Question

Two simple harmonic motions of same angular frequency ω

x₁ = a₁ sin ωt,

x₂ = a₂ sin (ωt + ϕ)

act on a particle along the x-axis simultaneously. Find the resultant motion.

Answer 1

The resultant displacement is

X = x₁+ x₂ = sin ωt [a₁ + a₂ cos ϕ] + cos ωt [a₂ sin ϕ].

which is also simple harmonic along the x-axis with the same angular frequency ω. The amplitude R and the phase angle θ of the resultant motion are given by Eqns. (2.8) and (2.9) respectively.

Special Cases

In this case, the resultant amplitude is zero when a₁ = a₂ and one motion is destroyed by the other.