Answer
The angular velocity is $~~18~rad/s$
Work Step by Step
We can find the initial rotational inertia of the system:
$I_0 = \frac{1}{2}mr^2+\frac{1}{2}(10~m)(3.0~r)^2$
$I_0 = \frac{1}{2}mr^2+\frac{1}{2}~(90~m~r^2)$
$I_0 = \frac{91~m~r^2}{2}$
We can find the final rotational inertia of the system:
$I_f = \frac{1}{2}mr^2+m(2r)^2+\frac{1}{2}(10~m)(3.0~r)^2$
$I_f = \frac{9}{2}mr^2+\frac{1}{2}~(90~m~r^2)$
$I_f = \frac{99~m~r^2}{2}$
We can use conservation of angular momentum to find the final angular velocity:
$I_f~\omega_f = I_0~\omega_0$
$\omega_f = \frac{I_0~\omega_0}{I_f}$
$\omega_f = \frac{\frac{91~m~r^2}{2}~\omega_0}{\frac{99~m~r^2}{2}}$
$\omega_f = \frac{91~\omega_0}{99}$
$\omega_f = \frac{(91)~(20~rad/s)}{99}$
$\omega_f = 18~rad/s$
The angular velocity is $~~18~rad/s$
