Hydrodynamics : Calculate force exerted by fluid on Tube

[Blandongstein]

Homework Statement

A tube of uniform cross-section [itex]A[/itex] is bent to form a circular arc of radius [itex]R[/itex], forming three quarters of a circle. A liquid of density [itex]\rho[/itex] is forced through the tube with a linear speed [itex]v[/itex]. Find the net force exerted by the liquid on the tube.

2. The attempt at a solution

The centripetal force acting on a small element of [itex]dm[/itex] mass will be

[tex]dF=\frac{v^2}{R}dm=\frac{\rho v^2}{R}dV=\frac{\rho A v^2}{R}dl[/tex].

Integrating with proper limits, I got

[tex]F=\frac{\rho A v^2}{R}\int_{0}^{3\pi R/2}dl=\frac{3\pi A \rho v^2}{2}[/tex]

However the answer provided in my book is [itex]\sqrt{2} \rho Av^2[/itex] .What am I doing wrong. Can someone guide me?

 

[haruspex]

It says 'net force', but you added all the outward radial forces by magnitude, not as vectors. In fact, the shape doesn't matter. All you need to know is the direction of the flow at entry and exit. What is the change in momentum?