# A rid of length l and mass is fixed at point O as shown in the figure and it is free to rotate about that point . Find the frequency (w) for small angular oscillation

$(a)\;\sqrt{\large\frac{3g}{2L}}\qquad(b)\;\sqrt{\large\frac{g}{L}}\qquad(c)\;\sqrt{\large\frac{4g}{5L}}\qquad(d)\;\sqrt{\large\frac{3g}{L}}$

Answer : (a) $\;\sqrt{\large\frac{3g}{2L}}$
Explanation :
As it is displaced by small angle $\;\theta$
$mg \sin \theta\times \large\frac{L}{2}=I \alpha$
$mg \sin \theta\times \large\frac{L}{2}=\large\frac{mL^3}{3} \alpha \quad$ for small values of $\;\theta \; \sin \theta \approx \theta$
$mg \theta \times \large\frac{L}{2}=\large\frac{L}{3}\times \alpha$
$\alpha=\large\frac{3g}{2L}\;\theta$
$w=\sqrt{\large\frac{3g}{2L}}$
edited Jan 12