Question

Which of the following represent simple harmonic motion (1)
(a) y = 1 + ωt.
(b) y = sin.ωt + cos.ωt.
(c) y = sin.ωt + cos ωt

Answer 1

condition of simple harmonic wave : acceleration, a ∝ - displacement, y

let's check the options !

(a) y = 1 + ωt

dy/dt = ω

and d²y/dt² = 0

acceleration, a = d²y/dt² is not proportional to -y. so, it is not a simple harmonic wave.

(b) y = sinωt + cosωt .........(1)

differentiating both sides,

dy/dt = ωcosωt -ωsinωt

again, differentiating both sides,

d²y/dt² = -ω²sinωt - ω²cosωt

= - ω² [sinωt + cosωt ]

from equation (1),

d²y/dt² = -ω²y

here it is clear that acceleration, a = d²y/dt² ∝ -y

so, it represents a simple harmonic wave.