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
JEE MAIN 2013, Diffcult
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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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