question_answer

A large open tank has two small holes in its vertical wall as shown in figure. One is a square hole of side 'L' at a depth '4y' from the top and the other is a circular hole of radius 'R' at a depth y from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same.

The, 'R' is equal to:

A) \[\frac{L}{\sqrt{2\pi }}\]

B) \[2\pi \,L\]

C) \[\sqrt{\frac{2}{\pi }}\,\,.\,\,L\]

D) \[\frac{L}{2\pi }\]

Correct Answer: C

Solution :

Let \[{{v}_{1}}\] and \[{{v}_{2}}\] be the velocity of efflux from square and circular hole respectively. \[{{S}_{1}}\] and\[{{S}_{2}}\] be cross-section areas of square and circular holes.

\[{{v}_{1}}=\sqrt{8gy}\]           and       \[{{v}_{2}}=\sqrt{2g\,(y)}\]

The volume of water coming out of square and circular hole per second is

\[{{Q}_{1}}={{v}_{1}}{{S}_{1}}=\sqrt{8gy}\,{{L}^{2}}\,\,;\,\,{{Q}_{2}}={{v}_{2}}{{S}_{2}}=\sqrt{2gy}\,\pi {{R}^{2}}\]

\[\because \]       \[{{Q}_{1}}={{Q}_{2}}\]

\[\therefore \]      \[R=\sqrt{\frac{2}{\pi }}\cdot L\]