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Discuss the Composition of Two S.H.M.S Along the Same Path Having Same Period. Find the Resultant Amplitude and Intial Phase. - Physics

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Sum

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

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Solution

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.

Concept: LC Oscillations
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