Now, let t denote the time of one revolution of the shaft, the variable horizontal distance of each ball from that shaft, « as usual = 3∙141593 : then will the velocity of each ball be = and (art. 209.) its centrifugal force = <+x= The balls being operated upon simultaneously by the centrifugal force and the force of gravity, of which one operates horizontally, the other vertically, the resultant of the two forces is, evidently, always in the actual position of the handle CD, CF. It follows, therefore, that the ratio of the gravity to the centrifugal force, is that of cos. ica to sin. icq, or that of the vertical distance of a below c to its horizontal distance from Aa. Call the former d, the latter being x : then d:x::g: Hence, the periodic time varies as the square root of the altitude of the conic pendulum, let the radius of the base be what it may. Hence, also, when ICQ = ICP = 45°, the centrifugal force of each ball is equal to its weight. ON THE CENTRES OF PERCUSSION, 216. The Centre of Percussion of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest, as it were, in equilibrio, without acting on the centre of suspension. 217. The Centre of Oscillation is that point, in a body vibrating by its gravity, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension. 218. The Centre of Gyration is that point, in which if the whole mass be collected, the same angular velocity will be generated in the same time, by a given force acting at any place, as in the body or system itself. 219. The angular motion of a body, or system of bodies, is the motion of a line connecting any point and the centre or axis of motion; and is the same in all parts of the same revolving body. And in different unconnected bodies, each revolving about à centre, the angular velocity is as the absolute velocity directly, and as the distance from the centre inversely; so that, if their absolute velocities be as their radii or distances, the angular velocities will be equal. 220. PROP. To find the centre of percussion of a body or system of bodies. Let the body revolve about an axis passing through any point s in the line SGO, passing through the centres of gravity and percussion, G and o. Let MN be the section of the body, or the plane in which the axis soo moves. And conceive all the particles of the body to be reduced to this plane, by perpendiculars let fall from them to the plane: a sup. position which will not affect the centres 6, o, nor the angular motion of the body. M B S Let A be the place of one of the particles, so reduced; join sa, and draw AP perpendicular to As, and a perpendicular to sGo: then AP will be the direction of a's motion as it revolves about s; and the whole mass being stopped at o, the body a will urge the point r, forward, with a force proportional to its quantity of matter and velocity, or to its matter and distance from the point of suspension s; that is, as A. SA; and the efficacy of this force in a direction perpendicular to so, at the point P, is as A. sa, by similar triangles; also, the effect of this force on the lever, to turn it about being as the length of the lever, is as a sa. Po= A. sa. (80 SP)=A. sa. so A. sa SPA. sa SOA. SA?. In like manner, the forces of в and c, to turn the system about o, are as - B. SB2, and c. sc. 80 — c. sc2, &c. 0, But, since the forces on the contrary sides of o destroy one another, by the definition of this force, the sum of the positive parts of these quantities must be equal to the sum of the negative parts, that is, A. sa. so + B. sb. so + c. sc. so, &c. = A. SA2+ B. SB2c. sc", &c; and SA+B Sp2 + c sc2, &c. which is the distance of the centre of percussion below the axis of motion. And here it must be observed that, if any of the points a, b, &c. fall on the contrary side of s, the corresponding product A. sa, or B. sb, &c. must be made negative. 221. Corol. 1. Since, by art. 105, A + B + c, &c. or the body b X the distance of the centre of gravity, sg, is = a. sa + B. sb+c. sc, &c. which is the denominator of the value of so; therefore the distance of the centre of percus. A. SAB. SB2 + c. Sc2 &c. sion, is so = 222. Corol. 