First slide

Power

Question

A pump motor is used to deliver water at a certain rate from a given pipe. To obtain ‘n’ times water from the same pipe in the same time, by what amount the power of the motor should be increased?

Moderate

A

$n^{2} times$
B

$n^{3} times$
C

n times
D

$n^{\frac{1}{2}} times$

Solution

If a liquid of density $ρ$ is flowing through a pipe of cross-section A at speed V, the mass coming out per second will be:
$(\frac{dm}{dt}) = AVρ$
In order to get n times water in the same time.
${(\frac{dm}{dt})}^{'} = n (\frac{dm}{dt}) i . e ., A' V^{'} ρ^{'} = nAVρ$
But as pipe and liquid are same, so $ρ^{'} = ρ, A^{'} = A$
$v^{'} = NV$
So $\frac{F^{'}}{F} = \frac{V^{'} (\frac{dm}{dt})}{V (\frac{dm}{dt})} = \frac{nV (\frac{ndm}{dt})}{V (\frac{dm}{dt})} = n^{2}$
or $F^{'} = n^{2} F$
$\frac{P^{'}}{P} = \frac{F^{'} V^{'}}{FV} = \frac{(n^{2} F) (nV)}{FV} or P^{'} = n^{3} P$

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