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, if the quantity of water flowing out per second from both the holes is same, then $R$ is equal to

• A. $\frac{L}{\sqrt{2\pi }}$
• B. $2\pi L$
• C. $L$
• D. $\frac{L}{2\pi }$

Answer 1

The correct option is A $\frac{L}{\sqrt{2\pi }}$

Let $a_1$ be the area of square of side $L$ and $a_2$ be the area of circular hole of radius $R$Velocity of efflux through $a_1$, $V_1=\sqrt{2gy}$
Velocity of efflux through $a_2$, $V_2=\sqrt{2g\left(4y\right)}=\sqrt{8gy}$
Since the discharge from both square and circular hole is same and same liquid is flowing through both the holes,
$a_1{V}_1=a_2{V}_2$
$L^2\sqrt{2gy}=\pi R^2\sqrt{8gy}$
$\Rightarrow R=\frac{L}{\sqrt{2\pi }}$