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Rewriting SHM equation from a function of sin(wt) and cos(wt) to just cos(wt-∅)

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Rewrite the equation of motion x(t) = 2cos(wt) - 2sin(wt) in the form x(t) = C cos(wt-∅)


    2. Relevant equations


    3. The attempt at a solution
    I am able to work out the constant C as it is just the magnitude of the sin and cos function,

    C = (2^2 + 2^2)^1/2 = √8

    However I am unable to find ∅, can anyone point me in the direction of how to do this?

    Many thanks

    James
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 20, 2012 #2
    Re: Rewritting SHM equation from a function of sin(wt) and cos(wt) to just cos(wt-∅)

    Hint:

    General equation of motion is: x=Asin wt+Bcos wt. If started from position x0 with velocity v0

    x = v0/w sin wt + x0cos wt

    Make substitution:

    x0 = X sin theta
    v0/w = X cos theta
     
  4. May 20, 2012 #3
    Re: Rewritting SHM equation from a function of sin(wt) and cos(wt) to just cos(wt-∅)

    Thanks that was all I needed!

    Essentially ∅ = arctan(b/a)

    Thanks for the help!
     
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