A large open tank has two holes in the wall. One is a square hole of side $L$ at a depth $y $ from the top and the other is a circular hole of radius $ R$ at a depth $ 4y $ from the top. When the tank is completely filled with water the quantities of water flowing out per second from both the holes are the same. Then $ R$ is equal to
IIT 2000,AIEEE 2012, Diffcult
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(b)Velocity of efflux when the hole is at depth h, $v = \sqrt {2gh} $
Rate of flow of water from square hole
${Q_1} = {a_1}{v_1}$= ${L^2}\sqrt {2gy} $
Rate of flow of water from circular hole
${Q_2} = {a_2}{v_2}$= $\pi {R^2}\sqrt {2g(4y)} $
According to problem ${Q_1} = {Q_2}$
==> ${L^2}\sqrt {2gy} = \pi {R^2}\sqrt {2g(4y)} $ $ \Rightarrow $$R = \frac{L}{{\sqrt {2\pi } }}$
Rate of flow of water from square hole
${Q_1} = {a_1}{v_1}$= ${L^2}\sqrt {2gy} $
Rate of flow of water from circular hole
${Q_2} = {a_2}{v_2}$= $\pi {R^2}\sqrt {2g(4y)} $
According to problem ${Q_1} = {Q_2}$
==> ${L^2}\sqrt {2gy} = \pi {R^2}\sqrt {2g(4y)} $ $ \Rightarrow $$R = \frac{L}{{\sqrt {2\pi } }}$
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