Meriam, Kraige & Bolton — Kinetics of Systems of Particles: Problem 4_54

4/54 The sprinkler is made to rotate at the constant angular velocity \(\omega\) and distributes water at the volume rate \(Q\). Each of the four nozzles has an exit area \(A\). Water is ejected from each nozzle at an angle \(\phi\) that is measured in the horizontal plane as shown. Write an expression for the torque \(M\) on the shaft of the sprinkler necessary to maintain the given motion. For a given pressure and thus flow rate \(Q\), at what speed \(\omega_0\) will the sprinkler operate with no applied torque? Let \(\rho\) be the density of water.