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that the motion, pro-A
E duced by two forces acting together, is not so great as that which is produced by the separate action of each; for the diagonal AF is obvious
F ly not equal to AEC and EF added together.
6. Motion in a curve line is always produced by the action of at least two forces. When a stone is projected from the earth, in any other than a perpendicular direction, it moves in a curve line of a particular description called a parabola. You will naturally ask, what two forces operate in this case? In truth, the stone is acted upon by no fewer than three forces at the same time; namely, the projecting force that set it in motion; the force of gravity that brings it to the ground; and the resistance of the air that impedes its course. The resistance of the air, however, has no effect upon
the direction, but only upon the velocity of the body; which moves forward, accordingly, in a direction compounded of that given it by the projecting force, and of that which it receives from gravity. The motion, in this instance, is not (as in those formerly mentioned) in a straight diagonal, because the velocity, which is communicated to the stone by the force of gravity, is not uniform but accelerated; in consequence of which, it falls sooner to the ground than if it had only been acted upon by gravitation once, at the commencement of its course. Circular motion is the result of two forces acting upon one body at the same time, by one of which it is im. pelled in a straight line, and by the other, is drawn to a fixed point. Thus, when you whirl a stone round in a sling, it is acted upon both by that force, by which, if it were set free, it would fly off in a straight line, and by that which confines it to your hand, and prevents its escape. The former of these is called the centrifugal force, and the latter the centripetal force. The wings of a windmill, for example, would be driven forward by the wind in a straight line, were they not fixed to a
centre, and so compelled to move in a circle. It is, in like manner, by means of the combined action of a centripetal and centrifugal force, that this earth and the other planets are retained in their orbits. By the force of gravity they are attracted towards the sun: this is their centripetal force. By another force, which was given them at the time of their creation, they are all of them propelled in a straight line; this is their centrifugal force. Their motion, it is true, is not strictly circular but elliptical, arising from other circumstances, which astronomers have now no difficulty in explaining. The point, round which any body moves, is called its centre of motion. Thus the sun is called the centre of revolution of the planets. When a body spins round like a top, the line real or imaginary, about which it revolves (which line is not always in the cen. tre of the body) is called its axis of motion. Thus, the imaginary line about which the earth performs its diure nal revolution, which occasions the succession of day and night, is called its axis.-If two or more bodies move quite round the same centre, at different distances, within the same time, that which is most remote from the centre moves with the greatest velocity, be. cause it is carried round in a large circle, in the very same time, in which the others are carried round only in smaller circles. For the same reason, when a body revolves round its own axis, in proportion to the distance of any part of the body from the axis, the greater is the velocity of that part.
You have often seen machines called round-abouts, on which children are carried round at fairs ; in these (upon the principle which we are now explaining), the children who are placed in the outer seats, get a much longer ride, than those who are placed next the centre of motion. This is a principle, which it will be necessary for you to keep carefully in mind, as it is a fundamental one in the construction of machinery.
In entering upon the consideration of the mechanical powers, it will be necessary for you to keep in recollec
tion the leading principle formerly explained, that, by increasing the velocity of a lighter and naturally weaker body, we may render its momentum much greater than that of a heavier and stronger one. It is a leading object in machinery, to produceas great adisproportion as possible between the velocity of the moving force, and that of the weight to bemoved, and thus to compensate for the want of strength in the former; so as to enable it to accomplish what either could not otherwise have been done at all, or at least without the greatest difficulty. There are six mechanical powers, one or more of which enter into the composition of every
machine,-the LEVER, WHEEL AND AXLE, PULLEY, INCLINED PLANE, WEDGE, AND SCREW.-1. Of these the LEVER is the most simple. It is an inflexible bar of iron, or the like, which, by moving upon a prop or fulcrum (as it is called), is of use in raising weights to a small height. Its operation depends upon the principle formerly explained, that where two bodies perform complete revolutions round the same centre, within the same time, that which is more remote from the centre, has proportionally greater velocity than the other. The Lever is of three kinds. The 1st kind is that in which the fulcrum is placed between the weight and the power. It is often used by workmen in the removal of heavy pieces of timber. For this purpose, they force one end of the bar beneath the timber, and resting it upon a block of wood or stone as a fulcrum, they apply their whole strength to the further extremity of it, by which the timber is at length removed. The manner in which this lever operates you will easily understand. Let AB represent a lever of this class, moving upon its fulcrum F, and having its arm AF, to which the force is to be applied, four times as
D great as its other arm FB,
А to which the weight is sus
B pended. Then, because the point A is four times as much removed from F, the centre of motion, as the point B, it must have four times its velocity; and accordingly it actually does pass
through the larger space AC, in the very same time that B passes through BD, which is only the fourth part of AC; it therefore follows, from what you
have Jearned of momentum, that a weight of one pound will have as much force at the point A, as a weight of four pounds would have at B; or, which is the same thing, the strength exerted by a man or a horse at A will have four times the force that the same strength would have at B. It is clear, accordingly, that, by means of this lever, a fourfold force has been acquired. The ordinary balance for weighing goods is generally accounted a lever of this first kind. It is very true that, where the fulcrum is placed in the middle of the beam (as is always the case in a true balance of that kind), no power is gained; because then AF would be equal to FB, and the point A would pass through a space AC just equal to BD, the space through which В moves; and, therefore, in that case the beam does not act as a mechanical force. But, if the fulcrum be placed in any other point of the beam between the two extremities, or if one of the weights be placed at any other point than the extremity, the beam then obviously acts as a lever of the first kind, and affords excellent illustrations of the mode of its operation. Thus, if the arms of a just balance of this kind be each of them divided into the same number of equal parts, an ounce applied to the ninth division from the fulcrum on one side will balance three ounces applied to the third division on the other; and two ounces at the sixth division will balance three at the fourth. In this way you see how a dishonest tradesman may cheat his employers, by using a balance with one arm longer than the other. From what has been said, you will also understand that there may be, and indeed are, two kinds of balances. In the one, various weights are employed ; in the other, all articles are weighed by the same weight, but placed at different distances from the fulcrum. Of this last kind is the steelyard used by butchers. This is a lever having two armsof very unequal lengths. At the extremity of the shorter arm is suspended the article to be weighed. The longer arm is divided into a number of parts, each of which is equal to the shorter arm. A pound weight, placed in the first division from the fulcrum of the longer arm, will balance an article of that weight suspended at the extremity of the other; the same weight placed at the 2d division will balance an article of 2 pounds weight; and when placed at the 10th division, for example, will balance an article of 10 pounds weight.-Levers of the first kind are in daily use for a variety of common purposes. Every poker is such a lever, of which the bar of the grate is the fulcrum, the hand is the power, and the coal the weight to be raised. Every pair of scissars, snuffers, pincers, &c., is composed of two levers of this kind acting against each other : in which you will accordingly observe that the longer the handles, and the shorter the points, the less exertion will be required in using them. The 2d kind of lever, is that in which the fulcrum is placed at one extremity, the power is applied at the other, and the weight to be raised is between the fulcrum and the power. In this lever, the power gained is just so much the greater, as the distance between the point, at which the power is applied, and the fulcrum, is greater than the distance between the point at which the weight is suspended, and the fulcrum. Thus let AF represent a lever of this kind, having its fulcrum at the extremity F, and a force applied at the other extremity A, for the purpose of raising a weight suspended at B between the other two points. Thus, because the point A is four times as much removed from F, the centre of motion, as the point B is, it has four times its velocity, and passes through the larger space AC, in the C very same time that B passes through the space BD, which is only a fourth