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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 $3 p$. The complete arrangement is placed in a liquid of density $2 p$ 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.
$(B,C)$
$(B,D)$
$(A,D)$
$(C,D)$
Solution
On small sphere
$\frac{4}{3} \pi R^3(\rho) g+k x=\frac{4}{3} \pi R^3(2 \rho) g$
on second sphere (large)
$\frac{4}{3} \pi R^3(3 p) g=\frac{4}{3} \pi R^3(2 p) g+k x$
by equation $(i)$ and $(ii)$
$x=\frac{4 \pi R^3 \rho g}{3 k}$
