# 2.25 Problem Sets - Section 5

## Problem 5.26: Lawn Sprinkler

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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 *k**w* 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, 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 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.