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 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 (- ω² t² ) 

(f) 1 + ω t + ω² t²

Answer 1

(a) sin ω t – cos ω t 

= 

(b) sin³ ω t = \(\frac{1}{4}\)(3 sin ωt – sin 3ωt) both terms are in SHM hence sin³ ωt is periodic.

period =2π/ω

(c) 3 cos (π/4 – 2 ω t) S H M with period = π/ω

(d) cos ω t + cos 3 ω t + cos 5 ω t Superposition of 3 periodic motion,period = 2π/ω

(e) exp (- ω² t² ) non periodic motion

(f) 1 + ω t + ω² t² non periodic motion.