Question

A hollow wooden cylinder of height $h$, inner radius $R$ and outer radius $2R$ is placed in a cylindrical container of radius $3R$. When water is poured into the container, the minimum height $H$ of the container for which cylinder can float inside freely is

• A. $\frac{h{\rho }_{water}}{{\rho }_{water}+{\rho }_{wood}}$
• B. $\frac{h{\rho }_{wood}}{{\rho }_{water}}$
• C. $h$
• D. $\frac{h^2}{R}$

Answer 1

The correct option is B $\frac{h{\rho }_{wood}}{{\rho }_{water}}$

For the hollow wooden cylinder to float freely,
Weight of the wooden cylinder = Weight of the liquid displaced
Now, weight of the hollow wooden cylinder,$mg=V{\rho }_{wood}g=\pi \left[\right(2R^2)-R^2]h{\rho }_{wood}g=\left(3\pi R^{2}h\right){\rho }_{wood}g$

So, by law of floatation,
$\left(3\pi R^{2}h\right){\rho }_{wood}g=\left(3\pi R^{2}H\right){\rho }_{water}g$
$\Rightarrow \frac{h{\rho }_{wood}}{{\rho }_{water}}=H$