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)
Solution
(a) SHM
The given function is,
sin ωt – cos ωt
This function represents SHM as it can be written in the form of a sin (ωt + Φ).
Its period is: 2π/ωThe given function is,
sin ωt – cos ωt
(b) Periodic but not SHM
The given function is,
sin3 ωt = 1/4 [3 sin ωt - 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.
It has a period of 2π/ω
(c) SHM
The given function is:
This function represents simple harmonic motion because it can be written as,
a cos (ωt + Φ)
Its period is: 2π/2ω = π/ω
(d) Periodic, but not simple harmonic motion.
The given function is cosωt + cos3ωt + cos5ωt.
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(-ω2t2) 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.
The given function is:
This function represents simple harmonic motion because it can be written as,
a cos (ωt + Φ)
Its period is: 2π/2ω = π/ω
(d) Periodic, but not simple harmonic motion.
The given function is cosωt + cos3ωt + cos5ωt.
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(-ω2t2) is an exponential function.
Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.
