A liquid of density ρ is filled in a cylindrical vessel whose base radius is R. A wooden cylinder of height h, base radius R2 and density ρ2 is floating in the liquid. Minimum work to be done by external agent in completely immersing the cylinder into liquid is nπR2ρgh2128. Find the value of n.

Let block be displaced by x, liquid rises by y F=πR24(x+y)gρ∫dw=πR2ρg4∫(x+y)dxy=x3w=πR2ρg4×43∫03h/8 xdxw=3128πR2ρgh2

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A liquid of density $ρ$ is filled in a cylindrical vessel whose base radius is $R$ . A wooden cylinder of height $h$ , base radius $\frac{R}{2}$ and density $\frac{ρ}{2}$ 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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Detailed Solution

Let block be displaced by x, liquid rises by y $\begin{aligned} F = \frac{π R^{2}}{4} (x + y) g ρ \int d w = \frac{π R^{2} ρ g}{4} \int (x + y) d x \\ y = \frac{x}{3} w = \frac{π R^{2} ρ g}{4} \times \frac{4}{3} \int_{0}^{3 h / 8} x d x w = \frac{3}{128} π R^{2} ρ g h^{2} \end{aligned}$