Question

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):

(a) sin ωt – cos ωt

(b) sin³ ωt

(c) 3 cos (π/4 – 2ωt)

(d) cos ωt + cos 3ωt + cos 5ωt

(e) exp (–ω²t²)

(f) 1 + ωt + ω²t²

Answer 1

(a) SHM

The given function is:

This function represents SHM as it can be written in the form:

Its period is:

(b) Periodic, but not SHM

The given function is:

${\sin }^3\omega t=\frac{1}{4}\left[3\sin \omega t-\sin 3\omega t\right]$

The terms sin ωt and sin ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.

(c) SHM

The given function is:

This function represents simple harmonic motion because it can be written in the form:

Its period is:

(d) Periodic, but not SHM

The given function is . Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.

(e) Non-periodic motion

The given function is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

(f) The given function 1 + ωt + ω²t² is non-periodic.