Question

Which of the following function of time represent (a)Periodic and non-periodic motion ? give the period for each case of periodic motion .[w is any positive constant] give reason also
(i) sin wt+cos wt (ii)sin wt + cos 2wt +sin 4wt (iii)e^-wt (iv)log(wt)

Answer 1

$$\begin{gathered} \left(a\right)isaperiodicmotion.Itisbasicallysuperpositionof2SHM \\ y=\sin \omega t+\cos \omega t \\ =\sin \omega t+\sin (\frac{\pi }{2}-\omega t) \\ =2\sin \left(\frac{\omega t+\frac{\pi }{2}-\omega t}{2}\right)\cos \left(\frac{\omega t-\frac{\pi }{2}+\omega t}{2}\right) \\ =2\sin \left(\frac{\pi }{4}\right)\cos (\omega t-\frac{\pi }{4}) \\ sotheperiodicityisT=\frac{2\pi }{\omega }withSHMangualrfrequency\omega \\ \left(b\right)nonpeiodicanditissuperpositionof3wavesofdifferentfrequencies \\ \left(c\right)isafunctionwithadecaynature.Thesearecallednon-periodicTranscendentalfunctions. \\ \left(d\right)isanonperiodicfunction. \end{gathered}$$