Question

A pump motor is used to deliver water at a certain rate from a given pipe. To obtain thrice as much water from the same pipe at the same time, the power of the motor has to be increased to:
A. 2 times
B. 9 times
C. 27 times
D. 81 times

Answer 1

Hint:Specific pump power is the measurement of electric power needed to operate a pump, relative to the volume flow rate. Also, \[1{\text{ }}hp = 746{\text{ }}Watts\]
Power is defined as the rate of doing work; it is the work done in a unit of time. The standard unit of power is “Watt”.
In this question, the power of the motor is given, which is delivering the water from a pipe. First, find the power required to deliver water and then compare the equation with the equation of the power required to deliver three times more of the water.

Complete Step by Step Answer:
Power is the rate at which energy is consumed; it can be written as
\[P = \dfrac{W}{t} = \dfrac{{\dfrac{1}{2}m{v^2}}}{t} - - - (i)\]
Now the rate and amount of flow of water is increased by 3 times from the same pipe, and at the same time, hence we get the equation
\[P' = \dfrac{{\dfrac{1}{2}3m{{\left( {3v} \right)}^2}}}{t} - - - (ii)\]
Hence by comparing equation (i) and (ii), we can write,
\[
P' = \dfrac{{\dfrac{1}{2}3m{{\left( {3v} \right)}^2}}}{t} \\
 = 3 \times 9\dfrac{{\dfrac{1}{2}m{v^2}}}{t} \\
 = 27P \\
 \]
Hence we can say to obtain water which is three times more from the same pipe, the power of the motor should be increased by 27 times.
Option (C) is correct.

Note:The potential energy is the same as the work done in the same way kinetic energy is. To deliver the water, the pump needs to operate with power, and as the water increases, the power required to deliver water must also be increased.