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) `sinomegat+cosomegat`
(ii) `sinomegat+cos2omegat+sin4omegat`
(iii) `e^(-omegat)`
(iv) `log(omegat)`

Explanation

`sin omega t + cos omega t = sqrt(2) [(1)/(sqrt(2))sin omega t +(1)/(sqrt(2)) cos omega t]`
`= sqrt(2) [sin omega t xx (1)/(sqrt(2)) + cos omega t xx (1)/(sqrt(2))]`
`= sqrt(2) [sin omega t cos "" (pi)/(4) + cos omega t sin (pi)/(4)]`
`= sqrt(2) [sin (omega t +(pi)/(4))]" " [therefore sin alpha cos beta pm cos alpha sin beta = sin (alpha pm beta)]`
`sin omega t+ cos omega t= sqrt(2) sin [omega t+(pi)/(4)]`
Comparing it with general equation of SHM,
`y= A sin [(2pi)/(T)t+ phi]`
This shows periodic motion. The periodic time `T= (2pi)/(omega)` and angular frequency `omega = (2pi)/(T)`.