120

The Mechanical Powers.

from a depth. By a combination of solid parts, or a machine, these energies may be made to answer the purpose required. A small stream of water falling a great height may do the work of a large volume falling a less height; or a horse may be used for drawing water, and so perform each day the work of ten men.

But in no case is there anything more than a remoulding of the Energy supplied by nature, a mere translation of it from one form into another.

222. Machines which recast the simpler form of Energy, the motions of visible or molar masses-translating a long-continued motion of a small mass into a less motion of a larger mass, or the reverse-are usually called the Simple Machines, and sometimes the Mechanical Powers. The latter term, however, is based on a false and very misleading notion, namely, that these machines increase the quantity of force applied to them or are in themselves somehow a source of power.

223. It is seen, for instance, that one pound at the end of a beam

Fig. 31.

C

already explained in Section II.

just balances two pounds at b (fig. 31), half the distance on the other side of the axis, or four pounds, at c, a quarter of the distance; and many persons believe that the beam or lever itself begets a force equal to the difference of the weights so balanced. The explanation of the apparent paradox follows at once from

the notions of force and Work,

The same amount of force which gives a certain velocity to four pounds is just that required to give four times the velocity to one pound; and, owing to the connection of the two weights through the beam, no motion downwards by gravity can occur in the four pounds without causing a motion upwards just four times as great in the one pound. These two tendencies being equal and directly opposed to each other, must exactly balance, and no motion whatever of the beam will be produced.

224. To illustrate this further, suppose a weighing-beam, xy (fig. 32), with one pound hanging at the end, . Now if a spring, issuing from the fixed box at E with a force of one pound, be made to push

Machines merely modify Power.

Ꮳ

Fig. 32.

121

y

B

at the other end of the beam, y, it will just balance the weight; and if it be in the slightest degree stronger than the weight, it will push the end of the beam, y, down to B, say two inches, and will raise the weight to F. If, instead of this single spring, two similar springs be applied at half-way from the centre, so as to press at A, where there is just half the extent of motion, or room to act, as at B, exactly the same effect will follow. Now, because one spring at the end of the beam is seen here doing the same Work as two similar springs, or a single spring of double strength at the middle, it might at first appear that there was a saving of power by using the single spring and longer lever; but let it be observed, that the two middle springs have each issued from their box only one inch, while the single spring at the end has issued two inches: in both cases, therefore, exactly two inches of one-pound spring have been used.

Each atom of matter may be considered as held to the earth by its thread of attraction, and if one atom rise or fall ten inches, just as much of the supposed thread of attraction will be drawn out or returned as if ten atoms rise or fall one inch. And so, where a weight of one pound is made to do any Work, in place of a weight of two pounds, there is no more saving than in giving away two yards of single rope instead of one yard of double rope; and in like manner for all other differences of intensity.

225. If a man were to exert a force of one hundred pounds at A (fig. 32), in order to lift the weight, a boy at B, with a force of fifty pounds, might do just the same work; but the man would only have worked or pressed down through one foot, while the boy would have worked through two; and therefore, although the boy with the assistance of the lever, seemed to become as strong as the man, the case would merely be, again, that of the one-pound spring unbending two inches, to produce an effect equal to that of the twopound spring unbending one inch. The boy would be using two feet of his smaller force, where the man used one foot of his greater force; and if the work had to be long continued, the boy would have completely exhausted himself when the man remained yet fresh; and there would be no economy in employing the boy's services instead of the man's.

[22

A Perpetual Motion Impossible.

B

226. A case of the lever, exhibited in fig. 33, serves well to explain the general principle of the so-called mechanical powers. Suppose A B a bar, with the arm, c B, four times as long as the arm, CA, but the two arms equipoised so as not to disturb the action of weights subsequently attached to them. Then one pound at the end, B, would just balance four pounds at the end, A. Let us suppose also the arc, B b, to have been fixed to the long arm of the

a

Fig. 33.

bar or lever with the four shelves here shown, on which balls of one pound each might rest. If one of the four balls from the plane, d, were to roll upon the first shelf, it would just balance A, and, with one grain more, would descend to the level of the plane e, one inch below, and then roll off; while a second ball of one pound would occupy the second shelf, and would descend in the same way, to be followed by a third, and afterwards by a fourth. Now, when the whole four had fallen from d to e, they would just have lifted the four-pound mass, at the other end of the lever, one inch. So that, although one pound were seen here lifting a weight of four pounds, it would only have lifted that one-fourth part as far as it fell itself; and the whole resolves itself into an exchange of four pounds falling one inch at the long end of the lever for four pounds rising through the same distance of one inch at the short end.

