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

(i)The given function is:`sinomegat-cosomegat` (ii) Periodic, but not SHM.The given function is:
`sin^(3)omegat=(1)/(4)[3sinomegat-sin3omegat]`
The terms`sinomegat` and `sin omegat`individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.
(iii) SHM `3cos[(pi)/(4)-2omegat]=[2omega-(pi)/(4)]`The given function is:
This function represents simple harmonic motion because it can be written in the form: `a cos(omegat+phi)`
Its period is: `(2pi)/(2omega)=(pi)/(omega)`
(iv) Periodic, but not SHM
The given function is `cosomegat+cos3omegat+cos5omegat`. Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.
(v) Non-periodic motion
The given function`exp(-omega^(2)t^(2))` is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.
(vi) The given function`1+omegat+omega^(2)t^(2)` is non-periodic.