to use a combination of wheels of moderate size. In

g

h

α

Fig. 43.

the adjoining cut (fig. 43) three wheels are thus connected. The

132

Wheels.--Cranes.-Bands.

teeth on the axle, d, of the first wheel, c, acting on six times the number of teeth in the circumference of the second wheel, g, turn it only once for every six times that c turns; and in the same manner the second wheel, by turning six times, turns the third wheel, h, once; the first wheel, therefore, turns thirty-six times for one turn of the last; and as the diameter of the wheel, c, to which the power is applied, is three times as great as that of the axle, f, which bears the resistance: 3 times 36, or 108, is the proportion of velocity, and therefore of intensity, between weights or forces that will balance here. An axle with teeth upon it, as d or e, is called a pinion.

On the principle of combined wheels, cranes are made, with which one man, by working a long time, can lift many tons. It is even possible to make an engine, by means of which a tiny windmill, of a few inches in diameter, should eventually tear up a strong oak by the roots.

The most familiar instances of wheel-work are in our clocks and watches. A few turns of the axle on which the watch-key is fixed, are rendered equivalent, by the train of wheels, to more than ninety thousand beats of the balance-wheel; and thus the exertion during a few seconds, of the hand which winds up, gives steady motion for more than twenty-four hours. By increasing the number of wheels, a time-piece might be made to go for years.

Wheels may be connected by bands as well as by teeth. This is

a

Fig. 44.

seen in the common spinning-wheel, turning lathes, grind-stones, &c. A spinning-wheel, as a c (fig. 44), of twenty inches in diameter, turns by its band a bobbin or spindle of half an inch diameter, b, forty times for every turn of itself (or nearly

so, for there is always more or less slipping of the band to be allowed for).

244. The inclined plane is the third means of balancing, by solid media, forces of different intensities.

When a force is applied to move a weight from c to d (fig. 45), by acting along the whole length of the plane cd, it has to raise it through only the perpendicular height e ✅; and if the plane be twice as long

The Inclined Plane.

133

as it is high, one pound at b, acting over the pulley d, would balance two pounds at a, or anywhere on

a

Fig. 45

A horse drawing on a road where there is a rise of one foot in twenty, is thus really lifting onetwentieth of the load, as well as overcoming the friction and other resistance of the carriage. Hence the importance of making roads as level as possible; and our forefathers often erred in carrying their roads directly over hills, for the sake of straightness considered vertically, where by going round the bases of the hills they would scarcely have had greater distance, and would have avoided all rising and falling. A road up a very steep hill is usually made to wind or zig-zag all the way; the ease to the horses being greater exactly as the road is made longer. The fatigue of ascending a high column is lessened by making the ascent an inclined plane winding round and round, and lessened just in proportion as the route is made longer than the height of the column.

245. Railways offer a beautiful illustration. When the line is perfectly level, the steam-engine which draws the train has just to overcome the friction of the carriages and the resistance of the air; but if there is a rise of 1 foot in 40, or I in 60, or I in 120, the engine has then the additional work to do of lifting vertically upth, orth, orth of the weight of the whole train.

A hogshead of merchandize, which ten men could not lift directly, may be rolled into or out of a waggon by one or two men, who have the assistance of two connected beams forming an inclined plane. There are some canals where, in particular situations, it is found convenient to have the loaded boats drawn up by machinery on inclined planes, instead of being raised by water in locks, as elsewhere.

It is supposed that the ancient Egyptians must have used the inclined plane to enable them to put in position those immense blocks of stone, which still remain marvels of their gigantic archi

tecture.

Our common stairs are inclined planes; but, being so very steep, they require to be notched into steps, that they may afford a firm footing.

We may here recall that a body falling freely, in obedience to gravity, descends sixteen feet and a fraction in the first second (Art. 138), and that if made to roll down an inclined plane, it moves just as much less quickly (besides the loss from friction and rotation) as the height of the plane is less than the length. On a plane sloping one foot in sixteen of its length, a body would descend only one foot in the first second.

The descent of a pendulum in its arc is completely explained by the laws of the inclined plane. And the laws of the inclined plane itself depend on those of falling bodies and of the resolution of forces already explained.

a

୯

d

246. The wedge

is merely an inclined plane forced in between resistances to separate or overcome them, instead of, as in the last case, being stationary, while the resistance is forced along its surface. Theoretically, a pressure acting through the distance, c d (fig. 46), or the length of the wedge, is converted into a more intense pressure acting through the shorter distance, c b, or half the breadth of the wedge. But practically the rule is of little use in this case; because mere pressure is never employed as the force, but percussion or the blow of a hammer, which renders the estimation of the effect of this machine very complicated.

Fig. 46.

Its force-transforming power is surprisingly great. The wedge is used for many purposes; as for splitting blocks of stone and wood; for squeezing strongly, as in the oil-press; for lifting great weights, as when a ship in dock may be raised a little by wedges driven under the keel, &c.

An engineer, who had built a very lofty and capacious chimney, found after a time that, owing to a defect in the foundation, it was beginning to incline. He then, by driving wedges under one side of it, restored it perfectly to the vertical position.

247. Nails, awls, needles, &c., are examples of the wedge; as also all our cutting instruments-knives, razors, the axe, the chisel, the plane, &c., and attention to the principles of the wedge guides the mechanic in giving the proper shape to the cutting edge. These tools are often used somewhat in the manner of a saw by pulling them lengthwise at the same time that they are pressed directly against the object to be cut. The edge of a knife, when viewed through a