Liquid of density `rho` flows along a horizontal pipe of uniform area of cross section a with a velocity v through a right angled bend. What force should be applied to the bend to hold it in equilibrium?

Explanation

Change in momentum of mass `Delta m` of liquid as it passes through the bend
`Delta p=P_(t)-P_(i)`
`=Delta mv sqrt(2)`
`F=(Delta P)/(Delta t)=(sqrt(2))v(Delta m)/(Delta t) , [as Delta m = rho A Delta L]`
`F=sqrt(2)v((rho.A Delta L))/(Delta t), [as Delta L//Delta t=v]`
`F = sqrt(2)rho A v^(2)`
`therefore` So the force to be applied at the bend to hold it in position is `sqrt(2)rho A v^(2)` in the direction as shown in fig. (Otherwise pipe will move in opposite direction.)