2. it is SA2 and SB2 and sc2 = and, by cor. 5, art. 101, the sum of the last terms is nothing, 2 namely, 2SG ca + 2SG Gb+2SG GC &c. = 0; therefore the sum of the others, or a . SA2+ B. BB2 &c. is = (A + B &C.). sg2 + a . GA2 + B. GB2 + C GC2 &c. b. SG2 + A. GAB. GR2 + C. GC2 &c; which being substituted in the numerator of the foregoing value of so, gives or= SO= b. SG2 + A. GA2 + B GB3+ &c. or so = SG + b. so A. GA2 + B. GB+c. GC2 &c. 223. Corol. 3. Hence the distance of the centre of per. cussion always exceeds the distance of the centre of gravity, A. GA2 + B. GB2 &C. and the excess is always Go = b. sG • GO= A. GA + B. GB3 &C. ; the body b 224. And hence also, sc that is so. Go is always the same constant quantity, where. ever the point of suspension s is placed; since the point o and the bodies A, B, &c. are constant. Or Go is always reciprocally as so, that is co is less, as sc is greater; and consequently the point o rises upwards and approaches towards the point G, as the point s is removed to the greater distance; and they coincide when so is infinite. But when s coincides with G, then Go is infinite, or o is at an infinite distance. 225. PROP. If a body ▲, at the distance sa from an axis passing through s, perpendicular to the plane of the paper, be made to revolve about that axis by any force acting at P in the line SP, perpendicular to the axis of motion: it is required to determine the quantity or matter of another body, Q, which being placed at P, the point where the force acts, it shall be accelerated in the same manner, as when a revolved at the distance sa; and consequently, that the angular velocity of a and Q about s, may be the same in both cases. By the nature of the lever, sa: SP:: f: SP SA f, the effect of the force f, acting at P, on the body at A; that is, the force f acting at P, will have the same effect on the body a, as SP the forcef, acting directly at the point a. SA S A P But as ASP revolves altogether about the axis at s, the abso lute velocities of the points A and s, or of the bodies a and Q, will be as the radii sa, sp, of the circle described by them. Here then we have two bodies A and Q, which being urged SP directly by the forces f and f, acquire velocities which are SA as sp and SA. And since the motive forces of bodies are as their mass and velocity: therefore SP =f:f:: A.SA: Q. SP, and sp2 : sa2 : : a : Q = SA SA 8p2 which therefore expresses the mass of matter which, being placed at r, would receive the same angular motion from the action of any force at P, as the body a receives. So that the resistance of any body A, to a force acting at any point P, is directly as the square of its distance sa from the axis of motion, and reciprocally as the square of the distance sp of the point where the force acts. 226. Corol. 1. Hence the force which accelerates the point P, is to the force of gravity, as to A. SA2. f. sp2 A. to 1, or as f. sp2 227. Corol. 2. If any number of bodies A, B, C, be put in motion, about a fixed A axis passing through s, by a force acting at P; the point P will be accelerated in the same manner, and consequently the whole system will have the same angular velocity, if instead of the bodies A, B, C, placed at the distances SA, SB, sc, there be substituted the B P ANGULAR MOTION. Spac; these being collected into the point SA2 SB2 SC2 bodies A, B, Sp2 Sp2 P. And hence, the moving force being f, and the matter A. SA2+ B. SB2+ c. Sc2 moved being f.spa A SAB. SBC. Sc2 : therefore is the accelerating force; which therefore is to the accelerating force of gravity, as f. sp2 to A. SA2+ B. SB2+ c. sc2. 228. Corol. 3. The angular velocity of the whole system of bodies, is as f. sp A. SA2+ B. SB2 + c. For the abso lute velocity of the point P, is as the accelerating force, or directly as the motive force f, and inversely as the mass A. 2 SA2 &c. : but the angular velocity is as the absolute velo. city directly, and the radius SP inversely; therefore the an-gular velocity of P, or of the whole system, which is the same f. sp thing, is as A. SA* + B. SB+C. SC. 229. PROP. To determine the centre of oscillation of any compound mass, or body мN, 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 G be the centre of gravity, sp, B. C. sr; therefore, by cor. 3, art. 228, the angular T motion generated by all these forces is sp + B. sq-c . sr A.SA+ B. SB+c.sc" |