In the figure shown, the instantaneous speed of end I of the rod is v to the left. The angular velocity of the rod of length L must be

The velocities of the ends A and B along the length of rod should be the same.
$\text{ Hence, }{\mathrm{v}}_{\mathrm{A}}\cos {30}^{\circ }={\mathrm{v}}_{\mathrm{B}}\cos {30}^{\circ }\Rightarrow {\mathrm{v}}_{\mathrm{A}}={\mathrm{v}}_{\mathrm{B}}=\mathrm{v}$
Hence, the angular velocity of the rod is
$\mathrm{\omega }=\frac{{\left({\mathrm{v}}_{\mathrm{AB}}\right)}_{\perp }}{\mathrm{l}}=\frac{2\mathrm{vsin}{30}^{\circ }}{\mathrm{l}}\Rightarrow \mathrm{\omega }=\frac{\mathrm{v}}{\mathrm{l}}$