Question

The displacement of a harmonic oscillator is given by x = α sin ωt + β cos ωt. What is the amplitude of the oscillation?

Answer 1

Given:  x = α sin ωt + β cos ωt

Let   α = rcos θ and β = r sin θ

Then  x = rcos θ sin ωt + rsin θ cos ωt

= rsin(ωt + θ )

∴ The amplitude of oscillations is r

α² = rcos² θ    (1)

β² = rsin² θ   (2)

From (1) and (2)

\(r=\sqrt{\alpha^2\,+\,\beta^2}\)