A cylindrical vessel filled with water up to a height h weighs w and is resting on a horizontal plane. The side of wall of the vessel has a plugged circular hole touching the bottom. The coefficient of friction between the bottom of the vessel and the plane is μ. The minimum diameter of the hole so that the vessel just begins to move on the floor after the removal of the plug is

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A cylindrical vessel filled with water up to a height h weighs w and is resting on a horizontal plane. The side of wall of the vessel has a plugged circular hole touching the bottom. The coefficient of friction between the bottom of the vessel and the plane is $μ$ . The minimum diameter of the hole so that the vessel just begins to move on the floor after the removal of the plug is

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$\sqrt{\frac{πμw}{ρhg}}$

$\sqrt{\frac{μw}{2 πρhg}}$

$\sqrt{\frac{μw}{πρhg}}$

$\sqrt{\frac{2 μw}{πρhg}}$

answer is D.

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Detailed Solution

velocity of water through the hole: $u = \sqrt{2 gh}$
rate of mass flow: $R = ρAu$ [A = area of the hole]
reaction force on the wall: $F = R (u - 0) = {ρAu}^{2}$
For cylinder to just move,