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.