A cylindrical vessel with a cross sectional area A has a small hole of area a at the bottom. A liquid is filled in the vessel upto a height H. Find the time taken to empty the tank.
A
t=aA√2Hg
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B
A√2Hg
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C
Aa√2Hg
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D
Aa√Hg
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Solution
The correct option is CAa√2Hg
Let at any time t, the height of the liquid be y, and in (t+dt) time, the level of the liquid will be (y−dy).
As per equation of continuity, A1v1=A2v2 ⇒a√2gy=A(−dydt) where, dydt is the rate of decrease in the level of liquid. y=0∫y=Hdy√y=−aA√2gt=T∫t=0dt ⇒[2√y]0H=−aA√2g[t]T0 ⇒T=Aa√2Hg is the time taken to empty the tank. Thus, option (c) is the correct answer.