Question

A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ 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. $\pi\sqrt{\dfrac{\rho A}{Mg}}$
• C. $2\pi\sqrt{\dfrac{M}{\rho A g}}$
• D. $\pi\sqrt{\dfrac{2M}{\rho A g}}$

Hint:

Small vertical oscillations of floating bodies always execute simple harmonic motion due to buoyant restoring force.

Answer 1

The Correct Option is C

Step 1: Identifying restoring force.
When the block is displaced downwards by a small distance $x$, an additional buoyant force acts upward equal to:
\[ F = \rho A g x \] Step 2: Writing equation of motion.
\[ M\frac{d^2x}{dt^2} = -\rho A g x \] Step 3: Comparing with SHM equation.
For simple harmonic motion:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \] Thus,
\[ \omega^2 = \dfrac{\rho A g}{M} \] Step 4: Calculating time period.
\[ T = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\dfrac{M}{\rho A g}} \] Step 5: Final conclusion.
The time period of oscillation is $2\pi\sqrt{\dfrac{M}{\rho A g}}$.