Define an ideal simple pendulum. Show that, under certain conditions, simple pendulum performs linear simple harmonic motion.

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Solution

Ideal simple pendulum is defined as a heavy point mass suspended from a rigid support by a weightless and inextensible string and set oscillating under gravity through a small angle in a vertical plane.
Let a simple pendulum of length $L$ suspended from a rigid support $O$ . Let it is displaced by a small angle $θ$ in a vertical plane and released. Resolving $m g$ into horizontal and vertical components at point $B$ as $m g cos θ$ and $m g sin θ$ respectively.
We see that restoring force,
$F = - m g sin θ$
If ' $θ$ ' is small then
$sin θ = θ = \frac{x}{L}$
$F = - m g θ$
$= - m g \frac{x}{L}$
We see that $F \propto (- x)$
Since $F$ is directly proportional to negative of displacement so motion of a simple pendulum is in linear S.H.M.
So acceleration $= \frac{F}{m}$
$= \frac{- g x}{L}$
acceleration per unit displacement
$∣ ∣ \frac{a}{x} ∣ ∣ = \frac{g}{L}$
$T = \frac{2 π}{\sqrt{a c c e l e r a t i o n p e r u n i t d i s p l a c e m e n t}}$
$= \frac{2 π}{\sqrt{\frac{g}{L}}}$
$T = 2 π \sqrt{\frac{L}{g}}$

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