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

The correct option is C

$27$ times

Explanation for the correct option:

Power is defined as the rate of doing work. It is the work done in unit time. The SI unit of power is Watt ($\mathrm{W}$) which is joules per second (${\mathrm{Js}}^{-1}$).

Thus,

$\mathrm{P}=\frac{\mathrm{W}}{t}$

Where $\mathrm{P}$ is the power, $\mathrm{W}$ is the work done and $t$ is the time taken.

We know that work done is the transfer of energy and the kind of work that a pump does is imparting kinetic energy on the water.

Thus, here, $\mathrm{W}=\mathrm{KE}=\frac{1}{2}m{v}^2$, where $m$ is the mass of the object and $v$ is its velocity.

Substituting,

$\mathrm{P}=\frac{{\displaystyle \frac{1}{2}}m{v}^2}{t}$

To pump three times as much water, we need three times as much velocity as well, while time is being kept constant. Thus, new power required will be,

$\begin{array}{rcl}\mathrm{P}\text{'}& =& \frac{{\displaystyle \frac{1}{2}}\left(3m\right)\left(3v^2\right)}{t}\\ \mathrm{P}\text{'}& =& \frac{27\times {\displaystyle \frac{1}{2}}m{v}^2}{t}\\ \therefore \mathrm{P}\text{'}& =& 27\mathrm{W}\end{array}$

Therefore, option C is correct.