Q: 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. Then, ‘R’ is equal to :
(A) $\frac{L}{\sqrt{2\pi}}$
(B) $2 \pi L$
(C) $\sqrt{\frac{2}{\pi}} L$
(D) $\frac{L}{2\pi}$
Ans: (C)
Solution: Let v1 and v2 be the velocity of efflux from square and circular hole respectively. A1 and A2 be cross-section areas of square and circular holes.
$v_1 = \sqrt{8 g y}$ ; $v_2 = \sqrt{2 g y}$
The volume of water coming out of square and circular hole per second is
$ Q_1 = v_1 A_1 = \sqrt{8 g y} L^2 $
$ Q_2 = v_2 A_2 = \sqrt{2 g y} \pi R^2 $
$Q_1 = Q_2 $
$ R = \sqrt{\frac{2}{\pi}} L$