Question

A spray pipe has a cylindrical tube of radius \(R\). It has \(n\) small holes of radius \(r\) at one end. The liquid flows through the tube with velocity \(V\). The velocity of the liquid through the holes is

• A. \( \dfrac{VR^2}{nr^2} \)
• B. \( \dfrac{Vr}{nR} \)
• C. \( \dfrac{VR}{nr} \)
• D. \( \dfrac{Vr^2}{nR^2} \)

Hint:

Always use the continuity equation for liquid flow problems involving pipes and holes.

Answer 1

The Correct Option is A

Step 1: Applying equation of continuity.
For incompressible flow, volume flow rate remains constant.
\[ A_1 v_1 = A_2 v_2 \]
Step 2: Area of the tube.
\[ A_1 = \pi R^2 \]
Step 3: Total area of the holes.
Each hole has area \( \pi r^2 \). For \(n\) holes:
\[ A_2 = n \pi r^2 \]
Step 4: Substituting in continuity equation.
\[ \pi R^2 \cdot V = n \pi r^2 \cdot v \]
Step 5: Solving for velocity through holes.
\[ v = \frac{VR^2}{nr^2} \]
Step 6: Conclusion.
The velocity of liquid through the holes is \( \dfrac{VR^2}{nr^2} \).