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or to their radii, which are their respective distances from it. Now their weights being multiplied into their velocities or distances from the common center producing equal momenta or products, the force of one body to cause the whole system to descend is exactly balanced by the other; so that the system is at rest, and the said point is the center of gravity.

This is the fundamental proposition of mechanics; and enables us to compute the force of any machine, whether simple or compound. You may find this point by the following

RULE.

By the sum of the weights, divide the product of the distance and one of the weights, and the quotient will be the distance of the other weight from the center of gravity.

OF THE MECHANIC POWERS.

THERE are but five simple mechanic powers; and of these all the machines in the world are compounded. The lever, the pulley, the wheel and axle, the inclined plane or wedge, and the screw.

They are called mechanic powers, because by the use of them our power over matter is increased; and by a small exertion of our natural strength we are thereby able to move large quantities of matter. But since the momentum of matter arises from its vis inertiæ or quantity, and its velocity of motion; and the first of these being unalterable, and the other variable, any contrivance, to diminish the velocity of its motion, will give us an advantage over its weight; but from what we have already said, we see that there is no other means to accomplish this, than by increasing the velocity of our own exertions exactly in the same

proportion that we diminish that of matter to be moved. For the products of the weight and velocity of matter, and of our strength and velocity of exertion must still be equal, when there is an equilibrium between them. Now the human force, being generally less than the quantity of matter we wish to move, may be represented by a small weight, which we are able to raise without the intervention of any machine, and is therefore by the writers on this subject, usually called the power; while the larger quantity of matter to be moved is denominated the weight. The power and the weight, thus understood, are always applied to the same machine, at different places; and the point on which the machine turns, which we have hitherto described under the name of the center of gravity, is commonly called the fulcrum; because by the distances from this point the velocities of the power and weight are respectively measured and determined.

THE LEVER.

The lever, the most simple of all the mechanic powers, and that to which all the rest may be referred, is an inflexible rod, supposed, in theory, to be without weight, and supported on an immovable point, called the fulcrum; while the power is applied at one end, and the weight to be moved at the other. Now, in order to increase the velocity of the power above that of the weight, it must be applied to that end which is most distant from the fulcrum, that it may pass over a larger space than the weight in the same time. As there will be an equilibrium between the power and weight, when their momenta are equal; it follows, by turning this equation into a proportion, that the power will be to the weight, inversely as their

velocities or distances from the fulcrum, or perpendicular ascent and descent in the same time: and this proposition holds true in all the mechanic powers.

The hammer, for drawing nails out of timber, is no more than a bended lever, where the hand is the power, the resistance of the wood is the weight, and the point on which the hammer rests in drawing the nail, is the fulcrum. Levers are of four different kinds: 1. Where the power and weight are applied to the ends; 2. Where the fulcrum is at one end, the power at the other, and the weight between them; 3. Where the fulcrum is at one end, the weight at the other and the power between them; 4. When the fulcrum is placed at equal distances between the power and weight, and then it is called the balance, whose use is not to raise a weight, but to compare one weight with another, called a standard. If the arms of the balance be unequal in length, and this inequality be compensated by the greater weight of the shorter arm, or by the difference of weight in the scales, so that the beam may hang in equilibrio with the scales, it will be deceitful; and this deceit may be detected by changing the scales, or the articles put into them, from one scale to the other.

When the arms are nicely equated in length and weight, and divided into equal parts, it becomes the proportional balance, to be used without scales: and when the products of the different weights into their respective velocities, added together on one side, are equal to the sum of the like products on the other, there will be an equilibrium formed. The steelyard is a lever, which shows the comparative weight of bodies by a common weight to be moved to different distances from the fulcrum.

The effect of the compound lever is computed, in the same way, as that of the simple lever.

THE PULLEY.

The pulley is a small wheel, movable round a fixed center, while a rope passes round a groove on its circumference, to one end of which the weight is fixed, and the power to the other. If the pulley be single, and fixed to a beam above, it contributes nothing to the raising of the weight, for the power and weight must be equal, to hang in equilibrio; since they have equal velocities as they pass over equal spaces in equal times. This pulley only changes the direction of the power.

But if one end of the rope were fixed to a beam above, and from thence passed round the pulley below, and the weight fastened to the pulley, the beam would sustain half of the weight, and the power would sustain the other half. For while the weight is raised one foot, each rope is shortened one foot, and the power of consequence passed over two feet. Therefore to constitute an equilibrium, the power being to the weight as the velocity of the weight to the velocity of the power, one will be double to the other. If the rope therefore instead of being held in the hand above, pass over a pulley fixed to the beam at top, and from thence pass round a second pulley fixed in the lower block, to which the weight is fastened; this second pulley would give the same advantage, with the first, in raising the weight. Now, while the weight rises one foot, each of the four ropes that go round the two pulleys of the lower block will be shortened one foot, and the hand or power will pass over four feet, and have four times the velocity of the

weight; and therefore need be no more than one fourth of the weight to stand in equilibrio with it. So that, let there be ever so many pulleys in the lower block, each of them affording an advantage equal to the first, in raising the weight, the velocity of the power will be increased beyond that of the weight, in the proportion of unity to the number of ropes that pass round the lower block. There will then be an equilibrium, when the power is to the weight, inversely as their velocities, or, as unity is to the number of ropes going to and from the lower block.

There are various other combinations of pulleys, which are easily understood from what has been said. For in all of them, the power is to the weight inversely as their velocities.

THE WHEEL AND AXLE.

The axle and wheel is another simple mechanic power, for raising a weight fixed by a cord to the axis, while the power is applied to the circumference of the wheel by another cord wound about the wheel in the contrary way. Now that the power and weight may be in equilibrio, the power must be to the weight, as the circumference of the axis to the circumference of the wheel, or as their diameters, because these are as their velocities. For while the power is passing over a space equal to the circumference of the wheel, the weight has passed over a space equal to the circumference of the axis, by the cord winding once round it. Now, these spaces being as the velocities of the power and of the weight, it is evident, that the same proportion holds good in this machine, as in the lever and pulley; viz. The power is to the weight inversely as their velocities. The wheel and axis is

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