1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    rock.freak667

    User Avatar
    Homework Helper

    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

    rock.freak667

    User Avatar
    Homework Helper


    you'd get b=Asinφ and a = Acosφ

    consider what a2+b2, gives. Since tanφ=b/a, then φ exists since a,b≠0
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Simple harmonic motion proof
  1. Simple Harmonic Motion (Replies: 1)

Loading...