Ans.

Let us consider a rectangular plate having edges of lengths $a$ and $b$ respectively. This rectangular plate hangs vertically from the edge of length $a$. Let $M$ be the total mass of the rectangular plate and $G$ be the centre of gravity of the plate.

(i) Time period for small oscillation:

Let $I$ be the moment of inertia of the plate about the horizontal axis through the point of suspension $O$ and perpendicular to the plate lying in the vertical plane.

$\displaystyle{I=\frac{1}{3}Mb^2}$
[ to know the derivation of the above equation (CLICK HERE) ]

Now the time period of small oscillation is

$\displaystyle{T=2\pi\sqrt{\frac{I}{Mg\left(OG\right)}}}$
[ to know the derivation of the above equation (CLICK HERE) ]

where, $g$ is the acceleration due to gravity.

From Fig.1 we get, $OG=\frac{b}{2}$

therefore, $\displaystyle{T=2\pi\sqrt{\frac{I}{Mg\left(\frac{b}{2}\right)}}}$

or, $\displaystyle{T=2\pi\sqrt{\frac{\frac{1}{3}Mb^2}{Mg\left(\frac{b}{2}\right)}}}$

or, $\displaystyle{T=2\pi\sqrt{\frac{2b}{3g}}}\tag{1}$

(ii) Length of the equivalent simple pendulum:

Let, $L$ be the length of the equivalent simple pendulum, then

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

Comparing with the equation (1), we get

$\displaystyle{L=\frac{2}{3}b}$