A thin walled hemispherical shell of mass m and radius R is pressed against a smooth vertical wall. Through a very small aperture at its top water of density ρ is filled in it completely. Minimum magn

Question

A thin walled hemispherical shell of mass m and radius R is pressed against a smooth vertical wall. Through a very small aperture at its top water of density ρ is filled in it completely. Minimum magnitude of force is to be applied to the shell for liquid not to escape from it. Weight of hemispherical shell is mg and total mass of water is M = 3 2 ​ π R 3 ρ :-

TestHub · Accepted Answer

Horizontal force = thrust on the circular base at depth R: $F_{H} = (ρ g R) π R^{2} = ρ g π R^{3} = \frac{3 M g}{2}$ (as $M = \frac{2}{3} π R^{3} ρ$ ) - so (A) incorrect

Vertical component supports the total weight: $F_{V} = (M + m) g$ - (C) correct

$F_{n e t} = (M + m)^{2} g^{2} + \frac{9 M ^{2} g ^{2}}{4} = M g \frac{9}{4} + (1 + \frac{m}{M})^{2}$ - (B) correct

$tan θ = \frac{F _{V}}{F _{H}} = \frac{2 ( M + m )}{3 M}$ - (D) incorrect

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