Equivalent simple pendulum:

We know that the time period of oscillation of a compound pendulum is

$\displaystyle{T=2\pi\sqrt{\frac{K^2+l^2}{gl}}}\tag{1}$
[ To know the derivation of this equation (CLICK HERE) ]

where, $g$ is the acceleration due to gravity, $l$ is the distance between the centre of gravity $G$ and point $O$, $K$ is the radius of gyration.

If $L$ be the length of the simple pendulum then the time period of oscillation of the simple pendulum is given by

$\displaystyle{T=2\pi\sqrt{\frac{L}{g}}}\tag{2}$

From equations (1) & (2) we get

$\displaystyle{L=\frac{K^2+l^2}{l}}$

or, $\displaystyle{L=\frac{MK^2+Ml^2}{Ml}}$

or, $\displaystyle{L=\frac{I}{Ml}}$

Where, $I$ is the moment of inertia of the compound pendulum about an axis through the point O.

So $L$ which is represented by OO’, is the length of the simple pendulum, equivalent to the compound pendulum.

O is called centre of suspension and O’ is called the centre of oscillation.