292. By the method of Fluxions, the centre of oscillation, for a regular body, will be found from cor. 1. But for an irregular one; suspend it at the given point; and hang up also a simple pendulum of such a length, that making them both vibrate, they may keep time together. Then the length of the simple pendulum, is equal to the distance of the centre of oscillation of the body, below the point of suspension. 293. Or it will be still better found thus: Suspend the body very freely by the given point, and make it vibrate in small arcs, counting the number of vibrations it makes in any time, as a minute, by a good stop watch; and let that number of vibrations made in a minute be called n shall the distance of the centre of oscillation, be so = Then 140850 inches. For the length of the pendulum vibrating seconds, or 60 times in a minute, being 39 inches; and the lengths of pendulums being reciprocally as the square of the number of vibrations made in the same time; therefore n* :60% :: 39%: 602 × 39 n n 140850 : the length of the pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion. 294. The foregoing determination of the point, into which all the matter of a body being collected, it shall oscillate in the same manner as before, only respects the case in which the body is put in motion by the gravity of its own particles, and the point is the centre of oscillation: but when the body is put in motion by some other extraneous force, instead of its gravity, then the point is different from the former, and is called the Centre of Gyration; which is determined in the following manner : `PRO PROPOSITION LVII. 295. To determine the Centre of Gyration of a Compound Body or of a System of Bodies. LET R be the centre of gyration, or the point into which all the particles A, B, C, &c. being collected, it shall receive the same angular motion from a force facting at P, as the whole system receives. Now, by cor. 3 pr. 54, the angular velocity generated in the system by the f. SP force fis as . A SA2B. SB2 &c. B CROE O C P by the same, the angular velocity of the system placed in R, is f. SP (A+B+C &c.). SR2 sions equal to each other, the equation gives SR = : then, by making these two expres for the distance of the A. SA2B.SB2+ c. A+B+C centre of gyration below the axis of motion. SG. So .b, 296. Corol. 1. Because A. SA2+ B. SB2 &c. where G is the centre of gravity, o the centre of oscillation, and b the body A+B+C &c; therefore SR2 = SG. SO; that is, the distance of the centre of gyration, is a mean proportional between those of gravity and oscillation. 297. Corol. 2. If denote any particle of a body b, at d distance from the axis of motion; then SR2 sum of all the pd2. PROPOSITION LVIII = body b 298. To determine the velocity with which a Ball moves, which being shot against a Ballistic Pendulum, causes it to vibrate through a given Angle. The Ballistic Pendulum is a heavy block of wood MN, suspended vertically by a strong horizontal iron axis at s, to which it is connected by a firm iron stem. This problem is the application of the last proposition, or of prop 54, and was invented by the very ingenious Mr. Robins, to determine the initial velocities of military projectiles; a circumstance very useful in that science; and it is the best method yet known for determining them with any degree of accuracy. Let G, R, o be the centres of gravity, gyration, and oscillation, as determined by the foregoing propositions; and let p be the point where the ball strikes the face of the pendulum ; the momentum of which, or the product of its weight and velocity, is expressed by the force ƒ, acting at P, in the foregoing propositions. Now, Put = the whole weight of the pendul. the weight of the ball, gG the dist. of the cen of grav, u = that of the point of impact P, c = chord of the arc described by o. M be placed all at R, the penmotion from the blow in the 2 By prop. 56, if the mass dulum will receive the same point P and as sp2: SR2 ; :/ : for-porn.(prop.54), the mass which being placed at P, the pendulum will still receive the same motion as before. Here then are two S:: 2 go quantities of matter, namely, band, the former moving with the velocity v, and striking the latter at rest; to determine their common velocity u, with which they will jointly proceed forward together after the stroke. In which case, by the law of the impact of non-elastic bodies, we have bii cop p+bb:: vu, and therefore v = bii u the velocity of the ball in terms of u, the velocity of the point P, and the known dimensions and weights of the bodies. But now to determine the value of u, we must have recourse to the angle through which the pendulum vibrates; for when the pendulum descends down again to the vertical position, it will have acquired the same velocity with which it began to ascend, and, by the laws of falling bodies, the velocity of the centre of oscillation is such, as a heavy body would acquire by freely falling through the versed sine of the arc described by the same centre o. But the chord of that arc is c, and its radius is o; and, by the nature of the circle, the chord is a mean proportional between the versed sine and diameter, therefore 20: c :: c: the versed sine of the arc described by o. Then, by the laws of falling bodies cc 20 the velocity acquired by the point o in descending through the arc whose chord is c, 16 feet: and therefore o:i::c√ ci 2a where a = which is the velocity u. of the point P 2a : Then, by substituting this value for u, the velocity of the ball before found, becomes v = bii+gop bio 24 So that the velocity of the ball is directly as the chord of the arc described by the pendulum in its vibration. SCHOLIUM. 299. In the foregoing solution, the change in the centre of oscillation is omitted, which is caused by the ball lodging in the point P But the allowance for that small change, and that of some other small quantities, may be seen in my Tracts, where all the circumstances of this method are treated at full length. 300. For an example in numbers of this method, suppose the weights and dimensions to be as follow: namely, P = 570lb, Then b = 18oz. ldr. bü+gof xc= = 1.131lb, bio 1131 x 94847 Therefore 656-56 × 2:1337 or 1401 feet, is the velocity, per second, with which the ball moved when it struck the pendulum. OF HYDROSTATICS. 301. HYDROSTATICS is the science which treats of the pressure, or weight, and equilibrium of water and other Auids, especially those that are non-elastic. 302. A fluid is elastic, when it can be reduced into a less volume by compression, and which restores itself to its former bulk again when the pressure is removed; as air. And it is non-elastic, when it is not compressible by such force; as water, &c. VOL. IF Dd PRO. PROPOSITION LIX. 303. If any Part of a Fluid be raised higher than the rest, by any Force, and then left to itself; the higher Parts will descend to the lower Places, and the Fluid will not rest, till its Surface be quite even and level. FOR, the parts of a fluid being easily moveable every way, the higher parts will descend by their superior gravity, and raise the lower parts, till the whole come to rest in a level or horizontal piane. 304 Corol. 1. Hence, water that communicates with other water, by means of a close canal or pipe,will stand at the same height in both places. Like as water in the two legs of a syphon, 305. Corol. 2. For the same reason, if a fluid gravitate towards a centre; it will dispose itself into a spherical figure, the centre of which is the centre of force. Like the sea in respect of the earth. PROPOSITION LX. 306. When a Fluid is at Rest in a Vessel, the Base of which is Parrallel to the Horizon; Equal Parts of the Base are Equally Pressed by the Fluid. FOR, on every equal part of this base there is an equal column of the fluid supported by it. And as all the columns are of equal height, by the last proposition they are of equal weight, and therefore they press the base equally; that is, équal parts of the base sustain an equal pressure. 307. Corol. 1. All parts of the fluid press equally at the same depth. For, if a plane parallel to the horizon be conceived to be drawn at that depth; then the pressure being the same in any part of that plane, by the proposition, therefore the parts of the fluid, instead of the plane, sustain the same pressure at the same depth. 308. Corol. 2. The pressure of the fluid at any depth, is as the depth of the fluid. For the pressure is as the weight, and the weight is as the height of the fluid. 309. Corol. |