Which of the following functions of time represent (a) periodic and (b) non-periodic motion. Give the period for each case of periodic motion (ω is any positive constant).

(i) sin ωt + cos ωt

(ii) sin ωt + cos 2ωt + sin 4ωt

(iii) e^-ωt

(iv) log(ωt)

(i) sin ωt + cos ωt is a periodic function. It can also be written as √2 sin(ωt + π/4).

√2 sin(ωt + π/4) = √2 sin(ωt + π/4 + 2π) = √2 sin[ω(t + 2π/ω) + π/4]

∴ The periodic time of the function is 2π/ω.

(ii) This is an example of a periodic motion. Each term represents a periodic function with a different angular frequency.

sin ωt has a period 2π/ω; cos 2ωt and sin 4ωt have periods π/ω and π/2ω respectively.

The period of the first term is a multiple of the last two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is 2π/ω.

∴ Period = 2π/ω

(iii) The function e^-ωt is not periodic. It decreases monotonically with increasing time and tends to zero as t tends to infinity, thus never repeats its value.

(iv) The function log(ωt) increases monotonically with time t. It never repeats its value and is a non-periodic function.