Page:Elementary Text-book of Physics (Anthony, 1897).djvu/90

of oscillation is the same as that of the whole pendulum. Its distance from the point of suspension is the length sought.

In determinations of the value of $g$ by observations upon the time of oscillation of a pendulum, the length of the equivalent simple pendulum may be found in either of two ways:

(1) The pendulum may be constructed in such a manner that its moment of inertia and the position of its centre of gravity may be calculated. From these data the required length is readily obtained.

To show this, we consider any mass swinging as a pendulum about a horizontal axis. The force which sets it in oscillation is its weight $Mg$. The effect of this force in producing rotation about the axis is given by $MgR\sin {\phi }=I\alpha$ (§ 39), where $I$ is its moment of inertia about that axis and $R$ is the distance from the axis to its centre of gravity. As in the case of the simple pendulum, when the oscillations are infinitesimal, $\sin \phi$ may be replaced by $\phi$. Now and a represent the angular displacement and the angular acceleration of any point of the pendulum, and the actual displacement and acceleration are proportional to them; and since the displacement and acceleration are proportional to each other, every point in the pendulum has a simple harmonic motion of the same period. The actual acceleration of the centre of mass equals $R\alpha ={\frac {MgR}{I}}\cdot$ Now $R\phi$ is the displacement of the centre of mass, and therefore from the formula connecting acceleration and displacement in simple harmonic motion, used in §61, we obtain ${\frac {4\pi ^{2}}{T^{2}}}={\frac {MgR}{I}}\cdot$. Hence $T=2\pi {\sqrt {\frac {I}{MRg}}}\cdot$ Or, if we designate by $t$ the time of oscillation from one extremity of the arc to the other, we have

$t=\pi {\sqrt {\frac {I}{MRg}}}\cdot$

(42)

We may replace $I$ by its equivalent $I'+MR^{2}$, where $I'$ is the moment of inertia about an axis parallel to the axis of suspension and passing through the centre of gravity. By comparison of this