Answer
$ \omega_B=12 rad/s \circlearrowleft$
Work Step by Step
We can determine the required angular velocity as follows:
$\alpha=\frac{d\omega}{dt}$
$\implies d\omega=\alpha dt$
$\implies \int_{\omega_{\circ}}^{\omega_A}=\int_0^t \alpha_A dt=\int_0^t 4t^3dt$
This simplifies to:
$\omega_A=t^4+20$
We plug in the known values to obtain:
$\omega_A=(2)^4+20=36rad/s$
We know that
$\omega_B r_B=\omega_A r_A$
This can be rearranged as:
$\omega_B=\frac{r_A}{r_B}\omega_A$
We plug in the known values to obtain:
$\omega_B=\frac{0.15}{0.05}(36)$
$\implies \omega_B=12 rad/s \circlearrowleft$
