Question

A hydrometer executes SHM when pushed into a liquid of density \( \rho \). If the mass of hydrometer is \( m \) and the radius of the tube is \( r \), then the time period of oscillation is:

• A. \( T = 2\pi \sqrt{\frac{m}{\pi r^2 \rho g}} \)
• B. \( T = 2\pi \sqrt{\frac{\pi r^2 \rho g}{m}} \)
• C. \( T = \frac{1}{2\pi} \sqrt{\frac{m}{\pi r^2 \rho g}} \)
• D. \( T = \frac{1}{2\pi} \sqrt{\frac{\pi r^2 \rho g}{m}} \)

Hint:

For SHM in fluids, use buoyant restoring force \( F = -\rho g A x \) and \( T = 2\pi \sqrt{\frac{m}{k}} \).

Answer 1

The Correct Option is A

Restoring force due to buoyancy: \( F = \rho g A x \), where \( A = \pi r^2 \).
\[ F = -\rho g \pi r^2 x \Rightarrow F = -k x \Rightarrow k = \rho g \pi r^2 \] Now, time period of SHM is: \[ T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{m}{\rho g \pi r^2}} \]