No mechanical power or machine generates force more than is done in this case.

227. What an infinity of vain schemes-and some of them displaying great ingenuity-for perpetual motion, and new mechanical engines of power, would never have been attempted had the great truth been generally understood, that no form or combination of machinery ever did or ever can increase, in the slightest degree, the amount of power applied! Ignorance of this is the hinge on which most of the dreams of mechanical projectors have turned. The delusion of a perpetual motion which even men of talent have often fallen under, owing to their imperfect knowledge of this branch of natural philosophy is a remarkable phenomenon in human

nature.

The Lever.

228. The mechanical powers, usually enumerated as the lever, the pulley, the inclined plane, the wheel and axle, the wedge, and the screw, will now be considered in detail.

"Lever."

123

229. A beam or rod of any kind, resting at one part on a prop ot axle as a centre of motion, is a lever; and it has been so called, probably because such a contrivance was first employed for lifting weights (levo, to lift, in Latin).

Fig. 34 represents a lever employed to move a heavy block : a is the end to which the power or force is applied, ƒ is the prop or fulcrum, and the mass, b, is the weight or resistance.

According to the rule already given and explained at page 122, the power may be as much less intense than the resistance, as it is farther from the fulcrum.

b

Fig. 34.

A man at a, therefore, twice as far from the prop as the centre of gravity of the stone b is, will be able to lift a stone twice as heavy as himself; but he will lift it only one inch for every two inches that he descends and two men would be required, acting at half the distance, to do the same work.

There is no limit to the difference, as to intensity, of forces which may be made to balance each other by the lever, except the length and strength of the material of which levers may be formed. Archimedes said, "Give me a lever long enough, and a prop strong enough, and with my own weight I will lift the world." But it is a matter of simple arithmetic to show that he would have required to move with the velocity of a cannon-ball for millions of years, to alter the position of the earth by a small part of an inch.

230. To calculate the effect of a lever in practice, we must always take into account the weight of the lever itself, and the fact of its bending more or less; but, theoretically, it is usual to consider, first, what would be the result, if the lever were a rod without weight and without flexibility.

The rule, that the opposing forces, to balance each other, must be greater or less, exactly as they act nearer to or farther from the centre, holds in all cases, whether the forces be on different sides of the prop or both on the same side, and whether the force nearest to the prop have the office of power or of resistance; it holds also, whether the lever be straight or crooked, provided we reckon the

124

The Three Kinds of Lever.

distances from the fulcrum along the perpendiculars from that point on the lines of action of the balancing forces.

231. The lever is commonly described as of the first kind, if the fulcrum be placed between the power and the weight or resistance (P. F. W.); of the second kind, if the fulcrum be beyond the weight, so that the power and weight are on one side, the weight nearest the fulcrum (P. W. F.); and of the third kind if the power be nearer the fulcrum than the weight (W. P. F.).

232. The following are examples of the first kind of levers, i.e., with the prop between the forces.

The handspike, represented in page 123, is a lever moving a block of stone. The same form when made of iron, with the extremity formed into claws, is called a crow-bar. These are used generally for lifting and moving heavy masses through small spaces, as the materials of the mason, the shipbuilder, the warehouseman, &c. A short crow-bar is the instrument used by housebreakers for wrenching open locks or bolts, tearing off hinges, &c.

ance.

The common claw-hammer, for drawing nails, is another example. A boy who cannot exert a direct force of fifty pounds, may yet, by means of this kind of hammer, extract a nail to which half a ton might be quietly suspended without drawing it, because his hand moves through perhaps ten inches, to make the nail move a small part of an inch. The claw-hammer also proves that it is of no consequence whether the lever be straight or crooked, provided it produces the required difference of velocity between power and resistThe part of the hammer resting on the plank is the fulcrum. In the pincers or forceps we have a double lever, of which the hinge is the common prop or fulcrum. In drawing a nail with steel forceps or nippers, we have a good example of the advantages of using a tool: 1. The nail is seized by the steel teeth instead of by the soft fingers: 2. Instead of the griping force of the extreme fingers only, there is the force of the whole hand conveyed through the handles of the nippers: 3. The force is rendered many times more effective by the lever-length of the handles and 4. By making the nippers, in drawing the nail, rest on one shoulder as a fulcrum, it acquires all the advantages of the lever or claw-hammer for the same purpose.

:

Common scissors are double levers, as are also those stronger shears with which, under the power of a steam-engine, bars and plates of iron are now cut as easily as paper is cut by the force of the hand.