Q 11-13-132JEE MainJEE Main 2019 (10 Jan, Shift 2)Medium
A cylindrical plastic bottle of negligible mass is filled with $310\ \text{ml}$ of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega$. If the radius of the bottle is $2.5\ \text{cm}$ then $\omega$ is close to: (density of water $= 10^3\ \text{kg/m}^3$)
Answer: (C) $7.9\ \text{rad s}^{-1}$
Pushing the bottle down by $y$ adds buoyancy $\rho gAy$, so $k = \rho gA$. The mass is $m = \rho V$.
$$\omega = \sqrt{\frac{\rho gA}{\rho V}} = \sqrt{\frac{gA}{V}} = \sqrt{\frac{9.8\times\pi(0.025)^2}{310\times10^{-6}}} = \sqrt{62.1} \approx 7.9\ \text{rad s}^{-1}$$
Solution by Sreeraj P, M.Sc Physics