Question

A rod of length 1m and mass 5 kg is attached to two springs as shown in figure in vertical plane. The rod is hinged about H point. If there is small angular displacement of rod from vertical position forcing it to do oscillation about H with angular frequency $\omega$, then value of [$\omega$] is : [Given K = 100 N/m and [.] is greatest integer function] ___

Answer 1

${\tau }_H=\left(KL\theta \right)L+\left(K\frac{L}{2}\theta \right)\frac{L}{2}-Mg\frac{L}{2}\theta$

$I_H\alpha =\frac{5K{L}^2\theta }{4}-\frac{MgL}{2}\theta \propto =\left[\frac{\frac{5K{L}^2}{4}-\frac{MgL}{2}}{\frac{M{L}^2}{3}}\right]\theta \therefore {\omega }^2=\frac{\frac{5KL}{4}-\frac{Mg}{2}}{\frac{ML}{3}}\omega =\sqrt{\frac{\frac{5KL}{4}-\frac{Mg}{2}}{\frac{ML}{3}}}$