Engineering Mechanics: Statics & Dynamics (14th Edition)

$\omega_A=32 rad/s$
We can determine the required angular velocity as follows: We know that $\vec{v_B}=\vec{v_E}+\vec{\omega_A}\times \vec{r_{B/E}}~~$[eq(1)] The velocity of $E$ is given as $\vec{v_E}=\vec{\omega_{B}}\times \vec{r_E}$ $\implies \vec{v_E}=2\hat k\times (-20\hat j)=40\hat i$ The velocity from $E$ to $B$ is given as $\vec{v_{B/E}}=5\hat j$ $\vec{v_B}=\vec{\omega_{BC}}\times \vec{r_{BC}}$ $\implies \vec{v_B}=-8\hat k\times (-15\hat j)$ $\implies \vec{v_B}=-120\hat i$ We plug in the known values in eq(1) to obtain: $-120\hat i=40\hat i+(\omega_A)\hat \times 5\hat j$ $\implies -120\hat i=(40-5\omega_A)\hat i$ Equating $i$ components on both sides, we obtain: $-120=40-5\omega_A$ This simplifies to: $\omega_A=32 rad/s$