Question

Which of the following functions of time represent (a) periodic and $\left(b\right)$ non-periodic motion? Give the period for each case of periodic motion. [$\omega$ is any positive constant].

• A. $\frac{2\pi }{\omega }$
• B. $\sin \omega t+\cos 2\omega t+\sin 4\omega t$
• C. $e^{-\omega t}$
• D. $\log \left(\omega t\right)$

Answer 1

The correct option is A $\frac{2\pi }{\omega }$

$\begin{array}{c}TheexplanationofoptionA\\ Thegivenfunctionis\\ \sin \omega t-\cos \omega t\\ =\sqrt{2}\left[\frac{1}{\sqrt{2}}\sin \omega t-\frac{1}{\sqrt{2}}\cos \omega t\right]\\ =\sqrt{2}\left[\sin \omega t\times \cos \frac{\pi }{4}-\cos \omega t\times \sin \frac{\pi }{4}\right]\\ =\sqrt{2}\sin \left(\omega t-\frac{\pi }{4}\right).\\ ThisfunctionrepresentsSHMasitcanbewrittenintheform:a\sin \left(\omega t+\varphi \right)\\ Itsperiodis\frac{2\pi }{\omega }.\end{array}$

Hence, the option $A$ is the correct answer.