Question

Water flows steadily through a horizontal pipe of a variable cross section. If the pressure of the water is $p$ at a point where the speed of the flow is $v$, what is the pressure at another point where the speed of the flow is $2v$; let the density of water be $\rho$

• A. $p + (3/2)\rho v^2$
• B. $p - 2\rho v^2$
• C. $p + 2\rho v^2$
• D. $p - 3\rho v^2$
• E. $p - (3/2)\rho v^2$

Hint:

In fluid dynamics, wherever the velocity of a fluid increases, the internal pressure decreases. This is the fundamental reason why the pressure is lower at the second point.

Answer 1

Answer: (E)

Concept:
For steady flow in a horizontal pipe, Bernoulli's Principle states: \[ P + \frac{1}{2}\rho v^2 = \text{constant} \]

Step 1: Apply Bernoulli's equation to the two points.
Let $P_2$ be the pressure at the second point. \[ p + \frac{1}{2}\rho v^2 = P_2 + \frac{1}{2}\rho (2v)^2 \]

Step 2: Solve for $P_2$.
\[ p + \frac{1}{2}\rho v^2 = P_2 + \frac{1}{2}\rho (4v^2) \] \[ p + \frac{1}{2}\rho v^2 = P_2 + 2\rho v^2 \] \[ P_2 = p + \frac{1}{2}\rho v^2 - 2\rho v^2 = p - \frac{3}{2}\rho v^2 \]