Question

An unsymmetrical sprinkler shown in the top view of the setup has frictionless shaft and equal fluid flows through each nozzle with a velocity of $10\, m / s$ relative to the nozzle. Its angular speed (in $rad / s$ ) of rotation is

Answer 1

Given: $A_{1} \,v_{1}=A_{2} \,v_{2}$
No external torque is acting,
$\therefore F_{1} r_{1}-F_{2} r_{2}=0$
$\rho A_{1} \,v_{1}^{2} \,r_{1}-\rho A_{2}\, v_{2}^{2} \,r_{2}=0$
$\left(\because \rho A_{1} \,v_{1}=\rho A_{2} \,v_{2}\right)$
or $v_{1} r_{1}-v_{2} r_{2}=0$
$\because v_{2}=10-\omega, v_{1}=(10+0.5 \omega)$
$\Rightarrow (10+0.5 \omega) \times 0.5-(10-\omega) \times 1=0$
$\Rightarrow 10+0.5 \omega-20+2 \omega=0$
$ 2.5 \omega=10 $
$\omega=4 \,rad / s$