Question

A cylindrical wooden block of area of cross section $A$, height $H$ and density $\rho$ is floating in water of density ${\rho }_w$. If block is slightly pushed down into water and released, it starts performing simple hormonic motion (SHM). Time period of SHM of the block is independent of

• A. $\rho$
• B. $H$
• C. $A$
• D. ${\rho }_w$

Answer 1

The correct option is C $A$

$F_B={\rho }_{w}Ayg\Rightarrow {\rho }_{w}Agy=-m\frac{d^{2}y}{d{t}^2}$
or $\frac{d^{2}y}{d{t}^2}=-\frac{{\rho }_{w}Agy}{m}=-\frac{{\rho }_{w}Ag}{\rho AH}y\Rightarrow \frac{d^{2}y}{d{t}^2}=-\left(\frac{{\rho }_{w}g}{\rho H}\right)y$

$\omega =\sqrt{\frac{{\rho }_{w}g}{\rho H}}$