JEE Main & Advanced Physics Simple Harmonic Motion Simple Pendulum

Simple Pendulum

Category : JEE Main & Advanced

(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}}$

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