Which of the following functions of time represents (a) simple harmonic motion and (b) periodic motion? Given the period for each case.
i. `Sin omega t - cos omegat` ii. `Sin^(2)omegat`

Explanation

i. `sin omegat - cos omegat = sqrt(2) [(1)/(sqrt(2))Sin omegat-(1)/(sqrt(2))Cos omegat]`
`= sqrt(2) [Sin omegat cos.(pi)/(4) -cos omega t "sin"(pi)/(4)]`
`= sqrt(2) sin (omegat - pi//4]`
This fuctions represent a simple harmonic motion having a period `T = (2pi)/(omega)` and a phase angle `(-pi//4)` or `(7pi//4)`.
ii) `Sin^(2) omegat =(1-cos.(2omegat))/(2) = (1)/(2) -(1)/(2) cos (2 omegat)`
The function is periodic having a period `T = pi//omega` It is not `SHM`
Note: If displacement of a particle moving along straight line is given by `y = A sin^(2)omegat` then it represents `SHM`.