Example

Question:

Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case.
(1) \(\sin \omega t - \cos \omega t\)
(2) \(\sin^2 \omega t\)

Solution:

(a) \(\sin \omega t - \cos \omega t\)
This can be written as \[ \sin \omega t - \cos \omega t = \sqrt{2} \sin(\omega t - \pi/4) \] This is a simple harmonic motion with period \(T = \frac{2\pi}{\omega}\) and phase angle \(-\pi/4\) or \(7\pi/4\).

(b) \(\sin^2 \omega t = \frac{1}{2} - \frac{1}{2} \cos 2\omega t\)
This function is periodic, with period \(T = \frac{\pi}{\omega}\). It also represents a harmonic motion where the equilibrium occurs at \(1/2\) instead of zero.