

Problem 5.26: Lawn Sprinkler 

The sketch shows a lawn sprinkler with two horizontal arms of radial length R, at the termination of which are nozzles with exit area A_{2} and outward normal vectors in a horizontal plane pointing outward at an angle q relative to the tangent of a circumferential line, as shown.The sprinkler is free to rotate, but the bearing on which it is mounted exerts a torque kw in the direction opposing the rotation, w being the angular rate of rotation. A constant volume flow rate Q passes through the sprinkler, the water being incompressible at density r. (a) Derive an expression for the steadystate angular velocity w_{¥} of sprinkler in terms of the given quantities R, A_{2}, q, Q, r and k. In this steady state, what is the velocity vector of the fluid emerging from the nozzles, as seen by an observer in the nonrotating reference frame? What is the fluid velocity relative to the ground at the nozzle exit planes if the bearing is frictionless (k=0)? Comment. 