Problem Statement
Known
Water enters a rotating sprinkler through the center stem and leaves through identical nozzle openings at the arm tips.
Treat the water inside the sprinkler as incompressible and use a control volume that moves with the sprinkler head.
Find
Determine the exit speed of the water relative to each nozzle, then compare it with the tangential exit speed seen from a fixed frame.
Analysis
For steady incompressible flow through a moving control volume, the total inflow must equal the total outflow. With \(N\) identical exits,
\[ Q = N A_2 W_2 \]where \(W_2\) is measured relative to the nozzle. Therefore,
\[ W_2 = \frac{Q}{N A_2} \]The rotating nozzle has tangential speed \(U=\omega R\). For this sprinkler orientation, the fixed-frame tangential exit component is modeled as
\[ V_{\theta 2}=W_2-U \]