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 *ω**t *– cos *ω**t*

(b) sin^{3} *ω**t*

(c) 3 cos (π/4 – 2*ω**t*)

(d) cos ω*t *+ cos 3*ω**t** *+ cos 5*ω**t*

(e) exp (–*ω*^{2}*t*^{2})

(f) 1 + *ω**t *+ *ω*^{2}*t*^{2}

#### Solution 1

a) SHM

The given function is:

`sin omegat - cos omegat`

`= sqrt2[1/sqrt2 sin omegat - 1/sqrt2 cos omegat]`

`=sqrt2[sin omegat xx cos pi/4 - cos omegat xx sin pi/4]`

`= sqrt2 sin (omegat - pi/4)`

This function represents SHM as it can be written in the form: `asin (omegat + phi)`

Its period is : `(2pi)/omega`

b) Periodic, but not SHM

The given function is:

sin3ωt=143sinωt-sin3ωt

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

c) SHM

The given function is:

`3cos[pi/4 - 2omegat]`

`= 3cos[2omegat - pi/4]`

This function represents simple harmonic motion because it can be written in the form: `acos(omegat + phi)`

Its period is `(2pi)/(2omega) = pi/omega`

d)Periodic, but not SHM

The given function is `cos omegat + cos 3omegat + cos 5omegat` 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 `exp(-omega^2r^2)` is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

f) The given function 1 + *ω**t* + *ω*^{2}*t*^{2} is non-periodic.

#### Solution 2

The function will represent a periodic motion, if it is identically repeated after a fixed interval of time and will represent S.H.M if it can be written uniquely in the form of a cos

`((2pit)/T + phi) or a sin ((2pit)/T + phi)` , where T is the time period

a) `sin omegat - cos omegat = sqrt2[1/sqrt2 sin omegat - 1/sqrt2 cos omegat]`

`= sqrt2[sin omegat cos pi/4 - cos omegat sin pi/4]`

`= sqrt2 sin (omegat - pi/4)`

It is a S.H.M and its Period is `2 pi"/"omega`

b) `sin^3 omegat = 1/3 [3sin omegat - sin 3omegat]`

Here each term `sin omegat` and `sin 3 omegat` individually represent S.H.M. But ii which is the outcome of the superposition of two S.H.M will only be periodic but not SHMs..

Its time period is `2pi"/"omega `

c) `3 cos (pi/4 - 2omegat) = 3 cos(2omegat - pi/4)` [∵ `cos(-theta) = cos theta`]

Clearly it represent SHM and its time period is `2pi"/"2omega`

d) `cos omegat + cos 3 omegat + cos 5 omegat`. It represent the periodic but not S.H.M. Its time period is `2pi"/"omega`

e)e^{-w2t2} . It is an exponential function which never repeats itself. Therefore it represents non-periodic motion

f) 1 + wt + w^{2}t^{2} also represents non periodic motion.