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A cylindrical vessel filled with water upto height of $H$ stands on a horizontal plane. The side wall of the vessel has a plugged circular hole touching the bottom. The coefficient of friction between the bottom of vessel and plane is $\mu$ and total mass of water plus vessel is $M$. What should be minimum diameter of hole so that the vessel begins to move on the floor if plug is removed (here density of water is $\rho$ )
$\sqrt {\frac{{2\mu M}}{{\pi \rho H}}} $
$\sqrt {\frac{{\mu M}}{{2\pi \rho H}}} $
$\sqrt {\frac{{\mu M}}{{\rho H}}} $
none
Solution
Force exerted by water $=\rho \mathrm{AV}^{2}$
$A=$ area of hole
$\mathrm{V}=$ velocity of water through hole Friction force $=\mu \mathrm{Mg}$ for the vessel to just move
$\rho \mathrm{AV}^{2}=\mu \mathrm{Mg}$
$\rho \times \frac{\pi \mathrm{D}^{2}}{4} \times 2 \mathrm{g} \mathrm{H}=\mu \mathrm{Mg}$
$\Rightarrow \mathrm{D}=\sqrt{\frac{2 \mu \mathrm{M}}{\pi \mathrm{pH}}}$
