Part 299

One of these forces is usually produced by the re-action of a fixed body, called the fulcrum.

If two contrary forces be applied to a lever at unequal distances from the fulcrum, they will equiponderate when the forces are to each other in the reciprocal proportion of their distances. For, by the resolution of force it appears, that if two contrary forces be applied to a straight lever, at distances from the fulcrum in the reciprocal proportion of their quantities, and in directions always parallel to each other, the lever will remain at rest in any position.

Since of the three forces which act on the lever, the two which are applied at the extremes, are always in a contrary direction to that which is applied in the space between them: this last force will sustain the effects of the other two; or, in other words, if the fulcrum be placed between the weights, it will be acted upon by their difference.

On the principle of the lever are made, scales for weighing different quantities of various kinds of things; the steelyard, which answers the same purpose by a single weight, removed to different distances from the fulcrum on a graduated arm, according as the body to be weighed is more or less in quantity; and the bent lever balance, which, by the revolution of a fixed weight, increasing in power as it ascends in the arc of a circle, indicates the weight of the counterpoise.

On this principle also, depend the motions of animals; the overcoming or lifting great weights by means of iron levers, called crows; the action of nutcrackers, pincers, and many other instruments of the same nature.

Of the Axis or Axle, and Wheel, and of the Pulley or Tackle.

The axis and wheel may be considered as a lever, one of the forces being applied at the circumference of the axis, and the other at the circumference of the wheel, the central line of the axis being as it were the fulcrum.

For if the semidiameter of the axis, be to the semidiameter of the wheel, reciprocally as the power of A is to the power B, the first of which is applied in the direction of a tangent of the axis, and the other in the direction of the tangent of the wheel, they will be in equilibrium.

To this power may be referred the capstan or crane, by which weights are raised; the winch and barrel, for drawing water, and numberless other machines on the same principle.

The pully is likewise explained on the same principle of the lever. Suppose the line A. C. to be a lever, whose arms A. B. and B. C. are equidistant from the fulcrum B. consequently the two equal powers E. and F. applied in the directions of the tangents to the circle in which the extremities are moveable, will be in equilibrium, and the fulcrum B. will sustain both forces.

But, suppose the fulcrum is at C. then a given force at E. will sustain in equilibrium a double force at F. for in that proportion reciprocally are their distances from the fulcrum. Whence it appears, that considering E. as a force, and F. as a weight to be raised, no increase of power is gained, when the pulley is fixed, but that a double increase of power is gained, when the pulley moves with the weight.

A combination of pullies is called a tackle, and a box containing one or more pullies, is called a block.

This is a tackle composed of four pullies, two of which are in the fixed block A. and the other two in the block B. that moves with the weight F. Now, because the rope is equally stretched throughout, each lower pulley will be acted upon by an equal part of the weight; and because in each pully that moves with the weight a double increase 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 number of lower pullies is to one, so is the weight suspending force.

But if the extremity of the rope C. be affixed to the lower block, it will sustain half as much as a pulley; consequently the analogy will then be, as twice the number of lower pullies, more 2 is to 1, so is the weight suspended to the suspending force.

The pulley or tackle is of such general utility, that it would seem unnecessary to point out any particular instance.

Of the inclined Plane, and of the Wedge.

The inclined plane has in its effects a near analogy to the lever; and the forces by which the same weight tends downwards in the directions of various planes, will be as the sines of their inclinations.

The wedge is composed of two inclined planes joined together at their common bases, in the direction of which the power is impressed.

This instrument is generally used in splitting wood, and was formerly applied in engines for stamping watch plates. The force impressed is commonly a blow, which is found to be much more effectual than a weight or pressure. This may be accounted for on the principles which obtain when resisting bodies are penetrated, as if the mass and velocity vary, the depths to which the impinging body penetrates will be in the compound ratio of the masses and the squares of the velocities.

All cutting instruments may be referred to the wedge. A chizel, or an axe, is a simple wedge; a saw is a number of chizels fixed in a line: a knife may be considered as a simple wedge, when employed in splitting; but if attention be paid to the edge, it is found to be a fine saw, as is evident from the much greater effect all knives produce by a drawing stroke, than what would have followed from a direct action of the edge.

Of the Screw, and of mechanical Engines, in general.

The screw is composed of two parts, one of which is called the screw, and consists of a spiral protuberance, called the thread, which is wound round a cylinder; and the other called the nut, is perforated to the dimensions of the cylinder, and in the internal cavity is cut a spiral groove adapted to receive the thread.

It would be difficult to enumerate the very many uses to which the screw is applied. It is extremely serviceable in compressing bodies together, as paper, linen, &c. It is the principal organ in all stamping instruments for striking coins, or making impressions on paper, linen, or cards, and is of vast utility to the philosopher, by affording an easy method of measuring or subdividing small spaces. A very ordinary screw will divide an inch into 5,000 parts; but the fine hardened steel screws, that are applied to astronomical instruments, will go much farther.

It is easy to conceive, that when forces applied to mechanical instruments are in equilibrium, if the least addition be made to one of them, it will preponderate and overcome the effort. But the want of a perfect polish or smoothness in the parts of all instruments, and the rigidity of all ropes, which increases with the tension, are great impediments to motion, and in compounded engines are found to diminish about one fourth of the effect of the power.

The properties of all the mechanical powers depending on the laws of motion, and the action or tendency to produce motion of each of the two forces, being applied in directions contrary to each other, the following general rule for finding the proportion of the forces in equilibrium on any machine will require no proof.

If two weights applied to the extremes of any mechanical engine, be to each other in the reciprocal proportion of the velocities resolved into a perpendicular direction, (rejecting the other part) which would be acquired by each when put in motion for the same indefinitely small time, they will be in equilibrio.

Whence it may be observed, that in all contrivances by which power is gained, a proportional loss is suffered in respect of time. If one man by means of a tackle, can raise as much weight, as ten men could by their unassisted strength, he will be ten times as long about it.

It is convenience alone, and not any actual increase of force, which we obtain from mechanics. As may be illustrated by the following example: