RE-NEET 2026 Physics Simple Harmonic Motion MCQ Question

The time period of oscillation for a floating object is given by the formula:

$T = 2\pi \sqrt{\frac{m}{A}}$

where $m$ is the mass of the cork and $A$ is the effective restoring force per unit displacement. The restoring force is proportional to the weight of the liquid displaced, which depends on the density of the liquid.

For the first liquid, we have:

$A_1 \propto \rho_1$

For the second liquid, where the time period is $2T$:

$2T = 2\pi \sqrt{\frac{m}{A_2}}$

From this, we find:

$A_2 \propto \rho_2$

Since $2T = 2\sqrt{T}$, we can relate the densities:

$\frac{A_2}{A_1} = \frac{1}{4} \implies \frac{\rho_2}{\rho_1} = \frac{1}{4}$

Thus, the correct answer is $\frac{\rho_2}{\rho_1} = \frac{1}{4}$. The other options fail because they do not satisfy the derived relationship between the densities based on the time period.