The sketch shows a lawn sprinkler with two horizontal arms of radial length R, at the termination of which are nozzles with exit area A₂ 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 steady-state angular velocity w_¥ of sprinkler in terms of the given quantities R, A₂, 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 non-rotating reference frame? What is the fluid velocity relative to the ground at the nozzle exit planes if the bearing is frictionless (k=0)? Comment.

(b) Now consider the startup of the rotation. Let the rotating arms of the sprinkler have a total mass m per unit length in the radial direction (kg/m), including the solid parts and the water contained therein. Suppose the sprinkler is turned on at t=0 in a static state. Derive a differential equation for its angular velocity, making whatever approximations you consider appropriate, and obtain a solution for w(t).