A simple pendulum having a bob of mass $m$ undergoes small oscillations with amplitude $ heta_{0}$.

Question

A simple pendulum having a bob of mass $m$ undergoes small oscillations with amplitude $	heta_{0}$. Find the tension in the string as a function of the angle $	heta$ made by the string with the vertical.

Professor PC Thomas Classes · Answer

Suppose the speed of the bob at angle $	heta$ is $v$. Using conservation of energy between the extreme position and the position with angle $	heta$,\As the bob moves in a circular path, the force towards the centre should be equal to $m v^{2} / l$. Thus, 
$ T-m g \cos 	heta=m v^{2} / l .$
 Using (i), 
$ \begin{array}{ll} & T-m g \cos 	heta=2 m g\left(\cos 	heta-\cos 	heta_{0}ight) \ 	ext { or, } & T=3 m g \cos 	heta-2 m g \cos 	heta_{0} \end{array} $
 Now $\cos 	heta$ is maximum at $	heta=0$ and decreases as $|	heta|$ increases (for $|	heta|<90^{\circ}$ ). Thus, the tension is maximum when $	heta=0$, i.e., at the mean position and is minimum when $	heta= \pm 	heta_{0}$, i.e., at extreme positions.\