Question
A cylindrical cork of uniform density floats in a liquid of density $\rho_{1}$. If the cork is depressed slightly and released, it oscillates harmonically with time period $T$. If the same cork floats in another liquid of density $\rho_{2}$, then the similar oscillation has time period $2T$. The value of $\rho_{2}/\rho_{1}$ is:
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Solution
Sol. [Key Concept: Law of flotation — density and submerged fraction]
Time period of a cylinder oscillating inside the liquid
$$\begin{matrix} T & \ = 2\pi\sqrt{\frac{\rho_{s}l}{\rho_{l}g}} \\ \frac{T_{1}}{T_{2}} & \left. \ = \sqrt{\frac{\rho_{2}}{\rho_{1}}}\ \left\lbrack \because\rho_{s} \rightarrow \text{~}\text{constant}\text{~} \right\rbrack \right\rbrack \\ \frac{T}{2T} & \ = \sqrt{\frac{\rho_{2}}{\rho_{1}}} \\ \frac{\rho_{2}}{\rho_{1}} & \ = \frac{1}{4} \end{matrix}$$
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