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 $\mathrm{\alpha }$ is

 

When the rod falls through an angle $\mathrm{\alpha }$ the C.G. falls through a height h.
$$\begin{gathered} \text{ In }\mathrm{\Delta }{\mathrm{OB}}^{\mathrm{\prime }}\mathrm{B}\text{, } \\ \cos \mathrm{\alpha }=\frac{\left(\frac{\mathrm{L}}{2}-\mathrm{h}\right)}{\mathrm{L}/2} \\ \text{ i.e. }\mathrm{h}=\frac{\mathrm{L}}{2}(1-\cos \mathrm{\alpha }) \\ \mathrm{K}.\mathrm{E}. \mathrm{rotation} = \mathrm{Decrease} \mathrm{in} \mathrm{P}.\mathrm{E}. \\ \frac{{\displaystyle 1}}{{\displaystyle 2}}{\mathrm{l\omega }}^2=\mathrm{mgh} \\ \text{ i.e. }\frac{1}{2}\left(\frac{{\mathrm{mL}}^2}{3}\right){\mathrm{\omega }}^2=\mathrm{mg}\frac{\mathrm{L}}{2}(1-\cos \mathrm{\alpha })\text{ or }\mathrm{\omega }=\sqrt{\frac{6\mathrm{g}}{\mathrm{L}}}\sin \frac{\mathrm{\alpha }}{2} \end{gathered}$$