Rankers Physics
Topic: Solid and Fluids
Subtopic: Fluid Dynamics

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 quantities of water flowing out per second from the holes are both same. Then, R is equal to:
\[\frac{L}{\sqrt{2\pi}}\]
\[2\pi L\]
L
\[\frac{L}{2\pi}\]

Solution:

According to Torricelli's Law, the velocity of efflux from a hole at a depth h is given by $$v = \sqrt{2gh}$$.

Since the volume flow rate per second

\( Q = \text{Area}\times\text{Velocity} \) is the same for both holes:

$$A_{\text{square}} \cdot v_1 = A_{\text{circle}} \cdot v_2$$
$$L^2 \sqrt{2gy} = (\pi R^2) \sqrt{2g(4y)}$$
$$L^2 = \pi R^2 \cdot 2$$
$$R = \frac{L}{\sqrt{2\pi}}$$

Thus, the correct option is Option 1.

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