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

(a) `sin omegat - cos omegat`
`sqrt(2)((1)/(sqrt(2)) sin omegat - (1)/(sqrt(2)) cos omegat)`
`sqrt(2)( cos (pi)/(4)sin omegat - sin (pi)/(4) cos omegat)`
`sqrt(2)sin(omegat - (pi)/(4))`
the function is simple harmonic with period
`((2pi)/(omega))`
(b) `sin^(3)omegat = (1)/(4)(3 sin omegat - sin 3omegat)`
sin `omegat` and sin 3 `omegat` individually are simple harmonic but their combination is not simple harmonic but only periodic with period of `(2pi)/(omega)`.
( c) `3 cos ((pi)/(4) - 2omegat)`
`3 cos (2omegat - (pi)/(4))`
It represents simple harmonic function with time period of `(pi)/(omega)`.
(d) It is periodic but not simple harmonic.
(e ) exp`(omega^(2)t^(2))` never repeats itself so it is neither periodic nor simple harmonic.
(f ) It is niether periodic nor simple harmonic.