Consider a cuboidal vessel $(2 R \times 2 R \times R)$ with a hemispherical cavity of radius R, kept at a horizontal smooth surface as shown in the figure. The vessel has very small hole of cross – sectional area “ $a$ ”. Now water of density $ρ$ is poured into space developed due horizontal surface and vessel through hole very slowly. When the height of water level is h, the vessel lifts off the surface and liquid leaks through space generated. (R = 2m, $ρ = 10^{3} k g / m^{3}$ , m=mass of the vessel $= 9 π k g$ ). If the value of h is 5k cm, find the value of k. (Atmospheric pressure = $10^{5} N / m^{2}$ ).

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answer is 6.

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Detailed Solution

$m g = F_{B} = [π R^{2} h - \int_{0}^{h} π (R^{2} - z^{2}) d z] p g \Rightarrow m = \frac{1}{3} π h^{3} p$

$\Rightarrow h^{3} = \frac{3 m}{π p} = \frac{3 \times π \times 9}{π \times 10^{3}} \Rightarrow h = \frac{3}{10} = 0.3 m = 30 c m$

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