Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non periodic motion (omega is any positive constant):
.(i) `sinomegat-cosomegat`
(ii)`sin^(3)omegat`
(iii) `3cos((x)/(4)-2omegat)`
(iv)`cosomegat+cos3omegat+cos5omegat`
(v)`exp(-omega^(2)t^(2))`
(vi) `1+omegat+omega^(2)t^(2)`

Explanation

1. `sin omega t-cos omega t=sqrt(2)[sin omega t "cos "(pi)/4-cos omega t "sin"(pi)/4]`
`=sqrt(2)(omegat-(pi)/4)`
It is simple harmonic with a time period `T=(2pi)/(omega)`
2. `sin^(3) omegat` is a periodic function but not simple harmonic because `a alpha -y` condition ios not satisfied. Its time period is `T=(2pi)/(omega)`
3. `3cos (pi//4-2omegat)=3cos (2omegat-pi//4)` it is simple harmonic with a time period `T=(2pi)/(2omega0=(pi)/(omega)`
4. `cos omega t+cos 3 omegat + cos 5 omega t` is a periodic funtion 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 sum is periodic with time period `(2pi)/(omega)`
5. `e^(-omega^(2)r^(2))` is not periodic as t increases the function `e^(-omega^(2)t^(2))` decreases and tends to zero as `to to oo`
6. `1+omegat+omega^(2)t^(2)` is not periodic as function increases with increase in t with out repetition.