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 nay positive constant)
(1) `Sin^(3) omegat` (2) `3cos (pi//4 -2 omegat)`
(3) `cos omegat +cos 3 omegat + cos 5 omegat`
(4) `e^(-omega^(2)t^(2))` (5) `1 + omegat +omega^(2) t^(2)`

Explanation

(1) `sin^(3) omegat` is a periodic functions but not simple harmonic because `a prop -y` conditionis not satisfied. Its time period is `T = (2pi)/(omega)`
(2) `3cos (pi//4 -2omegat) = 3 cos (2 omegat - pi//4)` it is simple harmonic with a time period `T = (2pi)/(2omega) = (pi)/(omega)`
(3) `Cos omega t + cos 3omegat +cos 5 omegat` is a periodic function but not simple harmonic. The time periods of each periodic function are `(2pi)/(omega),(2pi)/(3omega)` and `(2pi)/(5omega)`.Since `(2pi)/(omega)` is the multiple of the other two periods. The given function is periodic with time period `(2pi)/(omega)`
(4) `e^(-omega^(2)t^(2))` is not periodic as `t` increases `e^(-omega^(2)t^(2))` decreasesa and tends to zero as `t rarr oo`
(5) `1+ omegat + omega^(2) t^(2)` is not periodic, as function increases with time without repetition.