Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (`omega` is any positive constant):
(a) `sinomegat-cosomegat`
(b) `sin^(3)omegat`
(c) `3cos(pi//4-2omegat)`
(d) `cosomegat+cos3omegat+cos5omegat`
(e) `exp(-omega^(2)t^(2))`
(f) `1+omegat+omega^(2)t^(2)`

Explanation

The function will represent a periodic motion, if it is identically repeated after a fixed interval of time and will represent SHM if it can be written uniquely in the form of `cos ((2pi)/(T)+phi)" or a "sin((2pi)/(T)+phi)`, where T is the time period.
`sin omega t - cos omega t = sqrt((1)/(sqrt(2)) sin omega t- (1)/(sqrt(2)) cos omega t`
`=[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 -cos alpha sin beta = sin (alpha -beta)]`
This is the form of `sin(omega t +phi)` hence it is SHM and its period `T= (2pi)/(omega)` and phase `phi = -(pi)/(4)rad " OR " phi = (7pi)/(4)`.