Engineering Mechanics: Statics & Dynamics (14th Edition)

$\omega_{AB}=2 rad/s$
We can determine the required angular velocity as follows: We know that $\vec{v_C}=\vec{v_B}+\vec{\omega_{BC}}\times \vec{r_{C/B}}$..eq(1) As $\vec{v_C}=-4\hat j$ and $\vec{r_{C/B}}=3cos30\hat i+3sin 30\hat j=2.597\hat i+1.5\hat j$ similarly, $\vec{v_B}=-(\omega_{AB}r_{AB})\hat j=-2\omega_{AB}\hat j$ We plug in the known values in eq(1) to obtain: $-4\hat j=-2\omega_{AB}\hat j+\omega_{BC} \hat k\times (2.598\hat i+1.5\hat j)$ $\implies -4\hat j=-1.5\omega_{BC}\hat i+(-2\omega_{AB}+2.598\omega_{BC})\hat j$ Comparing the $i$ component, we obtain: $0=-1.5\omega_{BC}$ $\implies \omega_{BC}=0 rad/s$ and comparing the $j$ component, we obtain: $-4=-2\omega_{AB}+2.598\omega_{BC}$ $\implies -4=-2\omega_{AB}+0$ This simplifies to: $\omega_{AB}=2 rad/s$