Question

The power of a motor pump delivering water at a constant speed through a hose of radius \(r\) is \(P\). If the radius of the hose is doubled, then the power of the pump becomes}

• A. \(P\)
• B. \(2P\)
• C. \(4P\)
• D. \(8P\)
• E. \(16P\)

Hint:

If speed stays constant, flow rate is proportional to cross-sectional area: \[ A\propto r^2 \] So doubling radius makes the area four times.

Answer 1

The Correct Option is C

Power is the rate at which work is done. For a pump delivering water at constant speed, power is proportional to the mass of water delivered per second. Mass flow rate is: \[ \dot{m}=\rho A v \] Since speed is constant and density is constant: \[ \dot{m}\propto A \] Area of cross section of the hose is: \[ A=\pi r^2 \] If radius is doubled: \[ r\to 2r \] Then new area becomes: \[ A'=\pi (2r)^2=4\pi r^2 \] So the mass flow rate becomes 4 times, and therefore power also becomes 4 times. Thus: \[ P' = 4P \]
Hence, the correct answer is: \[ \boxed{(C)\ 4P} \]