Question

A solid sphere of radius R and density $\rho$ is attached to one end of a massless spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density $3\rho$. The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement(s) is (are)

• A. The net elongation of the spring is $\frac{\left(4\pi R^3\rho g\right)}{3k}$
• B. The net elongation of the spring is $\frac{\left(8\pi R^3\rho g\right)}{3k}$
• C. The light sphere is partially submerged.
• D. The light sphere is completely submerged.

Answer 1

Answer: (A), (D)

The correct options are
A

The net elongation of the spring is $\frac{\left(4\pi R^3\rho g\right)}{3k}$

D

The light sphere is completely submerged.

Here, $T=kx$
For upper sphere at equilibrium, $kx+\frac{4}{3}\pi R^3\rho g=\frac{4}{3}\pi R^{3}2\rho g\Rightarrow x=\frac{4}{3k}\pi R^3\rho g$
The system is completely submerged as total weight is balanced by total buoyant force.