Question

A cylindrical cork of uniform density \(\rho_1\) floats in a liquid of density \(\rho_1\). If the cork is depressed slightly and released, it oscillates harmonically with time period \(T\). If the same cork floats in another liquid of density \(\rho_2\), then the similar oscillation has time period \(2T\). The value of \(\dfrac{\rho_2}{\rho_1}\) is:

• A. \(\dfrac{1}{4}\)
• B. \(4\)
• C. \(2\)
• D. \(\dfrac{1}{2}\)

Hint:

For oscillations of a floating body, \[ T\propto \frac{1}{\sqrt{\rho}} \] A denser liquid provides a stronger restoring force and hence a smaller time period.

Answer 1

The Correct Option is A

Concept:

• When a floating body is displaced vertically by a small distance, the restoring buoyant force produces SHM.

• The time period is given by \[ T=2\pi\sqrt{\frac{m}{A\rho g}} \] where \(A\) is cross-sectional area and \(\rho\) is the density of the liquid.

• Hence, \[ T\propto \frac{1}{\sqrt{\rho}} \] for the same cork.

Step 1: Write the proportionality relation.
\[ T\propto \frac{1}{\sqrt{\rho}} \] Therefore, \[ \frac{T_2}{T_1} = \sqrt{\frac{\rho_1}{\rho_2}} \]

Step 2: Substitute the given time periods.
Given, \[ T_1=T \] and \[ T_2=2T \] Thus, \[ \frac{2T}{T} = \sqrt{\frac{\rho_1}{\rho_2}} \] \[ 2 = \sqrt{\frac{\rho_1}{\rho_2}} \]

Step 3: Square both sides.
\[ 4 = \frac{\rho_1}{\rho_2} \] \[ \rho_2 = \frac{\rho_1}{4} \] Therefore, \[ \boxed{ \frac{\rho_2}{\rho_1} = \frac14 } \]

Step 4: Choose the correct option.
\[ \boxed{\text{Option (A)}} \]