Question

Obtain an expression for the resultant amplitude of, the composition of two S.H.M.’s having the same period along the same path.  

Answer 1

1. Consider a particle simultaneously subjected to two S.H.M.s having the same period and along the same path (let it be along the x-axis) but of different amplitudes and initial phases. The resultant displacement at any instant is equal to the vector sum of its displacements due to both the S.H.M.s at that instant.
2. Let the two linear S.H.M’s be given by equations,
   x₁ = A₁ sin (ωt + Φ₁) ....(1)
   x₂ = A₂ sin (ωt + Φ₂) ....(2)
   where A₁, A₂ are amplitudes; Φ₁, Φ₂ are initial phase angles, and x₁, x₂ are the displacement of two S.H.M’s in time ‘t’. ω is the same for both S.H.M’s.
3. The resultant displacement of the two S.H.M’s is given by,
   x = x₁ + x₂ ....(3)
   Using equations (1) and (2) , equation (3) can be written as,
   x = A₁ sin (ωt + Φ₁) + A₂ sin (ωt + Φ₂)
   = A₁ [sin ωt cos Φ₁ + cos ωt sin Φ₁] + A₂ [sin ωt cos Φ₂ + cos ωt sin Φ₂] 
   = A₁ sin ωt cos Φ₁ + A₁ cos ωt sin Φ₁ +A₂ sin ωt cos Φ₂ + A₂ cos ωt sin Φ₂ 
   = [A₁ sin ωt cos Φ₁ + A₂ sin ωt cos Φ₂] + [A₁ cosωt sin Φ₁ + A₂ cos ωt sinΦ₂] 
   ∴ x = sin ωt [A₁ cos Φ₁ + A₂ cos Φ₂] + cos ωt [A₁ sin Φ₁ + A₂ sin Φ₂] .…(4)
4. As A₁, A₂, Φ₁ and Φ₂ are constants, we can combine them in terms of another convenient constants R and δ as  
   A₁ cos Φ₁ + A₂ cos Φ₂ = R cosδ .…(5)
   and A₁ sin Φ₁ + A₂ sin Φ₂ = R sin δ .…(6)
5. Using equations (5) and (6), equation (4) can be written as,
   x = sin ωt. R cos δ + cos ωt.R sin δ = R [sin ωt cos δ + cos ωt sin δ]
   ∴ x = R sin (ωt + δ) ....(7)
   Equation (7) is the equation of an S.H.M. of the same angular frequency (hence, the same period) but of amplitude R and initial phase δ. It shows that the combination (superposition) of two linear S.H.M.s of the same period and occurring along the same path is also an S.H.M.
6. Resultant amplitude is,
   R = `sqrt(("R" sinδ)^2 + ("R" cosδ)^2)`
   Squaring and adding equations (5) and (6) we get, (A₁ cos Φ₁ + A₂ cos Φ₂)² + (A₁ sin Φ₁ + A₂ sin Φ₂)² = R² cos² δ + R² sin² δ 
   ∴ `A_1^2 cos^2 Φ_1 + A_2^2 cos^2 Φ_2 + 2A_1 A_2 cos Φ_1 cosΦ_2 + A_1^2 sin^2 Φ_1 + A_2^2 sin^2 Φ_2 + 2A_1A_2 sinΦ_1 sinΦ_2 = R^2 (cos^2 δ + sin^2 δ)` 
   ∴ `A_1^2 (cos^2 Φ_1 + sin^2 Φ_1) + A_2^2 (cos^2 Φ_2 + sin^2 Φ_2) + 2A_1A_2 (cos Φ_1 cos Φ_2 + sinΦ_1 sinΦ_2) = R^2`
   ∴ `A_1^2 + A_2^2 + 2A_1A_2 cos (Φ_1 − Φ_2) = R^2`
    ∴ R = ± `sqrt(A_1^2 + A_2^2 + 2A_1A_2cos(Φ_1 − Φ_2))`   ....(8)
   Equation (8) represents resultant amplitude of two S.H.M’s.