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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) - Physics

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– cos ωt

(b) sin3 ωt

(c) 3 cos (π/4 – 2ωt)

(d) cos ω+ cos 3ωt + cos 5ωt

(e) exp (–ω2t2)

(f) 1 + ωω2t2

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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:


The terms sin ωand 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.

  Is there an error in this question or solution?
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NCERT Class 11 Physics Textbook
Chapter 14 Oscillations
Q 4 | Page 358
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