power will depend on the weight of the body and its velocity; consequently, the power of any body in motion to overcome any resisting force or obstacle, is measured by the product of its weight into its velocity. This product is called its momentum. If two bodies move with equal velocities, their momenta will be as their quantities of matter, that is, as their weights; and if their weights are the same, then their momenta will be as their velocities; but if their weights and velocities both differ, their momenta will be in the ratio of the products of their weights multiplied by their velocities. For example, the momentum of a cannon-ball weighing 24 pounds, and moving with a velocity of 40 feet in a second, would be 24 times 40, or 960; and the momentum of another ball weighing 18 pounds, and moving with a velocity of 120 feet per second, would equal 120 times 18, or 2,160; consequently, the effects produced by these balls are in the ratio of 960 to 2,160, or as 4 to 9. From the above data, it is evident that the power requisite to impart to a body a certain velocity must be in proportion to the momentum, that is, to the product of the mass of the body multiplied by the required velocity. Thus, a power equal to 2,000 pounds is required to impart to a mass of 400 pounds the velocity of 5, because 400 times 5 equals 2,000; that is, the power which would impart to a mass of 1 pound a velocity of 2,000, will impart to a body weighing 400 pounds a velocity of only 5. If the moving force continues to act upon a body uniformly after it is set in motion, the velocity of the body will continue to increase, as illustrated in the case of falling bodies. See ¶ 72. COMPOUND MOTION is that produced by the action of two forces on a body at the same time. If two equal forces act upon a body in opposite directions at the same time, it will remain at rest; but if they act upon it at right angles to each other, the body will describe the diagonal of a square, whose sides will represent the spaces through which the body would have passed in the same time, following the impulse of each of these forces. If the two forces which act upon a body at right angles to each other be unequal, the body will describe the diagonal of a rectangle, whose sides will represent the rela tive power of the two forces; and if the directions of the two forces be not at right angles, then the body will describe the diagonal of a rhombus, or the diagonal of a rhomboid, according as the forces are equal or unequal; and the sides of the figure will be proportional to the spaces through which the body would have passed in the same time, following the impulse of each of the forces. For example, if two balls of the same weight, and moving with the same velocity, one moving north and the other east, strike a third ball at the same instant, it will move directly northeast; and if its weight be the same as that of each of the other two, its velocity will be equal to the square root of twice the velocity of one of the balls which generated its motion; but if one of the balls which impinged against the third possessed twice the momentum of the other, then the third would describe the diagonal of a parallelogram or rectangle, whose length would be equal to twice its breadth. CURVILINEAR MOTION is that produced by the continued action of two separate forces on a body in motion. This kind of motion is illustrated by the motions of the planets, which describe ellipses round the sun. The sun's attraction is called the centripetal force, and the tendency which a planet has to move forward in a straight line, is called the centrifugal or tangential force; and both forces together are called CENTRAL FORCES. The centrifugal force is measured by the product of the weight of the body into its velocity, and the centripetal by the force of attraction towards the point around which the body revolves; the former will therefore vary as the velocity, and the latter will increase or diminish inversely as the square of the distance from the centre of attraction: that is, if the revolving body be 10 times nearer the centre of attraction, the force of attraction will be 100 times greater; or if it be removed 5 times further from the centre of attraction, the centripetal force will be 25 times less. This law applies to both magnetic and electrical attraction. The motion of a body on an inclined plane is analogous to that of a falling body, the law being the same, and the only difference being in the degree of velocity, which is as much less as the length of the plane is greater than its perpendicular height. If, for example, a heavy body rolls down an inclined plane, the velocity it acquires in any given time, is to the velocity acquired by a body falling perpendicularly in the same time, as the height of the plane is to its length; and the space described by the body on the plane, is to the space described in the same time by a falling body, as the height of the plane to its length; and the time of a body's descending through the plane, is to the time of falling through the height of the plane, as the length of the plane is to its height; and the velocity acquired by a body in rolling down a plane, is the same as that which it would acquire in falling perpendicularly through its height. A curved surface may be regarded as composed of an infinite number of inclined planes; and consequently, a body in descending along any curved surface will acquire the same velocity as it would in falling perpendicularly through the same height. A body will descend through any chord of a circle in the same time that it would fall perpendicularly through the whole diameter. Thus the times of descending through all the chords of a circle, as EB, DB, CB, and through the diameter AB, are equal; and the velocities of the bodies, which descend the several chords, when they arrive at B, will be equal to that which they would have severally acquired in falling through the versed sines of the arcs cut off by the chords. Thus the velocity of the body descending the chord EB, at B, will be the same as that acquired by a body in falling from c to B. In the above statements and proportions it is supposed, of course, that there is no resistance from friction. D E A B b a T79. SIMPLE MECHANICAL POWERS. The simple machines are usually reckoned six in number; the lever, the wheel and axle, the inclined plane, the wedge, the screw, and the pulley. The lever and the inclined plane are sometimes called prime movers. The wheel and axle is only a modification of the lever; and the wedge and screw are only different applications of the inclined plane. The LEVER is an inflexible rod moveable about a fulcrum or prop, and having forces applied to two or more points in it. It may be regarded as the simplest of the mechanical powers, though not the least useful. The force applied to the lever, for the purpose of moving some object, is called the power, and the object to be moved is called the weight. The parts of the lever between the fulcrum and the power, and the fulcrum and the weight, are called the arms of the lever. Thus, (in Fig. 1,) FW and FP are the arms of the lever; W is the weight; F is the fulcrum; and P is the power. Levers are usually divided into three kinds, according to the relative situations of the power, the weight, and the fulcrum. The first kind has the fulcrum between the power and the weight, (Fig. 1.) The second has the weight between the fulcrum and the power, (Fig. 2.) The third has the power between the fulcrum and the weight. The general principle of the lever is this: When the power and weight are in equilibrium, the power is to the weight inversely as their distances from the fulcrum; that is, the power required to balance a given weight may be as many times less than the weight, as the distance between the weight and the fulcrum is less than the distance between the power and the fulcrum. Suppose PF (Figs. 1 and 2) equals 2 feet, and WF equals 1 foot; then a power of 1 pound acting at P will overcome a resistance, or balance a weight of 2 pounds at W; and if PF is 4 feet and FW 1 foot, then 1 pound at P will overcome a resistance of 4 pounds at W; and if PF is 30 feet, and FW 3 feet, then (because PF is 10 times greater than FW) 1 pound at P will bal ance 10 pounds at W. When the power and the weight are in equilibrium, the product of the weight into its distance from the fulcrum is equal to the product of the power into its distance from the fulcrum; and if several powers and weights are made to act upon the same lever, they will be in equilibrium, when the sum of the products of the several weights into their distances from the fulcrum, equals the sum of the products of the several powers into their distances from the fulcrum. In all cases, the weight divided by the power will show the relative length of the arms of the lever. Thus, suppose that it is required to raise, by means of a lever, a weight of 2,000 pounds with a power of 100 pounds; then, since 100 is contained in 2,000 twenty times, the relative length of the arms of the lever must be as 1 to 20; and if the lever used be 10 feet in length, then the shorter arm will be 5.7413 inches, and the longer arm will be of 10 feet, or of 10 feet, or |