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)

The correct options are
A the net elongation of the spring is $\frac{4\pi R^3\rho g}{3k}$
D the light sphere is completely submerged

Weight of the balls i.e system $=mg=[\rho \times \frac{4}{3}\pi R^3+3\rho \times \frac{4}{3}\pi R^3]gw=\frac{16\rho \pi R^{3}g}{3}$

Buoyant force$=V_{\rho g}{\rho }_1=2\rho =V=2\rho g$

Buoyant force$=$ weight of system

$V.2\rho g=\frac{16\rho \pi R^{3}g}{3}V=\frac{8\pi R^3}{3}$

Submerged volume$=\frac{8\pi R^3}{3}$

Total volume of system$=\frac{4\pi }{3}{R}^3+\frac{4\pi }{3}{R}^3=\frac{8\pi R^3}{3}$

Total volume $=$ submerged volume

it means both sphere are submerged completely

$A$ for spring elongation, on sphere of density

$Rx+\frac{4\pi }{3}{R}^3\times \rho \times g=\frac{4\pi }{3}{R}^3\times 2\rho \times gRx=\frac{4\pi R^3\rho g}{3}[2-1]\Rightarrow x=\frac{4\pi R^3\rho g}{3R}$