Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the period for each case of periodic motion [`omega` is any positive constant].
(i) `sinomegat+cosomegat`
(ii) `sinomegat+cos2omegat+sin4omegat`
(iii) `e^(-omegat)`
(iv) `log(omegat)`

Explanation

(i) sin `omegat + cos omegat` is a periodic function, it can also be written as `sqrt2 sin (omegat + pi/4)`. Now `sqrt2 sin (omegat + pi//4)= 2 sin (omegat + pi//4+2pi)`
`=sqrt2sin[omega(t+2pi//omega)+pi//4]`
The periodic time of the function is `2pi//omega`.
(ii) This is an example of a periodic motion. It can be noted that each term represents a periodic function with a different angular frequency. Since period is the least interval of time after which a function repeats its value, `sinomegat` has a period `T_(0)= 2pi//omega , cos 2 omegat` has a period `pi//omega =T_(0)//2`, and `sin 4omegat` has a period `2pi//4omega = T_(0)//4`. The period of the first term is a multiple of the periods of the last two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is `T_(0)`, and thus, the sum is a periodic function with a period `2pi//omega`.
(iii) The function e–wt is not periodic, it decreases monotonically with increasing time and tends to zero as `t to oo` and thus, never repeats its value.
(iv) The function `log(omegat)` increases monotonically with time t. It, therefore, never repeats its value and is a nonperiodic function. It may be noted that as `t to oo, log(omegat)` diverges to `oo`. It, therefore, cannot represent any kind of physical displacement.