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

Options

A. n-times
B. `n^(2)` -times
C. `n^(3)` times
D. `n^(4)` - times

Answer

C

Explanation

If the motor pumps water (density `rho` )
continuously through a pipe of
area of cross- section A with
velocity `upsilon`,s then mass flowing out per second.
` A upsilon rho`.....(i)
Rate of increase of kinetic energy
`=(1)/(2)m upsilon^(2)=(1)/(2)(A upsilon rho)v^(2)`
Mass m, flowing out per, sec, can be
increased to `m^(.)` by increasing `upsilon` be `upsilon^(.)` then power increases from P to `P^(.)`
`(P^(.))/(P)=((1)/(2)A rho upsilon^(.3))/(1/2 A rho upsilon^(3))or(P^(.))/(P)=((upsilon)/(upsilon))^(3)`
Now, `(m^(.))/(m)=(A rho upsilon^(.))/(A rho upsilon)=(upsilon^(.))/(upsilon)`
As, `m^(.)=nm, upsilon^(.)=n upsilon`
`(P^(.))/(P)=n^(3)`
or `P^(.)=n^(3)P.`