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
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 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:
`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 + ωt + ω2t2 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 + w2t2 also represents non periodic motion.
