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) sin^{3} ωt

(c) 3 cos (π/4 – 2ωt)

(d) cos ωt + cos 3ωt + cos 5ωt

(e) exp (–ω^{2}t^{2})

(f) 1 + ωt + ω^{2}t^{2}

The condition for a function to be periodic is that it must identically repeat itself after a fixed interval of time. For a function to represent an SHM, it must have the form of cos ( or sin ( with a time period T.

(a) sin ωt – cos ωt = √2(-)..(Multiply & divide by √2 )

= (sin ωt.cosπ/4 - cos ωt.sinπ/4)

= √2.sin (ωt - π/4)

Hence, it is an SHM with a period 2π/ω.

(b) sin^{3} ωt = 1/3(3sin ωt - sin3ωt)

Each term here, sin ωt and 3ωt represent SHM. But B. is the result of superposition of two SHMs, is only periodic not SHM. Its time period is 2π/ω.

(c) It can be seen that it represents an SHM with a time period of 2π/2ω.

(d) It represents periodic motion but not SHM. Its time period is 2π/ω.

(e) An exponential function never repeats itself. Hence, it is a non periodic motion.

(f) It clearly represents a non-periodic motion.

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