Question

Derive an expression for Time period of Oscillation of a simple pendulum.

Answer 1

Time period $T=$ time required to complete one oscillation.

At equilibrium $T_0=mg$.

For small displacement $\theta$

Restoring force $=-mg\sin \theta$

for small $\theta$ $\sin \theta \approx \theta$

$\Rightarrow fe=-mg\theta =-mg\left(\frac{x}{l}\right)$

acceleration $a=\frac{fe}{m}=\frac{-g}{l}.x$

We know for $SHM$, $a={-w}^{2}x$

$\Rightarrow$ On comparing we get $w=\sqrt{\frac{g}{l}}$

$\therefore$ Time period of oscillation $T=\frac{2\Pi }{w}=2\Pi \sqrt{\frac{l}{g}}$.