Question

A cylindrical block of density $d$ stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2$. The block is in equilibrium with half of it in liquid $1$ and the other half in liquid $2$ as shown in the figure. If the block is given a displacement downwards and released, then neglecting friction, which of the following statement(s) is/are correct?

• A. Block executes simple harmonic motion.
• B. The block's motion is periodic but not simple harmonic.
• C. The frequency of oscillation is independent of the size of the block.
• D. The motion of the block is symmetric about its equilibrium position.

Answer 1

Answer: (A), (D)

The motion of the block is symmetric about its equilibrium position.

Since liquid $2$ is below liquid $1$, liquid $2$ is denser than liquid $1$.
Let the area of cross-section of the cylindrical block be $A$ and it is displaced downwards by $y$. Then, the volume of liquid $2$ displaced will increase by $Ay$ and that of liquid $1$ will decrease by the same amount $Ay$.
Hence, net increase in upthrust on the block will be equal to $(Ay{d}_{2}g-Ay{d}_{1}g).$ This additional upthrust will try to restore the block to the original position.
Net restoring force $=Ay(d_2-d_1)g$.
Since this force is restoring and directly proportional to displacement $y$, the cylindrical block will execute $SHM$ along a vertical line.
Hence, option $\left(a\right)$ is correct and option $\left(b\right)$ is wrong.

If the mass of the block is equal to $m$, then its acceleration will be equal to $\frac{Ayg(d_2-d_1)}{m}$.
Since, its acceleration depends on mass $m$, frequency of oscillations will depend on the size of the cylindrical block.
Hence, option $\left(c\right)$ is wrong.

If the cylinder is displaced upward through $y$ from equilibrium position, then it will experience a net downward force equal to the calculated value above. This shows that its motion will be symmetric about its equilibrium position.
Hence, option $\left(a\right)$ and $\left(d\right)$ are correct.