Properties Of Solids And Liquids Question 197

Question: 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-JEE (Screening) 2000]

Options:

A) $ 2\pi L $

B) $ \frac{L}{\sqrt{2\pi }} $

C) L

D) $ \frac{L}{2\pi } $

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Answer:

Correct Answer: B

Solution:

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} $

therefore$ L^{2}\sqrt{2gy}=\pi R^{2}\sqrt{2g(4y)} $

$ \Rightarrow $ $ R=\frac{L}{\sqrt{2\pi }} $



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