JEE Main & Advanced Physics Simple Harmonic Motion Simple Pendulum

(1) An ideal simple pendulum consists of a heavy point mass body (bob) suspended by a weightless, inextensible and perfectly flexible string from a rigid support about which it is free to oscillate.

(2) But in reality neither point mass nor weightless string exist, so we can never construct a simple pendulum strictly according to the definition.

(3) Suppose simple pendulum of length \[l\] is displaced through a small angle \[\theta \] from it's mean (vertical) position. Consider mass of the bob is m and linear displacement from mean position is \[x\]

Restoring force acting on the bob

\[F=-mg\sin \theta \] or \[F=-mg\theta \]\[=-mg\frac{x}{l}\]      

(When \[\theta \] is small \[\sin \theta \tilde{-}\theta =\frac{\text{Arc}}{\text{Length}}=\frac{OP}{l}=\frac{x}{l}\])

\[\therefore \]\[\frac{F}{x}=\frac{-mg}{l}=k\,\] (Spring factor)

So \[T=2\pi \sqrt{\frac{\text{Inertia factor}}{\text{Spring factor}}}\]\[=2\pi \sqrt{\frac{m}{mg/l}}\]\[=2\pi \sqrt{\frac{l}{g}}\]