Question

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

• A. (\sqrt{2\pi} r)
• B. (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)