Page:Elementary Text-book of Physics (Anthony, 1897).djvu/91

§ 62] equation with the one giving the time of oscillation of a simple pendulum, it appears that the length $$l$$ of the simple pendulum which will oscillate in the same time as the physical pendulum, or, as it is called, the length of the equivalent simple pendulum, is given by A line drawn parallel with the axis of suspension, through a point at the distance $$l$$ from that axis and on the line drawn through the centre of gravity perpendicular to that axis, is called the axis of oscillation. It evidently contains the centre of percussion (§ 44).

A pendulum consisting of a heavy spherical bob suspended by a cylindrical wire was used by Borda in his determinations of the value of $$g$$. The moment of inertia and the centre of gravity of the system were easily calculated, and the length of the simple pendulum to which the system was equivalent was thus obtained.

(2) We may determine the length of the equivalent simple pendulum directly by observation. The method depends upon the principle that, if the axis of oscillation be taken as the axis of suspension, the time of oscillation will not vary. The proof of this principle is as follows:

Suppose the pendulum suspended so as to swing about the axis of oscillation as a new axis of suspension. The distance of the axis of oscillation from the centre of gravity is $$l - R$$, and the length $$l'$$ of the equivalent simple pendulum, in this case, is $$l' = \frac{I' + M(l - R)^2}{M(l - R)}\cdot$$ Now $$l = \frac{I' + MR^2}{MR}$$ or $$I' = MR (l - R)$$, and substituting this value in the equation for $$l'$$ and reducing, we obtain $$l' = l$$. That is, the length of the equivalent simple pendulum, and consequently the time of oscillation when the pendulum swings about its axis of suspension, is the same as that when it is reversed and swings about its former axis of oscillation.

A pendulum (Fig. 26) so constructed as to take advantage of this principle was used by Kater in his determination of the value of $$g$$; and this form is known, in consequence, as Kater's pendulum.