Question

A simple pendulum of length $l$ and mass (bob) $m$ is suspended vertically. The string makes an angle $\theta$ with the vertical. The restoring force acting on the pendulum is

• A. $mg\tan \theta$
• B. $-mg\sin \theta$
• C. $mg\sin \theta$
• D. $-mg\cos \theta$

Answer 1

Answer: (B)

When the bob is displaced to position $P$, through a small angle $\theta$ from the vertical, the various forces acting on the bob at $P$ are
(i) the weight $mg$ of the bob acting vertically downwards
(ii) the tension $T$ in the string acting along $PS$ Resolving $mg$ into two rectangular components, we get
(a) $mg\cos \theta$ acts along $PA$, opposite to tensions, we get
(b) $mg\sin \theta$ acts along $PB$, tangent to the arc $OP$ and directed towards $O$.
If the string neither slackens nor breaks but remains taut, then
$T=mg\cos \theta$
The force $mg\sin \theta$ tends to bring the bob back to its mean position $O$.
$\therefore$ Restoring force acting on the bob is
$F=-mg\sin \theta$