There is a rod initially vertical(length l) and standing on a frictionless surface. The rod is given a small push and it starts to fall down. What should be the angular velocity after time t ?

My attempt:

Let the linear downward velocity of c.o.m be v

Let’s say the rod has angular velocity w at an instant(and making an angle \$theta\$ with vertical). Then the velocity of the bottom point of the rod due to the angular velocity is wl/2. Since this point has no vertical velocity, the vertical component of wl/2 should be equal to that of velocity of c.o.m. So it becomes wlsin(\$theta\$)/2 = v . Hence w = 2v/lsin(\$theta\$) . I don’t know what to do now. If I am able to get v as a function of time, I can do the rest. How to proceed now? Am I correct till this point ?