A block of mass M rests on a smooth horizontal table. There is a small gap in the table under the block through which a pendulum has been attached to the block. The bob of the simple pendulum has mass m and length of the pendulum is L. The pendulum is set into small oscillations in the vertical plane of the figure. Calculate its time period. The table does not interfere with the motion of the string.

Consider the string at a small angle $\mathrm{\theta }$ to the vertical.

If T is tension in the string,
T sin $\mathrm{\theta }$ = Ma₀…..(i)
Now, we will consider the motion of the pendulum in the reference frame attached to the block.

For small oscillations, the path of pendulum bob can be approximated to be a straight line and $\mathrm{T}=\mathrm{mgcos}\mathrm{\theta }\simeq \mathrm{mg}$ 
The restoring force = $\mathrm{mgsin}\mathrm{\theta }+{\mathrm{ma}}_0$
$\begin{array}{l}=\mathrm{mgsin}\mathrm{\theta }+\mathrm{m}\frac{\mathrm{Tsin}\mathrm{\theta }}{\mathrm{M}}\\ =\mathrm{mgsin}\mathrm{\theta }+\frac{{\mathrm{m}}^2\mathrm{gsin}\mathrm{\theta }}{\mathrm{M}}\\ \simeq \mathrm{mg}\left[1+\frac{\mathrm{m}}{\mathrm{M}}\right]\mathrm{\theta }\end{array}$
$\begin{array}{l}\mathrm{m}\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}=-\mathrm{mg}\left[1+\frac{\mathrm{m}}{\mathrm{M}}\right]\mathrm{\theta }\\ \frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}=-\mathrm{g}\left[\frac{\mathrm{M}+\mathrm{m}}{\mathrm{M}}\right]\frac{\mathrm{x}}{\mathrm{L}}\left[\mathrm{SHM}\right]\\ {\mathrm{\omega }}^2=\frac{\mathrm{g}}{\mathrm{L}}\left(\frac{\mathrm{M}+\mathrm{m}}{\mathrm{M}}\right)\end{array}$
$\therefore \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{LM}}{\mathrm{g}(\mathrm{M}+\mathrm{m})}}$