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Homework Help: Simple harmonic motion proof

  1. Dec 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that there exists a number A>0 and \phi such that acos(ct)+bsin(ct)=Acos(ct-\phi).

    2. Relevant equations
    a,b,c are predermined constants where c>0. From this equation I can justify conclusions regarding the amplitude, frequency, and so forth of a simple harmonic ocillator.

    3. The attempt at a solution
    Obviously if a (or b) is 0, then A is equal b (or a, respectively) and \phi is 0. Thus, I can now assume that a and b are not 0. I try defining two different functions and proving that they are equal for every t using properties of the derivatives.
    Last edited: Dec 18, 2009
  2. jcsd
  3. Dec 18, 2009 #2


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    expand out Acos(ct-φ) and then equate coefficients. You should get two equations. Just try to relate them.

    Hint: sin2x+cos2x=1
  4. Dec 18, 2009 #3
    Got it. I expanded using the trig sums of angles formula for cosine. Thank you.
  5. Dec 19, 2009 #4
    how do you finish this?

    do you get b=-Asin(phi) and a = Acos(phi)

    then phi = arctan(b/a)

    and A = a/(cos(phi))

    do you say that there exists phi = arctan(b/a) > 0 which implies cos(phi) > 0 for 0<phi<pi/4. provided that a > 0 A > 0. etc? I dont see how you 'prove' this.
  6. Dec 19, 2009 #5


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    you'd get b=Asinφ and a = Acosφ

    consider what a2+b2, gives. Since tanφ=b/a, then φ exists since a,b≠0
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