Question

A pump is used to deliver water at a certain rate from a given pipe. To obtain twice the volume of water from the same pipe in the same time, by what factor must the power of the motor pump be increased?

• A. 4
• B. 8
• C. 16
• D. 32

Hint:

Volume flow rate $Q \propto v$. Power $P \propto Q \cdot v^2$, so $P \propto v^3$. If velocity doubles, power increases $2^3 = 8$ times.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To deliver twice the volume in the same time, the velocity of the water must be doubled. Power for a pump is related to both the mass flow rate and the kinetic energy imparted to that mass.
Step 2: Key Formula or Approach:
1. Mass flow rate \( \frac{dm}{dt} = \rho A v \).
2. Power \( P = \frac{dK.E.}{dt} = \frac{1}{2} \left(\frac{dm}{dt}\right) v^2 = \frac{1}{2} (\rho A v) v^2 \propto v^3 \).
Step 3: Detailed Explanation:
If the volume required is doubled in the same time, then the new velocity \( v' = 2v \). Since Power \( P \) is proportional to the cube of velocity (\( P \propto v^3 \)): \[ \frac{P'}{P} = \left( \frac{v'}{v} \right)^3 = \left( \frac{2v}{v} \right)^3 \] \[ \frac{P'}{P} = 2^3 = 8 \]
Step 4: Final Answer:
The power of the motor pump must be increased by a factor of 8.