Which of the following functions of time represent $(a)$ periodic and $(b)$ non-periodic motion? Give the period for each case of periodic motion $[\omega$ is any positive constant].
$(i)$ $\sin \omega t + \cos \omega t$
$(ii)$ $\sin \omega t + \cos 2\omega t + \sin 4\omega t$
$(iii)$ $e^{-\omega t}$
$(iv)$ $\log(\omega t)$

(N/A) $(i)$ $\sin \omega t + \cos \omega t$ can be written as $\sqrt{2} \sin(\omega t + \pi/4)$. Since $\sin(\omega t + \pi/4 + 2\pi) = \sin(\omega(t + 2\pi/\omega) + \pi/4)$,it is a periodic function with period $T = 2\pi/\omega$.
$(ii)$ $\sin \omega t + \cos 2\omega t + \sin 4\omega t$ is a periodic function. The individual periods are $T_1 = 2\pi/\omega$,$T_2 = 2\pi/(2\omega) = \pi/\omega$,and $T_3 = 2\pi/(4\omega) = \pi/(2\omega)$. The least common multiple of these periods is $T = 2\pi/\omega$. Thus,the sum is periodic with period $2\pi/\omega$.
$(iii)$ $e^{-\omega t}$ is a non-periodic function. It decreases monotonically as $t$ increases and tends to $0$ as $t \to \infty$,so it never repeats its value.
$(iv)$ $\log(\omega t)$ is a non-periodic function. It increases monotonically with time $t$ and diverges to $\infty$ as $t \to \infty$,so it never repeats its value.