Question

Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is \( P_1 \). The reading of the pressure gauge falls to \( P_2 \) when the valve is opened. The speed of water flowing in the pipe is proportional to:

• A. \( \sqrt{P_1 - P_2} \)
• B. \( (P_1 - P_2)^2 \)
• C. \( (P_1 - P_2)^4 \)
• D. \( P_1 - P_2 \)

Hint:

For fluids in motion, Bernoulli's equation relates the pressure difference to the speed of the fluid, often expressed as \( v \propto \sqrt{P_1 - P_2} \).

Answer 1

The Correct Option is A

To solve the problem of finding the relationship between the speed of water flowing in a pipe and the pressure difference, we can apply Bernoulli's principle. Bernoulli's principle states that in a streamline flow of an ideal fluid, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is a constant. 

The general form of Bernoulli's equation for a horizontal flow (neglecting height-related potential energy difference) is:

\(P + \frac{1}{2} \rho v^2 = \text{constant}\)

Where:

• \(P\) is the pressure exerted by the fluid.
• \(\rho\) is the density of the fluid.
• \(v\) is the velocity of the fluid.

Now, when the valve is closed, the pressure is \(P_1\) and the velocity is zero because the fluid is at rest. Thus, the Bernoulli equation becomes:

\(P_1 = \text{constant}\)

When the valve is opened, the pressure drops to \(P_2\) and the water starts to flow with velocity \(v\). Applying Bernoulli's equation, we get:

\(P_2 + \frac{1}{2} \rho v^2 = \text{constant}\)

Equating the two equations, we have:

\(P_1 = P_2 + \frac{1}{2} \rho v^2\)

Simplifying for the velocity \(v\), we get:

\(\frac{1}{2} \rho v^2 = P_1 - P_2\)

\(v^2 = \frac{2 (P_1 - P_2)}{\rho}\)

\(v = \sqrt{\frac{2 (P_1 - P_2)}{\rho}}\)

This shows that the velocity \(v\) is proportional to \(\sqrt{P_1 - P_2}\). Therefore, the correct answer is \(\sqrt{P_1 - P_2}\).

Answer 2

From Bernoulli's equation, the velocity \( v \) of the fluid is related to the pressure difference \( P_1 - P_2 \) by: \[ v = \sqrt{\frac{2(P_1 - P_2)}{\rho}}, \] where \( \rho \) is the density of the fluid. Thus, the speed of water is proportional to: \[ \sqrt{P_1 - P_2}. \]