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A solid sphere of radius R and density rho is attached to one end of a massless spring of stiffness k. The other end of the spring connects to a second solid sphere of radius R but density 3*rho. The whole assembly is placed in a liquid of density 2*rho and reaches equilibrium. Which statement(s) is/are correct?
- the net elongation of the spring is 4*pi*R³*rho*g/(3k).
- the net elongation of the spring is 8*pi*R³*rho*g/(3k).
- the light sphere is partially submerged.
- the light sphere is completely submerged.
Correct answer: the net elongation of the spring is 4*pi*R³*rho*g/(3k).
Solution
The heavy sphere (density 3 rho) sinks because it is denser than the liquid (2 rho), while the light sphere (density rho) is buoyant and floats up, fully submerged, stretching the connecting spring. Force balance on the light sphere: buoyancy = weight + spring tension. The net force imbalance is V*(2 rho - rho)*g = V*rho*g where V = (4/3) pi R³, giving spring elongation x = V*rho*g/k = 4 pi R³ rho g/(3k). The light sphere is completely submerged.
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