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Question 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 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
(a) SHM
The given function is:
This function represents SHM as it can be written in the form:
Its period is: 
(b) Periodic, but not SHM
The given function is:
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:
This function represents simple harmonic motion because it can be written in the form:
Its period is: 
(d) Periodic, but not SHM
The given function is 
. 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 
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.
