Here n 30, and the above expression for L becomes L Er. 2. What must be the length of a pendulum to vibrate 40 times in a minute? Here L= 391 × 602 88 inches, is the answer. 2. Having the length of the pendulum given, to find how many vibrations it will make in a minute. From the above proportion, viz. n2 : 602:: 394: L, we find 602 × 391 L Ex. 1. Let the number of vibrations which a pendulum 1561 inches long makes in a minute be required. 601×391 In this case L=1563, and n= =30, is the 1561 number of vibrations such a pendulum would make in a minute. What number of vibrations would a pendulum 88 inches long make in one minute? 3. The length of a pendulum vibrating seconds being known in any place, the space through which a body falls in a second by the force of gravity at that place may be found. For let S denote the space a body falls through in a second at the given place, then by art. 346, S: L:: 3,14159): 1; therefore S-LX3,14159)2. Thus, in the latitude of London, a pendulum vibrating seconds is known to be 39 inches; therefore, the space through which a heavy body descends from quiescence in the latitude of London may be easily determined; for in the above 391 formula, L is equal to -19%, and S=19% ×3,141592 193,07 inches, or 16 2 feet nearly is the space required. By the reverse of this method we may find the length of a second pendulum, having the space fallen through by a heavy body from quiescence in the first second of time: for 2S. since S-LX3,14159, we have L= EXAMPLES. 3,14159 Ex. 1. Find the length of a pendulum vibrating seconds, in the latitude of London, where a heavy body descends from rest through 16 feet in a second of time. Answ. 39,11 inches. Ex. 2. Find the length of a pendulum vibrating in half a second, also of one vibrating in a quarter of a second. It is plain that a pendulum vibrating in half a second makes 120 vibrations in a minute, and this number substituted for n, in the formula in art. 347, case 1, gives 9 inches, for the length of the half second pendulum. The length of the quarter second pendulum is found in the same manner to be 2 inches long. Er. 3. What difference will there be in the number of vibrations made by a pendulum 6 inches long, and one of 12 inches long, in an hour's time? Answ. 26921 Er. 4. A stone being dropped into a well, it was observed that a pendulum 18 inches long had made just 8 vibrations when the sound from the bottom returned. What was the depth of the well, supposing that sound moves uniformly at the rate of 1142 feet in one second of time? Answ. 412,61 feet, Ex. 5. If a ball vibrate in the arc of a circle 10° on each side the perpendicular, or a ball slide down the lowest 10 degrees of the arc; required the velocity at the lowest point, the radius of the circle, or length of the pendulum being 20 feet? Answ. 4,4213 feet per second. Er. 6. If a ball descend down a smooth inclined plane whose length is 100 feet, and altitude 10 feet; how long will it be in descending, and what will be the last velocity 2 Answ. The velocity 25,364 feet per second, and 7,8852" is the time. On the collision of Bodies. Defs. 348. The collision of two or more bodies, is the shock by which, when they come in contact, they alter each other's motion. The percussion of bodies is the same thing. The force of percussion, or collision, is the same as the momentum or quantity of motion, and is measured by the product arising from the mass moved, multiplied by the velocity; and that without any regard to the duration of the action. 349. Bodies are either hard, soft, or elastic. A body is said to be perfectly hard when its component particles cannot be separated, or moved among themselves, by any finite force. A soft body consists of particles which give way by the application of the least force, or impression, and do not restore themselves, but the form of the body remains altered. Bodies either perfectly hard, or perfectly soft, have, therefore no elasticity. An elastic body consists of particles which restore themselves in the same time, and with the same force, that was employed in compressing them. No bodies that we are acquainted with are either pefectly hard, soft or elastic; but all bodies whatever contain these properties in a greater or less degree. In steel balls, the elastic force, is to the force of compression, as 5 to 9; and in glass as 15 to 16; but, in most cases, the elastic force seems to depend, in some degree, on the diameter of the ball. 350. The impact of two bodies is said to be direct when their centres of gravity move in the right line which passes through the point of impact. In considering the effects of collision, the bodies are usually supposed to be spheres of uniform density; and in their actions upon each other, not to be affected by gravity or any other force; but by their inertia only. PROPOSITION LXXXVII. 351. When the impact of two hard, or non-elastic bodies is direct, after the stroke they will either remain at rest, or move on uniformly together. For, since the bodies have no elasticity, when they come in contact there is no force whatever to separate them again; they must, therefore, either remain at rest, or proceed with one common velocity, that is uniformly. PROPOSITION LXXXVIII. 352. Let the impact of two hard bodies be direct ; it is required to determine their common velocity after the stroke. 1. Let the bodies A and B (fig. 136.) move the same way, or from A towards B. Call the velocity with which A moves V, and that with which B moves v; then the momentum of A is VA, and that of B is vB, and the sum of their momenta is VA+vB; now if x represent their common velocity after the stroke, we have s= VA+vB A+B ; because the velocity is as the momentum divided by the quantity of matter moved. 2. When the bodies move the contrary way, or meet. |