question_answer

Two masses m and \[\frac{m}{2}\] are connected at the two ends of a massless rigid rod of length l. The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring torque is \[\tau \,\,=\,\,k\theta \] for angular displacement \[\theta \]. If the rod is rotated by \[{{\theta }_{0}}\] and released, the tension in it when it passes through its mean position will be: [JEE Main Online Paper (Held On 09-Jan-2019 Morning]

A) \[\frac{2\,k\,{{\theta }_{0}}^{2}}{l}\]

B) \[\frac{k\,{{\theta }_{0}}^{2}}{l}\]

C) \[\frac{3k\,{{\theta }_{0}}^{2}}{l}\]

D) \[\frac{k\,{{\theta }_{0}}^{2}}{2\,l}\]

Correct Answer: B

Solution :

\[\frac{1}{2}k{{\theta }_{0}}^{2}=\,\,\,\frac{1}{2}\,\left( \frac{m{{l}^{2}}}{9}\,\,+\,\,\frac{m}{2}\,\,\frac{4{{l}^{2}}}{9} \right){{\omega }^{2}}\] \[{{\omega }^{2}}\,\,=\,\,\,m\,\frac{3\,k\,{{\theta }^{2}}}{m{{l}^{2}}}\] \[T\,\,=\,\,m{{\omega }^{2}}\,\frac{l}{3}\,\,=\,\,m\frac{3k{{\theta }^{2}}}{m{{l}^{2}}}\,\,.\,\,\frac{l}{3}\,\,=\,\,\frac{k\,{{\theta }^{2}}_{0}}{l}\] Option is correct.