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Question

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 μ 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 ρ):

A
2μmπρh
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B
μm2πρh
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C
μmρh
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D
none
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Solution

The correct option is A 2μmπρh
From Toricelli's theorem we get the velocity of the water from the hole as v=2gh
Impulse of force acting from the side of the vessel on out flowing water is
FΔt=Δ(mv)=(Δm)v(here Δm=ρAvΔt)
or
F=ρAv2=ρA(2gh)
Now the vessel to move this force F should be greater than the frictional force. Thus
2ρghA>μmg
or
A>μm2ρh
or
r>μmπ2ρh
or
d>2μmπρh

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