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

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The net elongation of the spring is $\frac{4 π R^{3} ρ g}{3 k}$

The net elongation of the spring is $\frac{8 π R^{3} ρ g}{3 k}$

The light sphere is partially submerged

The light sphere is completely submerged

answer is A, D.

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Detailed Solution

At equilibrium, we have

$\frac{4}{3} π R^{3} 2 ρ g = \frac{4}{3} π R^{3} ρ g + k Δ l \Rightarrow k Δ l = \frac{4}{3} π R^{3} 2 ρ g \Rightarrow k Δ l = \frac{4}{3 k} π R^{3} ρ g$

For equilibrium of the complete system, net force of buoyancy must be equal to the total weight of the sphere which holds true in the given problem. So both the spheres are completely submerged.

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