Question

A cylindrical piece of cork of height $h$ and density ${\rho }_c$ floats vertically in a liquid of density ${\rho }_l$ The cork is depressed slightly and released. If viscous effects are neglected, the time period of vertical oscillations of the cylinder is given by

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Let A be the cross-sectional area of the cork and M its mass. Figure(a) shows the static equilibrium, the weight of the cork being balanced by the weight of the liquid it displaces. If the cork is depressed through a distance x, as shown in Fig(b), the buoyant force on it increases by ${\rho }_l$ Agx, because ${\rho }_l$ Ax is the mass of the liquid displaced by dipping, g being the acceleration due to gravity If viscous effects are neglected, the restoring force on the cork is given by
$F=-{\rho }_{l}Agx=-Kx$
Where K=${\rho }_l$Ag . since F $\propto -x$ the motion of the corkIs simple harmonic. The time period of the motion is
$T=2\pi \sqrt{\frac{M}{k}}$
Where M is the mass of the cork =$Ah{\rho }_c$ Hence
$T-2\pi \sqrt{\frac{Ah{\rho }_c}{{\rho }_{l}Ag}}=2\pi \sqrt{\frac{h{P}_c}{g{\rho }_l}}$