Question

A horizontal pipeline carries water in streamline flow. At $A_1 = 10$ cm$^2$, $v_1 = 1$ m/s and $P_1 = 2000$ Pa. The pressure at $A_2 = 5$ cm$^2$ is \dots [$\rho = 1000$ kg/m$^3$]

• A. 1000 Pa
• B. 750 Pa
• C. 500 Pa
• D. 250 Pa

Hint:

This is the Venturi effect in action! Wherever the pipe gets narrower, the fluid is forced to speed up (Continuity). As its kinetic energy increases, its static pressure must simultaneously drop to conserve total energy (Bernoulli).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We use the Equation of Continuity ($A_1 v_1 = A_2 v_2$) and Bernoulli’s Equation ($P + \frac{1}{2} \rho v^2 = \text{const}$).

Step 2: Formula Application:
Continuity: $10 \times 1 = 5 \times v_2 \implies v_2 = 2$ m/s. Bernoulli: $P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2$.

Step 3: Explanation:
$2000 + \frac{1}{2}(1000)(1)^2 = P_2 + \frac{1}{2}(1000)(2)^2$ $2000 + 500 = P_2 + 2000$ $P_2 = 500$ Pa.

Step 4: Final Answer:
The pressure at the second point is 500 Pa.