Physics - SHM Question with Solution | TestHub

Mass of the wooden block is $M=abcd{\rho }_w$, where ${\rho }_w$ is the density of water.

When the block is pushed down by a small distance $x$, the additional volume submerged is $bcx$.

The restoring force acting on the block is the additional buoyant force:

$F=-(bcx){\rho }_{w}g$

The acceleration of the block is:

$\alpha =\frac{F}{M}=\frac{-bcx{\rho }_{w}g}{abcd{\rho }_w}=-\frac{g}{ad}x$

Comparing this with the standard equation of S.H.M., $\alpha =-{\omega }^{2}x$, we get:

${\omega }^2=\frac{g}{ad}\Rightarrow \omega =\sqrt{\frac{g}{ad}}$

The time period of S.H.M. is:

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{ad}{g}}$

Answer: $2\pi \sqrt{\frac{\mathrm{ad}}{\mathrm{g}}}$