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)

$n^3$-times

Answer 2

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 $=\frac{1}{2} m v^{2}=\frac{1}{2}(A v p) v^{2}$ ...(ii) Mass $m$, flowing out per sec, can be increased to $m'$ by increasing $v$ to $v'$ then power increases from $P$ to $P'$. $\frac{p'}{p}=\frac{\frac{1}{2} A \rho v^{' 3}}{\frac{1}{2} A \rho v^{3}}$ or $\frac{p'}{p}=\left(\frac{v'}{v}\right)^{3}$ Now, $\frac{m'}{m}=\frac{A \rho v'}{A \rho v}=\frac{v'}{v}$ As $m'=n m, v'=n v$ $\therefore \frac{p'}{p}=n^{3}$ $\Rightarrow p'=n^{3} P$