What is a simple pendulum? Show that the motion of the pendulum is SHM and hence deduce an expression for the time period of pendulum. Also, define Second's pendulum.

Simple pendulum is a point mass body suspended by a weightless thread or string from a rigid support about which it is free to oscillate.

Now restoring force acting on body can be given by
F = -mgsin$\theta$ (from figure)

If $\theta$ is small
F = -mg$\theta$ ( $\because sin\theta \approx \theta$)

Let P be any point an the path of pendulum and makes $\angle OMP=\theta$ which is small and arc OP be x (displacemebt) then
$\theta =\frac{OP}{1}=\frac{X}{1}$
$\Rightarrow F=\frac{-mgx}{1}$ ........(1)

$\Rightarrow$ force $\propto$ displacement and opposite in direction hence an SHM

Now this of form F = -kx...............(2)
$\Rightarrow$ k = mg/I (from (1) and (2)
$T=2\pi \sqrt{\frac{inertialfactor}{springfactor}}$

$T=2\pi \sqrt{\frac{m}{mg/1}}$

$\Rightarrow T=2\pi \sqrt{\frac{1}{g}}$