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)
$$T = 2\pi \sqrt{\dfrac{inertial\ factor}{spring\ factor}}$$
$$T = 2\pi \sqrt{\dfrac{m}{mg/1}}$$
$$\Rightarrow T = 2\pi \sqrt{\dfrac{1}{g}}$$