question_answer

A rod of mass M and length L is hinged at its centre of mass so that it can rotate in a vertical plane. Two springs each of stiffness k are connected at its ends, as shown in the figure. The time period of SHM is 

A) \[2\pi \sqrt{\frac{M}{6k}}\]

B) \[2\pi \sqrt{\frac{M}{3k}}\]

C) \[2\pi \sqrt{\frac{ML}{k}}\]

D) \[\pi \sqrt{\frac{M}{6k}}\]

Correct Answer: A

Solution :

[a] The restoring torque \[\left( for\text{ }small\text{ }\theta  \right)\] \[{{\tau }_{rest}}=-\left[ \frac{kL\theta }{2}\times \frac{L}{2} \right]\times 2=\frac{k{{L}^{2}}}{2}(-\theta )\] \[\therefore \,\alpha =\frac{{{\tau }_{rest}}}{I}=\frac{k{{L}^{2}}/2}{M{{L}^{2}}/12}(-\theta )=\frac{6k}{M}(-\theta )\] \[\therefore \,\,T=2\pi \sqrt{\frac{M}{6k}}\].