A cylindrical block of mass M and area of cross section A is floating in a liquid of density ρ
Q. A cylindrical block of mass M and area of cross section A is floating in a liquid of density ρ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is _____

(A) \( 2\pi \sqrt{\dfrac{\rho A}{Mg}} \)

(B) \( \pi \sqrt{\dfrac{2M}{\rho Ag}} \)

(C) \( 2\pi \sqrt{\dfrac{M}{\rho Ag}} \)

(D) \( \pi \sqrt{\dfrac{\rho A}{Mg}} \)

Correct Answer: \( 2\pi \sqrt{\dfrac{M}{\rho A g}} \)

Explanation

At equilibrium, the weight of the block is balanced by the buoyant force.

\[ Mg = \rho V g \]

When the block is depressed downward by a small distance \(x\), the volume of liquid displaced increases by

\[ \Delta V = A x \]

This causes an additional buoyant force acting upward given by

\[ F = \rho g A x \]

This force acts as a restoring force, hence

\[ F = -kx \quad \text{where} \quad k = \rho g A \]

Thus, the motion of the block is simple harmonic motion.

For SHM, the time period is

\[ T = 2\pi \sqrt{\dfrac{M}{k}} \]

Substituting the value of \(k\),

\[ T = 2\pi \sqrt{\dfrac{M}{\rho g A}} \]

Therefore, the period of oscillation is \[ \boxed{2\pi \sqrt{\dfrac{M}{\rho A g}}} \]

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