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) | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-14 Oscillations

Q.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) sin3ωt (c) 3 cos (π/4 – 2ωt) (d) cos ωt + cos 3ωt+ cos 5ωt (e) exp (–ω2t2)

(a) SHM
The given function is:
sin ω– cos ωt

This function represents SHM as it can be written in the form: a sin (ωt + Φ)

Its period is: 2π/ω

(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.
Ites period is: 2π/ω

(c) SHM
The given function is:

This function represents simple harmonic motion because it can be written in the form: a cos (ωt + Φ)Its period is: 2π/2ω = π/ω

(d) Periodic, but not SHM
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.