Question 47: Oscillations

Question

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

Accepted Answer

Initially, the block is floating. The weight of the block is balanced by the buoyant force $\left(\mathrm{F}_{\mathrm{B}}ight)$.
Let h be the height of the submerged part of the block.
$$ \mathrm{Mg}=\mathrm{F}_{\mathrm{B}}=\mathrm{V}_{	ext {submerged }} 	imes ho_{	ext {liquid }} 	imes \mathrm{g} $$
$$\Rightarrow $$ $\mathrm{Mg}=(\mathrm{A} 	imes \mathrm{h}) ho \mathrm{g}$
When the block is depressed by a small distance x downwards, the volume of the displaced liquid increases by $A \cdot x$. This creates an extra buoyant force which acts as the restoring force.
The additional buoyant force is :
$$ \mathrm{F}_{	ext {restoring }}=-(	ext { Additional Volume }) 	imes ho 	imes \mathrm{g} $$
$$ \mathrm{F}_{	ext {restoring }}=-(\mathrm{Ax}) ho \mathrm{g} $$
Since the force is proportional to displacement x and acts opposite to it ( $\mathrm{F} \propto-\mathrm{x}$ ), the motion is Simple Harmonic Motion (SHM).
Using Newton's Second Law ( $\mathrm{F}=\mathrm{Ma}$ ) :
$$ \mathrm{Ma}=-(ho \mathrm{Ag}) \mathrm{x} $$
$$\Rightarrow $$ $$ a=-(ho A g M) x . $$
Standard equation for SHM acceleration is $\mathrm{a}=-\omega^2 \mathrm{x}$
Comparing the two equations:
$$ \omega^2=\frac{ho \mathrm{Ag}}{\mathrm{M}} \Rightarrow \omega=\sqrt{\frac{ho \mathrm{Ag}}{\mathrm{M}}} $$
The time period T is given by $\mathrm{T}=\frac{2 \pi}{\omega}$.
$$\Rightarrow $$ $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{M}}{ho \mathrm{Ag}}}$
Hence, the correct option is (C).

Answer

$2 \pi \sqrt{\frac{\rho A}{M g}}$

Answer

$\pi \sqrt{\frac{2 M}{\rho A g}}$

Answer

$2 \pi \sqrt{\frac{M}{\rho A g}}$

Answer

$\pi \sqrt{\frac{\rho A}{M g}}$

Oscillations

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

Choose the correct answer: