Question

A wooden cylinder of diameter $4 r$, height $h$ and density $\rho / 3$ is kept on a hole of diameter $2 r$ of a tank, filled with water of density $\rho$ as shown in the figure. The height of the base of cylinder from the base of tank is $H$.

Let the cylinder is prevented from moving up, by applying a force and water level is further decreased. Then, the height of water level $\left(h_{2}\right.$ in the figure) for which the cylinder remains in original position without application of force is

• A. $h / 3$
• B. $4 h / 9$
• C. $2 h / 3$
• D. $h$

Answer 1

Answer: (B)

The free-body diagram of the block gives

For the equilibrium of the wooden cylinder, we have
$P_{0} \pi\left(4 r^{2}\right)+\frac{4 \pi}{3} r^{2} \rho h g=P_{0}\left(\pi r^{2}\right)+\left(P_{0}+\rho g h_{2}\right) \pi 3 r^{2} $
$\frac{4 \pi}{3} r^{2} \rho h g=\rho g h_{2}\left(\pi 3 R^{3}\right) $
$\frac{4 h}{3}=3 h_{2} \Rightarrow h_{2}=\frac{4 h}{9}$