A liquid of density $ρ$ is filled in a cylindrical vessel whose base radius is R. A wooden cylinder of height h, base radius $R / 2$ and density $ρ / 2$ is floating in the liquid. Minimum work to be done by external agent in completely immersing the cylinder into liquid is $\frac{n π R^{2} ρ g h^{2}}{128}$ . Find the value of n.

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answer is 3.

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$π {(\frac{R}{2})}^{2} x = [π R^{2} - π {(\frac{R}{2})}^{2}] y \Rightarrow \frac{x}{y} = y \times \frac{3}{4}$

$x = 3 y \dots . . (1) & x + y = \frac{h}{2} \dots . (2) \Rightarrow 3 y + y = \frac{h}{2} \Rightarrow y = \frac{h}{8} \Rightarrow x = \frac{3 h}{8} W_{e x t} = Δ M E = Δ K + Δ U = 0 + Δ U = Δ U = π {(\frac{R}{2})}^{2} x ρ g [\frac{x}{2} + \frac{h}{2} + \frac{y}{2}] - π {(\frac{R}{2})}^{2} h \frac{ρ}{2} g x = \frac{π R^{2}}{4} ρ g \times \frac{3 h}{8} \times \frac{4 h}{16} = \frac{3}{128} π R^{2} ρ g h^{2} A n s . n = 3$