Answer
$$
\begin{aligned}
& \omega_A=47.8 \mathrm{rad} / \mathrm{s} \\
& \omega_B=7.78 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
Work Step by Step
$$
\begin{aligned}
& v_P=\omega_H r_H=100(50)=5000 \mathrm{~mm} / \mathrm{s} \\
& \omega_G=\frac{5000}{180}=27.78 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
Point $O$ is a fixed point of rotation for gears $A, E$, and $B$.
$$
\begin{aligned}
& \Omega=\omega_G+\omega_E=\{27.78 \mathbf{j}+30 \mathbf{k}\} \mathrm{rad} / \mathrm{s} \\
& \mathbf{v}_{P^{\prime}}=\Omega \times \mathbf{r}_{P^{\prime}}=(27.78 \mathbf{j}+30 \mathbf{k}) \times(-40 \mathbf{j}+60 \mathbf{k})=\{2866.7 \mathbf{i}\} \mathrm{mm} / \mathrm{s} \\
& \omega_A=\frac{2866.7}{60}=47.8 \mathrm{rad} / \mathrm{s} \\
& \mathbf{v}_{P^*}=\Omega \times \mathbf{r}_{P^{\prime \prime}}=(27.78 \mathbf{j}+30 \mathbf{k}) \times(40 \mathbf{j}+60 \mathbf{k})=[466.7 \mathbf{i}\} \mathrm{mm} / \mathrm{s} \\
& \omega_B=\frac{466.7}{60}=7.78 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
