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

Explanation

(i) `sin omega t + cos omega t` is a periodic function, it can also be written as `sqrt(2) sin (omega t + pi//4)`.
Now `sqrt(2) sin (omega t +pi//4) = sqrt(2) sin (omega t + pi//4 + 2 pi)`
`= sqrt(2) sin [omega(t + 2pi//omega)+pi//4]`
The peroidic time of the function is `2 pi//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, `sin omega t` has a period `T_(0) = 2pi//omega , cos 2 omega t` has a period `pi//omega = T_(0)//2` , and `sin 4 omega t` has a period `2pi//4 omega = T_(0)//4`. The period of the first term is a multiple of the periods of the last two terms. Therefore, the smallest interval two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is `T_(0)`, and thusm the sum is a periodic function with a period `2 pi//omega`
(iii) The function `e^(-omegat)` is not periodic, it decreases monotonically with increasing time and tends to zero as `t rarr infty` and thus, never repeats its value.
(iv) The function `log (omega t)` increases monotonically with time `t`. It, therefore, never repeats its value and is a non-periodic function. It may be noted that as `t rarr infty, log (omega t)` diverges to `infty`. It, therefore, cannot represent any kind of physical displacement.