A thin rod \(MN\), free to rotate in the vertical plane about the fixed end \(N\), is held horizontal . When the end \(M\) is released the speed of this end, when the rod makes an angle \(\alpha \) with the horizontal, will be proportional to ( see figure)

Displacement of center of mass \( = \frac{1}{2}\cos \,\alpha \)

\(mg\frac{1}{2}\cos \alpha  = \frac{l}{2}I{\omega ^2}\,\)

\(mg\frac{1}{2}\cos \alpha  = \frac{{m{l^2}}}{6}{\omega ^2}\) (M.I. of thin uniform rod about an axis passing through its center of mass and perpendicular to the rod \(I = \frac{{m{l^2}}}{{12}}\))

\( \Rightarrow \omega  = \sqrt {\frac{{3g\cos \alpha }}{l}} \)

speed of end \( = \omega  \times l = \sqrt {3g\cos \alpha l} \)

i.e., Speed of end, \(\omega \, \propto \,\sqrt {\cos \alpha } \)