Question

A horizontal pipe carries water in a streamline flow. At point along the pipe, where the cross-sectional area is \(A_1\), the velocity of water is \(V_1\) and the pressure is \(P_1\). What is the pressure of water at another point where the cross-sectional area is \(A_2\)?

• A. \(P_1-\frac{ρV_2^2}{2A_1^2}(A_2^2-A_1^2)\)
• B. \(P_1+\frac{ρV_1^2}{2A_2^2}(A_2^2-A_1^2)\)
• C. \(P_1ρV_1^2(A_1^2-A_2^2)\)
• D. \(P/ρV_1^2A_2^2\)

Hint:

Use the equation of continuity to get V2, then Bernoulli for a horizontal pipe.

Answer 1

The Correct Option is B

Step 1: Continuity
\(A_1V_1 = A_2V_2\), so \(V_2 = \frac{A_1V_1}{A_2}\).

Step 2: Bernoulli
For a horizontal pipe, \(P_1 + \frac12\rho V_1^2 = P_2 + \frac12\rho V_2^2\).

Step 3: Solve for P2
\[ P_2 = P_1 + \frac12\rho V_1^2\left(1 - \frac{A_1^2}{A_2^2}\right) = P_1 + \frac{\rho V_1^2}{2A_2^2}(A_2^2 - A_1^2) \]
Option (B).

Step 4: Check sense
If \(A_2 > A_1\), the water slows down and the pressure rises, which agrees with the sign above.

Final Answer:
The pressure at the second section is option B. \[ \boxed{\text{(B)}\ P_2=P_1+\frac{\rho V_1^2}{2A_2^2}(A_2^2-A_1^2)} \]