Question

A fluid flows through a pipe of diameter 5 cm with a velocity 2 m/s. If the pipe is constricted to a diameter of 3 cm, the velocity of fluid at the constriction is:

• A. 4.55 m/s
• B. 3.55 m/s
• C. 2.55 m/s
• D. 5.55 m/s

Hint:

In fluid dynamics, the continuity equation ensures that the flow rate is conserved. The velocity increases when the cross-sectional area decreases.

Answer 1

The Correct Option is D

We apply the continuity equation: \[ A_1 v_1 = A_2 v_2 \] Where: - \( A_1 = \pi r_1^2 \) and \( A_2 = \pi r_2^2 \) - \( v_1 = 2 \, \text{m/s} \) - \( r_1 = 2.5 \, \text{cm} \), \( r_2 = 1.5 \, \text{cm} \) Substitute the areas and velocities: \[ 6.25 \pi \times 2 = 2.25 \pi \times v_2 \] Cancel \( \pi \): \[ 12.5 = 2.25 v_2 \] Solve for \( v_2 \): \[ v_2 = \frac{12.5}{2.25} = 5.55 \, \text{m/s} \] Thus, the velocity at the constriction is 5.55 m/s.