Which of the following functions of time represent $(a)$ simple harmonic,$(b)$ periodic but not simple harmonic,and $(c)$ non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):
$(a)$ $\sin \omega t - \cos \omega t$
$(b)$ $\sin^3 \omega t$
$(c)$ $3 \cos (\pi/4 - 2 \omega t)$
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$
$(e)$ $\exp(-\omega^2 t^2)$
$(f)$ $1 + \omega t + \omega^2 t^2$

(A) $\sin \omega t - \cos \omega t = \sqrt{2} \sin(\omega t - \pi/4)$. This is $SHM$ with period $T = 2\pi/\omega$.
$(b)$ $\sin^3 \omega t = (3 \sin \omega t - \sin 3 \omega t)/4$. This is a superposition of two $SHM$s,hence periodic but not $SHM$. The period is $T = 2\pi/\omega$.
$(c)$ $3 \cos(\pi/4 - 2 \omega t) = 3 \cos(2 \omega t - \pi/4)$. This is $SHM$ with period $T = 2\pi/(2\omega) = \pi/\omega$.
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$. This is a superposition of three $SHM$s,hence periodic but not $SHM$. The period is $T = 2\pi/\omega$.
$(e)$ $\exp(-\omega^2 t^2)$ is a non-periodic motion as it decays to zero as $t \to \infty$.
$(f)$ $1 + \omega t + \omega^2 t^2$ is a non-periodic motion as it increases indefinitely with time.