A uniform cylinder of length \(L\) and mass \(M\) having crosssectional area \(A\) is suspended, with its length vertical, from a fixed point by a massless spring such that it is half submerged in a liquid of density \(\sigma\) at equilibrium position. The extension \(x_0\) of the spring when it is in equilibrium is

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Correct Answer

(C) \(\;\frac{{Mg}}{k}\left( {1 - \frac{{LA\sigma }}{{2M}}} \right)\)

Detailed explanation

From figure, \(k{x_0} + {F_B} = Mg\)

\(k{x_0} + \sigma \frac{L}{2}Ag = Mg\)

\(\left[ {mass = density \times volume} \right]\)

\( \Rightarrow k{x_0} = Mg - \sigma \frac{L}{2}Ag\)

\( \Rightarrow {x_0} = \frac{{Mg - \frac{{\sigma LAg}}{2}}}{k} = \frac{{Mg}}{k}\left( {1 - \frac{{LA\sigma }}{{2M}}} \right)\)

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