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 $\frac{4\pi R^3\rho g}{3k}$
• B. the net elongation of the spring is $\frac{8\pi R^3\rho g}{3k}$
• 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 $=mg$
$\begin{array}{l}=\left[\rho \times \frac{4}{3}\pi R^3+3\rho \times \frac{4}{3}\pi R^3\right]g\\ w=\frac{16\rho \pi R^{3}g}{3}\end{array}$
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 $=\frac{4\pi }{3}{R}^3+\frac{4\pi }{3}{R}^3$