Question

A cylindrical piece of cork of density $\rho$ of base area $A$ and height $h$ floats in a liquid of density $\rho^{\prime} .$ The cork is slightly depressed and then released. The time period of oscillation of the cork is

• A. $\pi \sqrt{\frac{g \rho^{\prime}}{g \rho}}$
• B. $\pi \sqrt{\frac{h \rho}{g \rho^{\prime}}}$
• C. $2 \pi \sqrt{\frac{h \rho^{\prime}}{g \rho}}$
• D. $2 \pi \sqrt{\frac{h \rho}{g \rho^{\prime}}}$

Answer 1

Answer: (D)

At equilibrium while floating,
Weight = Upthrust i.e $mg=\left(\rho' A y_{0}\right) g$ where $y_{0}$ is the immersed length of the cork
$\Rightarrow (\rho A h) g=\left(\rho' A y_{0}\right) g$
When the cork is further immersed by $y$ and released, then
$F=(m g)-\rho'A\left(y+y_{0}\right) g=\rho^{\prime} A y_{0} g-\rho' A y g-\rho' A y_{0} g$
$F=-\left(\rho' A g\right) y $
$\Rightarrow T=2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{\rho A h}{\rho' A g}}$
$=2 \pi \sqrt{\frac{h \rho}{g \rho'}}$