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  4. 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. The weight of hemispherical shell is m g and total mass of water is M=(2/3) π R3 ρ ⋅ The net external force is M g √(α/β)+(1+(m/M))2 Find (α+β).

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Q. 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 $\rho$ is filled in it completely. Minimum magnitude of force is to be applied to the shell for liquid not to escape from it. The weight of hemispherical shell is $m g$ and total mass of water is $M=\frac{2}{3} \pi R^{3} \rho \cdot$ The net external force is $M g \sqrt{\frac{\alpha}{\beta}+\left(1+\frac{m}{M}\right)^{2}}$ Find $(\alpha+\beta)$.Physics Question Image

Mechanical Properties of Fluids

Solution:

$F_{H}=(\rho g R) \pi R^{2}=\rho g \pi R^{3}=\frac{3 M g}{2}$
$F_{v}=(M+m) g$
$F_{\text {net }}=\sqrt{[(M+m) g]^{2}+\left(\frac{3 M g}{2}\right)^{2}}$
$F_{\text {net }}=M g \sqrt{\frac{9}{4}\left(1+\frac{m}{M}\right)^{2}}$