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. $m g \tan \theta$
• B. $-m g \sin \theta$
• C. $m g \sin \theta$
• D. $-m g \cos \theta$

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 $m g$ of the bob acting vertically downwards
(ii) the tension $T$ in the string acting along $P S$ Resolving $m g$ into two rectangular components, we get
(a) $m g \cos \theta$ acts along $P A$, opposite to tensions, we get
(b) $m g \sin \theta$ acts along $P B$, tangent to the arc $O P$ and directed towards $O$.
If the string neither slackens nor breaks but remains taut, then
$T=m g \cos \theta$
The force $m g \sin \theta$ tends to bring the bob back to its mean position $O$.
$\therefore $ Restoring force acting on the bob is
$F=-m g \sin \theta$