Question

A vessel completely filled with water has two holes ' P ' and ' Q ' at depths ' 2 h ' and ' 8 h ' from the top respectively. Hole ' P ' is square of side ' \(a\) ' and hole ' \(Q\) ' is a circle of radius ' \(r\) '. The water flowing out per second from both the holes is same, then side ' \(a\) ' of hole ' \(P\) ' is

• A. \(\sqrt{2\pi r}\)
• B. \(\mathrm{r}\sqrt{2\pi}\)
• C. \(2\sqrt{\pi r}\)
• D. \(2\pi r\)

Hint:

Rate of flow (Discharge) = Area $\times$ Velocity.

Answer 1

The Correct Option is B

Step 1: Velocity of Efflux
$v = \sqrt{2gh}$.
$v_P = \sqrt{2g(2h)} = 2\sqrt{gh}$.
$v_Q = \sqrt{2g(8h)} = 4\sqrt{gh}$.

Step 2: Rate of Flow Equality
$A_P v_P = A_Q v_Q$.
$(a^2)(2\sqrt{gh}) = (\pi r^2)(4\sqrt{gh})$.

Step 3: Solve for a
$2a^2 = 4\pi r^2 \implies a^2 = 2\pi r^2$.
$a = r\sqrt{2\pi}$.
Final Answer: (B)