Question

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less 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 $\frac{4 \pi R^{3} \rho g}{3 k}$
• B. the net elongation of the spring is $\frac{8 \pi R^{3} \rho g}{3 k}$
• C. the light sphere is partially submerged.
• D. the light sphere is completely submerged.

Answer 1

Answer: (A), (D)

Weight of the balls i.e system $=m g$
$ \begin{array}{l} =\left[\rho \times \frac{4}{3} \pi R ^{3}+3 \rho \times \frac{4}{3} \pi R ^{3}\right] g \\ w =\frac{16 \rho \pi R ^{3} g }{3} \end{array} $
Buoyant force $= V _{ \rho g }$
$ \begin{array}{l} \rho_{1}=2 \rho \\ = V =2 \rho g \end{array} $
Buoyant force= weight of system
$ \begin{array}{l} V \cdot 2 \rho g =\frac{16 \rho \pi R ^{3} g }{3} \\ V =\frac{8 \pi R ^{3}}{3} \end{array} $
Submerged volume $=\frac{8 \pi R ^{3}}{3}$
Total volume of system $=\frac{4 \pi}{3} R^{3}+\frac{4 \pi}{3} R^{3}$