14.4 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) Sin³ $ω$ t

(c) 3 cos ( $π$ /4 – 2 $ω$ t)

(d) Cos $ω$ t + cos 3 $ω$ t + cos 5 $ω$ t

(e) Exp (– $ω$ 2t2)

(f) 1 + $ω$ t + $ω$ ²t²

3 Views | Posted 5 months ago

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Vishal Baghel | Contributor-Level 10

5 months ago

(a) The given function is sin $ω$ t – cos $ω$ t

= $\sqrt 2 [\frac{1}{\sqrt 2} \sin ? ω t - \frac{1}{\sqrt 2} \cos ? ω t]$ = $\sqrt 2 [\frac{1}{\sqrt 2} \sin ? ω t \times \cos ? \frac{π}{4} - \frac{1}{\sqrt 2} \cos ? ω t \times \sin ? \frac{π}{4}]$

= $\sqrt 2 s i n [ω t - \frac{π}{4}]$ )

This function represents SHM as it can be written in the form: a sin ( $ω$ t + $θ$ )

Its period is $\frac{2 π}{ω}$ . It is periodic, but not SHM

(b) Sin³ $ω$ t = $\frac{1}{2} [3 \sin ? ω t - \sin ? 3 ω t]$

The terms $\sin ? ω$ t and $\sin ? 3 ω$ t individually represents simple harmonic motion. However, the superposition of two SHM is periodic but not simple harmonic.

(c) The given function is 3 cos ( $π$ /4 – 2 $ω$ t) = -3cos(2 $ω$ t - $π$ /4)

This function represents simple harmonic motion because it can be written in the form: a co

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(a) The given function is sin $ω$ t – cos $ω$ t

= $\sqrt 2 s i n [ω t - \frac{π}{4}]$ )

This function represents SHM as it can be written in the form: a sin ( $ω$ t + $θ$ )

Its period is $\frac{2 π}{ω}$ . It is periodic, but not SHM

(b) Sin³ $ω$ t = $\frac{1}{2} [3 \sin ? ω t - \sin ? 3 ω t]$

The terms $\sin ? ω$ t and $\sin ? 3 ω$ t individually represents simple harmonic motion. However, the superposition of two SHM is periodic but not simple harmonic.

(c) The given function is 3 cos ( $π$ /4 – 2 $ω$ t) = -3cos(2 $ω$ t - $π$ /4)

This function represents simple harmonic motion because it can be written in the form: a cos( $ω t + θ$ ), its time period is $\frac{2 π}{2 ω}$ = $\frac{π}{ω}$ . Periodic but not SHM

(d) The given function is cos $ω$ t + cos 3 $ω$ t + cos 5 $ω$ t – Each individual cosine function represents SHM, however the superposition of three SHM is periodic but not simple harmonic.

(e) The given function exp (– $ω$ ²t²) is an exponential function. Exponential functions do not repeat themselves, hence non-periodic motion.

(f) The given function 1 + $ω$ t + $ω$ ²t² is non-periodic.

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