A liquid of density p flows along a horizontal pipe of uniform cross - section A with a velocity v through a right angled bend as shown in fig. What force has to be exerted at the bend to hold the pipe in equilibrium?

Explanation

Change in momentum of mass `Deltam` of liquid as it passes through the bend

`Deltap =P_(f)-P_(i)`
`=Deltamv sqrt(2)`
`F =(DeltaP)/(Deltat) = sqrt(2) xx (Deltam)/(Deltat)`, [as `Deltam = rho ADeltaL`]
`F=sqrt(2)v(rho.ADeltaL)/(Deltat)`, [as `DeltaL//Deltat=v`]
`F=sqrt(2)rhoAv^(2)`
So the force to be applied at the bend to hold it in position is`sqrt(2)rhoAv^(2)` in the direction as shown in fig. (Otherwise pipe will move in opposite direction.)