The function will represent a periodic motion, if it is identically repeated after a fixed interval of time and will represent simple harmonic motion, if it can be written uniquely in the form of a $\cos \left( \frac{2 \pi}{T} t + \phi \right)$ or $\sin \left( \frac{2 \pi}{T} t + \phi \right)$whereT is the time period. (a) $\sin \omega t - \cos \omega t$ $= \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin \omega t - \frac{1}{\sqrt{2}} \cos \omega t \right)$$= \sqrt{2} \left( \sin \omega t \cos \frac{\pi}{4} - \cos \omega t \sin \frac{\pi}{4} \right)$ $= \sqrt{2} \sin \left( \omega t - \frac{\pi}{4} \right)$ ∴ It represents simple harmonicwith aperiod $T = \frac{2 \pi}{\omega}$ and a phase angle $\left(-\frac{\pi}{4}\right).$ (b) $\sin^{3} \omega t = \frac{1}{4} \left( 3 \sin \omega t - \sin 3 \omega t \right)$ It represents periodic but not simple harmonic motion. Its time period is $= \frac{2 \pi}{\omega}.$ (c) $3 \cos \left( \frac{\pi}{4} - 2 \omega t \right) = 3 \cos \left( 2 \omega t - \frac{\pi}{4} \right)$ It represents simple harmonic and its time period is $\frac{2 \pi}{2 \omega} = \frac{\pi}{\omega}$ (d) $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$ It represents periodic but not simple harmonic motion. Its time period is $\frac{2 \pi}{\omega}$ . It can be noted that each termrepresents a periodic function with a different angular frequency. Since period is the least interval of time after which a function repeats its value, $\cos \omega t$ has a period $T = \frac{2 \pi}{\omega}$ , $\cos 3 \omega t$ has a period $\frac{2 \pi}{3 \omega}$ , $\cos 5 \omega t$ has period $\frac{2 \pi}{5 \omega} = \frac{T}{5}$ , the last two forms repeat after any integral multiple of their period. Thus each term in the sum repeats itself after T, and hence the sum is a periodic function with a period $\frac{2 \pi}{\omega}$ (e) exp $(-\omega^{2} t^{2})$ : It is an exponential function which decreases monotonically with increasing time and tends to zero as $t \rightarrow \infty$ and thus never repeats itself. Therefore it represents non-periodic motion. (f) $1 + \omega t + \omega^{2} t^{2}$ It represents non-periodic motion (physically unacceptable because the function tends to infinity as $t \rightarrow \infty$).