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.
$(1)$ $\sin \omega t - \cos \omega t$
$(2)$ $\sin^2 \omega t$

(A) $(1)$ $\sin \omega t - \cos \omega t$
$= \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin \omega t - \frac{1}{\sqrt{2}} \cos \omega t \right)$
$= \sqrt{2} \sin (\omega t - \pi/4)$
This function represents a simple harmonic motion with period $T = 2\pi/\omega$.
$(2)$ $\sin^2 \omega t = \frac{1 - \cos 2\omega t}{2} = \frac{1}{2} - \frac{1}{2} \cos 2\omega t$
This function is periodic but not simple harmonic because it represents a motion about an equilibrium position shifted by $1/2$. The period is $T = \pi/\omega$.