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 quantity of water flowing out per second, from both the holes, is the 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 $\frac{L}{\sqrt{2\pi }}$

From continuity equation, we have,
$A_1{v}_1=A_2{v}_2$

The velocity of efflux from a tank is given by, $v=\sqrt{2gh}$
where $h$ is the depth of the outlet below the water surface.

$\Rightarrow A_1\sqrt{2g{h}_1}=A_2\sqrt{2g{h}_2}$

$A_1^2{h}_1=A_2^2{h}_2$

$\left(L^2{)}^2\right(y)=(\pi R^2{)}^2\left(4y\right)$

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

Hence, $\left(A\right)$ is the correct answer.

Why this Q ? Torricielli's law of efflux (velocity of efflux): The law states that the speed $v$ of efflux of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth $h$ is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height $h$, i.e. $v=\sqrt{2gh}$