CENTER of oscillation, that point in a pendulum, in which, if the weight of the several parts thereof were collected, each vibration would be performed in the same time as when those weights are separate. This is the point from whence the length of a pendulum is measured, which, in our latitude, in a pendulum that swings seconds, is 39 inches and I The center of suspension is the point on which the pendulum hangs.
A general rule for finding the center of oscillation. If several bodies be fixed to an inflexible rod suspended on a point, and each body be multiplied by the square of its distance from the point of suspension, and then each body be multiplied by its distance from the same point, and all the former products, when added together, be divided by all the latter products add ed together, the quotient which shall arise from thence will be the distance of the center of oscillation of these bodies from the said point.
Thus if C F (fig. 8) be a rod on which are fixed the bodies A, B, D, &c. at the several points A, B, D, &c. and if the body A be multiplied by the square of the dis tance C A, and B be multiplied by the square of the distance C B, and so on for the rest ; and then if the body A be mul tiplied by the distance C A, and B be multiplied by the distance C B, and so on for the rest ; and if the sum of the pro ducts arising in the former case be divid ed by the sum of those which arise in the latter, the quotient will give C Q the dis tance of the center of oscillation of the bodies A, B, D, &c. from the point C.
To determine the center of oscillation of the rectangle 11 I H S (fig. 9) suspended on the middle point A of the side 11I, and oscillating about its axis R I. Let R I 8 II a, A P = x, then will Pp=dx and the element or the area, consequently one weight = a d x and its momentum axds. Whertfore = 1 a x3 : axe = x, indefinitely ex presses the distance of the center of os cillation from the axis of oscillation in the segment It C D I. If then for x be sub stituted the altitude of the whole rec tangle R S = b, the distance of the ter of oscillation from the axis will be found =lb.
The center of oscillation in an equila teral triangle S A H oscillating about its axis R I, parallel to the base S H, is found at a distance from the vertex A equal to 1/- A E the altitude of the triangle.
The center of oscillation in an equila teral triangle S A H oscillating about its base S H, is found at a distance from the vertex A = A E.
For the centers of oscillation of para bolas and curves of the like kind oscillat ing about their axes parallel to their bases, they are found as follows. In the apol lonian parabola, the distance of the cen ter of oscillation from the axis = A E.
In the cubical paraboloid, the distance of the center from the axis'A A E. In a biquadratic paraboloid, the distance of the center from the axis = 9 A E.