A thin rod AB of length l is kept vertical on a horizontal floor such that end A is in contact with floor. If the rod is allowed to fall without slipping at end A, then its angular velocity just before hitting the ground is?

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Please find below the solution to the asked query:

If the rod is allowed to fall without slipping the point A, Let the angular velocity of the rod at the moment just before hitting the ground is ω and the speed of the COM of the rod is V_C.Then according to the conservation of energy,

$$\begin{gathered} ∆K=∆P \Rightarrow \frac{1}{2}m{V}_C^2+\frac{1}{2}I{\omega }^2 =mg\frac{l}{2} \Rightarrow \frac{1}{2}m{\left(\frac{l}{2}\omega \right)}^2+\frac{1}{2}\left(\frac{1}{3}m{l}^2\right){\omega }^2 =mg\frac{l}{2} \\ \Rightarrow \frac{1}{4}l{\omega }^2+\frac{1}{3}l{\omega }^2 =g \Rightarrow l{\omega }^2\left(\frac{1}{4}+\frac{1}{3}\right) =g \Rightarrow {\omega }^2 =\frac{12g}{7l} \\ \Rightarrow \omega =\sqrt{\frac{12g}{7l}} \end{gathered}$$

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