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 (a) the force and (b) power of the motor should be increased ?

Explanation

Let `a=` area of cross section of pipe, `v=` velocity of flow of water, `rho = ` density of water.
`:.` Mass of water flowing out `//` sec `=(dm)/(dt)`
`=avrho`
To get n times water in the same time,
`(dm//dt^('))=n(dm//dt):.a^(')v^(')rho^(')=n(avrho)`
But `rho^(')=rho and a^(')=a(` pipe is same )
`:. v^(') = nv` ...(i)
(a) As `F=v(dm)/(dt)`
`:. (F^('))/(F)=(v^(')(dm//dt)^('))/(v(dm//dt))=(nv(n(dm)/(dt)))/(vdm//dt)=n^(2)`
`:. F^(')=n^(2)F`
(b) Also, `(P^('))/(P)=(F^(')v^('))/(Fxxv)=(n^(2)Fxxnv)/(Fxxv)=n^(3)`
Therefore, `P^(')=n^(3)P`