A rod of uniform mass and of length L can freely rotate in a vertical plane about an axis passing through O. The angular velocity of the rod when it falls from position P and P' through and angle α is

When the rod falls through an angle α the C.G. falls through a height h. In ΔOB′B,  cos⁡α=L2−hL/2  i.e. h=L2(1−cos⁡α) K.E. rotation = Decrease in P.E. 12lω2=mgh  i.e. 12mL23ω2=mgL2(1−cos⁡α)  or ω=6gLsin⁡α2