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Question

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.

Solution
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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)
$$T = 2\pi \sqrt{\dfrac{inertial\ factor}{spring\ factor}}$$

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

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

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