Question

A large vessel completely filled with water has two holes ‘A’ and ‘B’ at depths \( h \) and \( 4h \) from the top. Hole ‘A’ is a square of side \( L \) and hole ‘B’ is a circle of radius \( R \). If from both the holes same quantity of water is flowing per second, then side of square hole is

• A. \( 2\pi R \)
• B. \( \sqrt{2\pi R} \)
• C. \( \sqrt{2\pi}\,R \)
• D. \( \dfrac{R}{2} \)

Hint:

In fluid flow problems, discharge depends on both area and depth of hole.

Answer 1

The Correct Option is C

Step 1: Write Torricelli’s theorem.
Velocity of efflux at depth \( h \): \[ v = \sqrt{2gh} \]
Step 2: Write discharge formula.
\[ Q = A v \]
Step 3: Write discharge from both holes.
For hole A (depth \( h \), area \( L^2 \)): \[ Q_A = L^2 \sqrt{2gh} \]
For hole B (depth \( 4h \), area \( \pi R^2 \)): \[ Q_B = \pi R^2 \sqrt{2g(4h)} = \pi R^2 \cdot 2\sqrt{2gh} \]
Step 4: Equate discharges.
\[ L^2 \sqrt{2gh} = 2\pi R^2 \sqrt{2gh} \]
Step 5: Solve for \( L \).
\[ L^2 = 2\pi R^2 \Rightarrow L = \sqrt{2\pi}\,R \]
Step 6: Conclusion.
The side of square hole is \( \sqrt{2\pi}\,R \).