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?

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: (dmdt) = AVρIn order to get n times water in the same time.(dmdt)' = n(dmdt) i.e.,A'V'ρ' = nAVρBut as pipe and liquid are same, so ρ'= ρ, A' = A  v' = NVSo F' F   = V'(dmdt)V(dmdt) = nV(ndmdt)V(dmdt)  = n2or F' = n2F  P'P= F'V'FV = (n2F)(nV)FV or P' = n3P