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 ($\frac{π}{4}$ – 2ωt)

(d) cos ωt + cos 3ωt + cos 5ωt

(e) exp (–ω 2 t 2 )

(f) 1+ωt+ω2 t2

Answer 1

SHM

The given function is:

$sin\, ωt-cos \,ωt$

$=\sqrt{2}[\frac{1}{\sqrt2}sin\,ωt-\frac{1}{\sqrt2}cos\,ωt]$

$=\sqrt2[sin\,ωt×cos\frac{\pi}{4}-cos\,ωt×sin\frac{\pi}{4}]$

$=\sqrt2\,sin\,(ωt-\frac{\pi}{4})$

This function represents SHM as it can be written in the form: a sin (ωt+ϕ)

Its period is: $\frac{2\pi}{ω}$

Periodic, but not SHM

The given function is:

$sin^3 ωt$

$=\frac{1}{2}[3\\.sin\,sin^3 ωt-sin\,3 \,sin^3 ωt]$

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.

**SHM **

The given function is:

$=3\,cos[\frac{\pi}{4}-2ωt]$

$=3\,cos[2ωt-\frac{\pi}{4}]$

This function represents simple harmonic motion because it can be written in the form: a (ωt+ϕ)

a cos (ωt+ϕ)

Its period is: $\frac{2\pi}{2ω}=\frac{\pi}{ω}$

Periodic, but not SHM

The given function is cos ω+cos 3ωt+ cos 5ωt. Each individual cosine function represents SHM.

However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.

Non-periodic motion

The given function exp(- ω2 t2) is an exponential function. Exponential functions do not repeat themselves.

Therefore, it is a non-periodic motion.

The given function 1+ωt+ω2 t 2 is non-periodic.