Q: Which of the following functions of time represents (a) simple harmonic motion and (b) periodic motion ? Give the period for each case. (i) Sin ωt – cos ωt (ii) Sin^{2} ωt

Sol: (i) Sin ωt – cos ωt

$\large = \sqrt{2}[\frac{1}{\sqrt{2}}sin\omega t – \frac{1}{\sqrt{2}}cos\omega t]$

$\large = \sqrt{2}[sin\omega t cos\pi/4 – cos\omega t sin\pi/4 ]$

$\large = \sqrt{2}sin(\omega t – \pi/4 ) $

This function represent a simple harmonic motion having a period T = 2π/ω and a phase angle (-π / 4) or (7π/4).

(ii) $\large sin^2 \omega t = \frac{1-cos2\omega t }{2}$

$\large = \frac{1}{2} – \frac{1}{2} cos2\omega t $

The function is periodic having a period T = π/ω.