Question

A motor is used to deliver water at a certain rate through a given horizontal pipe. To deliver $n$-times the water through the same pipe in the same time the power of the motor must be increased as follows

• A. $n$-times
• B. $n^2$-times
• C. $n^3$-times
• D. $n^4$-times

Answer 1

Answer: (C)

If the motor pumps water (density $=p$ ) continuously through a pipe of area of cross-section $A$ with velocity $v$, then mass flowing out per second
$m=Avp$ ...(i)
Rate of increase of kinetic energy
$=\frac{1}{2}m{v}^2=\frac{1}{2}(Avp)v^2$ ...(ii)
Mass $m$, flowing out per sec, can be increased to $m^{\prime }$ by increasing $v$ to $v^{\prime }$ then power increases from $P$ to $P^{\prime }$.
$\frac{p^{\prime }}{p}=\frac{\frac{1}{2}A\rho v^{{}^{\prime }3}}{\frac{1}{2}A\rho v^3}$
or $\frac{p^{\prime }}{p}={\left(\frac{v^{\prime }}{v}\right)}^3$
Now, $\frac{m^{\prime }}{m}=\frac{A\rho v^{\prime }}{A\rho v}=\frac{v^{\prime }}{v}$
As $m^{\prime }=nm,v^{\prime }=nv$
$\therefore \frac{p^{\prime }}{p}=n^3$
$\Rightarrow p^{\prime }=n^{3}P$