Question

A large tank of area A contains liquid of density $\rho$ . A cylinder of density $\frac{\rho }{4}$ and length ℓ, and area a $(a\ll A)$ is kept in equilibrium by applying an external vertically downward force. The cylinder is just submerged in liquid. At t = 0 the external force disappears instantaneously. Answer the following questions.

The time taken by the cylinder to just come out form the liquid

• A. $\frac{1}{2}\sqrt{\frac{l}{g}}si{n}^{-1}\left(\frac{1}{3}\right)$
• B. $\frac{\pi }{4}\sqrt{\frac{l}{g}}$
• C. $[\frac{\pi }{4}+\frac{1}{2}si{n}^{-1}\left(\frac{1}{3}\right)]\sqrt{\frac{l}{g}}$
• D. None of these

Answer 1

The correct option is C $[\frac{\pi }{4}+\frac{1}{2}si{n}^{-1}\left(\frac{1}{3}\right)]\sqrt{\frac{l}{g}}$

Time for $\frac{3l}{4}\Rightarrow \frac{1}{4}\frac{2\pi }{2}\sqrt{\frac{l}{9}}=\frac{\pi }{4}\sqrt{\frac{l}{g}}$
Time for $\frac{l}{4}\Rightarrow \frac{1}{2}\sqrt{\frac{l}{g}}si{n}^{-1}\left(\frac{1}{3}\right)$
Henc total time $\Rightarrow \frac{\pi }{4}\sqrt{\frac{l}{g}}+\frac{1}{2}\sqrt{\frac{l}{g}}si{n}^{-1}\left(\frac{1}{3}\right)\Rightarrow \sqrt{\frac{l}{g}}[\frac{\pi }{4}+\frac{1}{2}si{n}^{-1}\left(\frac{1}{3}\right)]$