Answer
The angular speed after the cockroach stops is $~~4.2~rad/s$
Work Step by Step
We can find the rotational inertia of the cockroach:
$I_c = m~r^2$
$I_c = (0.17~kg)(0.15~m)^2$
$I_c = 0.003825~kg~m^2$
We can find the initial angular velocity of the cockroach:
$\omega_c = \frac{v}{r}$
$\omega_c = \frac{2.0~m/s}{0.15~m}$
$\omega_c = 13.33~rad/s$
Note that since the lazy Susan rotates clockwise, the angular velocity is negative.
We can use conservation of angular momentum to find the angular velocity after the cockroach stops:
$L_f = L_i$
$I_f~\omega_f = L_i$
$\omega_f = \frac{L_i}{I_f}$
$\omega_f = \frac{(0.003825~kg~m^2)(13.33~rad/s)+(5.0\times 10^{-3}~kg~m^2)(-2.8~rad/s)}{0.003825~kg~m^2+5.0\times 10^{-3}~kg~m^2}$
$\omega_f = 4.2~rad/s$
The angular speed after the cockroach stops is $~~4.2~rad/s$.
