bodies A, B, C, placed at the distances SA, SB, SC, there be SA2 SB2 SC2 SP2 B, substituted the bodies A, theref. f sp2 sp2 c; these being col And hence, the moving force be- sp2 is the accelerating force; A SA2B. SB2 + c. sc2 which therefore is to the accelerating force of gravity, as f. SP2 to A. SA2+ B. SB2+c.sc2. 288. Corol. 3. The angular velocity of the whole system of bodies, is as A. SA2 &c SP2 f.sp A SAB. SB2+ c.sc2 For the absolute velocity of the point P, is as the accelerating force, or directly as the motive force f, and inversely as the mass : but the angular velocity is as the absolute velocity directly, and the radius SP inversely; therefore the angular velocity of P, or of the whole system, which is the same JSB thing, is as A.SA2+B. SB2 +C. SC2 PROPOSITION LVI. 289. To determine the Centre of Oscillation of any Compound Mass or Body MN, or of any System of Bodies A, B, C, &c. LET MN be the plane of vibration, to which let all the matter be reduced, by letting fall perpendiculars from every particle, to this plane. Let a be the centre of gravity, A. motion generated by all these forces is sp+ sq - c. sr A. SA2+ B. SB2+ c.sc2 Also, the angular veloc. any particle, placed in o, generates in the system, by its weight, is an p. s02 or sn, SG.SO because of the similar triangles sam, son. But, by the problem, the vibrations are performed alike in both cases, and therefore these two expressions must be equal to each other, A. sp+B.99 that is, Sm Sr ; and hence A. SA2 + B. SB2C. SC2 sm A SA2+ B SB2+c. Sc2 SG A. sp+B.sq - c. sr But, by cor. 2, pr. 41, the sum A. Sp+ B. sq (A + B + C). sm; therefore the distance so = A. SA2+ B SB2+ c. SC2 A. SA2+ B SB2 + C. SC2 SG (A + B + C) sb + c. sc by prop 42, which is the distance of the centre of oscillation o, below the axis of suspension; where any of the products A. sa, B so, must be negative, when a, b &c, lie on the other side of s. So that this is the same expression as that for the distance of the centre of percussion, found in prop. 54. Hence it appears, that the centres of percussion and of oscillation, are in the very same point. And therefore the properties in all the corollaries there found for the former, are to be here understood of the latter. 290. Corol. 1. If h be any particle of a body b, and d its distance from the axis of motion s; also G, o the centres of gravity and oscillation. Then the distance of the centre of oscillation of the body, from the axis of motion, is 291. Corol. 2. If b denote the matter in any compound body, whose centres of gravity and oscillation are G and o; the body P, which being placed at P, where the force acts as in the last proposition, and which receives the same motion from that force as the compound body b, is p For, by corol. 2, prop. 54, this body p is = A SA2B SB2+ c SC2 1 292. 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: Then 140850 shall the distance of the centre of oscillation, be so = nn 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 602 × 39 n% : 60 :: 39: = 140850 nn : 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 Bady 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 A SA2+ B. SB2 &C . and B G P by the same, the angular velocity of the system placed in R, is f. Sp : then, by making these two expres (A+B+C &c). SR2 sions equal to each other, the equation gives A. SA2+ B. SB2+ c. sc2 SR = A+B+C for the distance of the centre of gyration below the axis of motion. 296. Corol. 1. Because A. SA2+ B. SB2 &c SG. So. b, where G is the centre of gravity, o the centre of oscillation, and 6 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 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 f, acting at P, in the foregoing propositions. Now, Put the whole weight of the pendul. b g the weight of the ball, SG the dist. of the cen. of grav. 0 = so the dist. of the cen. of oscilla. u that of the point of impact P, c = chord of the arc described by o. N By prop. 56, if the mass be placed all at n, the pendulum will receive the same motion from the blow in the SR2 72 go horor, (prop. 54), 1 point P: and as sp2: SR2 ::/: the mass which being placed at P, the pendulum will still receive the same motion as before. Here then are two quantities of matter, namely, 6 and, 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 + gop +bb:::u, and therefore v⇒ u the velobir city 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 20 |
