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