Question

The motion of a simple pendulum is

• A. exactly simple harmonic
• B. exactly rectilinear
• C. approximately simple harmonic
• D. Simple harmonic for very small angular displacements

Answer 1

Answer: (C), (D)

The correct options are
C approximately simple harmonic
D Simple harmonic for very small angular displacements

In pendulum, $mglsin\left(\theta \right)$ is restoring torque.

$\Rightarrow Torque=-mglsin\left(\theta \right)$

$\Rightarrow I\alpha =-mglsin\left(\theta \right)$, Where $I=m{l}^2$ and $\alpha$ is angular acceleration.

$\because \theta$ is very small so $\sin \left(\theta \right)\approx \theta$,

$\Rightarrow I\alpha =-mgl\theta$ $\Rightarrow m{l}^2\alpha =-mgl\theta$ $\Rightarrow \alpha =-\frac{g}{l}\theta$

$\Rightarrow \alpha =-{\omega }^2\theta$,

If $\theta$ is very large then $sin\left(\theta \right)\ne \theta$,

So for very small angular displacement pendulum is simple harmonic. and we assume $sin\left(\theta \right)\approx \theta$ so it is approximately simple harmonic.