Question

Discuss the composition of two S.H.M.s along the same path having same period. Find the resultant amplitude and intial phase.

Answer 1

Analytical treatment:

i. Let the two linear S.H.M’s be given by equations,
x₁ = A₁ sin (ωt + α₁)          …(1)

x₂ = A₂ sin (ωt + α₂)           …(2)

where A₁, A₂ are amplitudes; α₁, α₂ are initial phase angles and x₁, x₂ are the displacement of two S.H.M’s in time ‘t’. ω is same for both S.H.M’s.

ii. The resultant displacement of the two S.H.M’s is given by,

x = x₁ + x₂                       ....(3)

iii. Using equations (1) and (2) , equation (3) can be written as,

x = A₁ sin (ωt + α₁) + A₂ sin (ωt + α₂)

   = A₁ [sin ωt cos α₁ + cos ωt sin α₁] + A₂ [sin ωt cos α₂ + cos ωt sin α₂]

   = A₁ sin ωt cos α₁ + A₁ cos ωt sin α₁ + A₂ sin ωt cos α₂ + A₂ cos ωt sin α₂

   = [A₁ sin ωt cos α₁ + A₂ sin ωt cos α₂] + [A₁ cos ωt sin α₁ + A₂ cos ωt sinα₂]

   ∴ x = sin ωt [A1 cos α₁ + A₂ cos α₂] + cos ωt [A₁ sin α₁ + A₂ sin α₂]                                …(4)

iv. Let A₁ cos α₁+ A₂ cos α₂ = R cos δ                           …(5)

     and A₁ sin α₁ + A₂ sin α₂ = R sin δ                                 …(6)

v. 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) represents linear S.H.M. of amplitude R and initial phase angle δ with same period.

Resultant amplitude (R):

Squaring and adding equations (v) and (vi) we get,

        (A₁ cos α₁ + A₂ cos α₂)² + (A₁ sin α₁ + A₂ sin α₂)² = R²cos²δ + R²sin²δ

∴       A₁²cos² α1+ A₂² cos² α₂ + 2A₁ A₂ cosα₁ cosα₂ +A₁² sin² α₁ + A₂²sin² α₂ + 2A₁A₂ sinα₁ sinα₂ = R² (cos² δ + sin² δ)

∴       A₁² (cos² α1 + sin² α₁) + A₂² (cos² α₂ + sin² α₂) +  2A₁ A₂ (cosα₁ cosα₂ + sinα₁ sinα₂) = R²

∴       A₁² + A₂² + 2A₁ A₂ cos (α₁ - α₂) = R²

∴       `"R"=+-sqrt(A_1^2+A_2^2+2A_1A_2cos(alpha_1-alpha_2))`                                   ...........(8)

Equation (8) represents resultant amplitude of two S.H.M’s.

Resultant (intial) phase (δ):

Dividing equation (6) by (5), we get

`(A_1sinalpha_1 +A_2sinalpha_2)/(A_1cosalpha_1+A_2cosalpha_2)=(Rsindelta)/(Rcosdelta)`

`therefore(A_1sinalpha_1 +A_2sinalpha_2)/(A_1cosalpha_1+A_2cosalpha_2)=tandelta`

`thereforedelta=tan^(-1)[(A_1sinalpha_1 +A_2sinalpha_2)/(A_1cosalpha_1+A_2cosalpha_2)]`                                                   ..................(9)

Equation (9) represents resultant or intial phase of two S.H.M’s.