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: Understanding the Question:
The volume of water flowing out per second (discharge rate) is the same for two holes of different shapes at different depths.
Step 2: Key Formula or Approach:
1) Torricelli's Law: Velocity of efflux \(v = \sqrt{2gh}\).
2) Discharge rate \(Q = A \times v\), where \(A\) is the cross-sectional area of the hole.
Step 3: Detailed Explanation:
For hole P (square):
Depth \(h_P = 2h\)
Area \(A_P = a^2\)
Velocity \(v_P = \sqrt{2g(2h)} = \sqrt{4gh} = 2\sqrt{gh}\)
Rate \(Q_P = a^2 \times 2\sqrt{gh}\)
For hole Q (circular):
Depth \(h_Q = 8h\)
Area \(A_Q = \pi r^2\)
Velocity \(v_Q = \sqrt{2g(8h)} = \sqrt{16gh} = 4\sqrt{gh}\)
Rate \(Q_Q = \pi r^2 \times 4\sqrt{gh}\)
Given \(Q_P = Q_Q\):
\[ a^2 \times 2\sqrt{gh} = \pi r^2 \times 4\sqrt{gh} \]
Cancel \(2\sqrt{gh}\) from both sides:
\[ a^2 = 2 \pi r^2 \]
Taking the square root:
\[ a = \sqrt{2\pi} r \]
Step 4: Final Answer:
The side '\(a\)' of hole 'P' is \(\sqrt{2\pi} r\).