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A Particle Undergoes SHM

  1. Jan 16, 2013 #1
    1. The problem statement, all variables and given/known data

    A particle undergoes simple harmonic motion. It has velocity v1 when the displacement is x1 and velocity v2 when the displacement is x2. Find the angular frequency ω and amplitude A in terms of the given quantities.

    2. Relevant equations

    x = A sin (ωt + ∅ )

    v = A ω cos ( ωt + ∅ )

    3. The attempt at a solution

    I tried starting x1 and v1 at t=0 s. This yields

    x = A sin ( ∅ )

    v = A ω cos ( ∅ )
    The equations for position 2 included the ωt. I have four equations: two position and two velocity and four unknowns: A, ω, ∅, and t. I just need A and ω. Is this the right direction?
     
  2. jcsd
  3. Jan 16, 2013 #2

    TSny

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    Welcome to PF! Your approach is fine. (Another approach is to consider energy relations.) What do you get if you divide your velocity equation above by ω and then square the equation? How does that compare with squaring the x equation? Can you see how to eliminate ∅ and t in one fell swoop?
     
  4. Jan 16, 2013 #3
    x2 = A2 sin2 (ωt + ∅ )

    (v/ω)2 = A2 cos2 ( ωt + ∅ )

    I could say

    A = √ (v/ω)2 + x2

    I don't see how I can eliminate t and ∅ though
     
  5. Jan 16, 2013 #4

    haruspex

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    Right. And you have two such equations, one for x1 and v1, and one for x2 and v2.
    You just did. :biggrin:
     
  6. Jan 16, 2013 #5
    Okay guys, I believe I have the right answer.

    I solved for A in each set of x and v.

    A = √ (v1/ω)2 + (x1)2 and A = √ (v2/ω)2 + (x2)2

    Then I solved for ω in the second equation ( 2 )

    I substituted this into the first equation. After a bunch of algebra, I obtained an answer in terms of the given values which makes me happy.

    A = √ [ ( (x1)2 (v2)2) ) - ( (x2)2) (v1)2) / ( (v2)2) - (v1)2 ) ]

    Then I just had to substitute this value for A in equation 1 to find ω.

    Thanks for the help TSny and haruspex
     
  7. Jan 16, 2013 #6

    TSny

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    Looks very good! You can save some effort by not taking the square roots. You have

    A2 = (v1/ω)2 + (x1)2

    A2 = (v2/ω)2 + (x2)2

    Subtracting these two equations should allow you to fairly easily find ω. Then you can find A.
     
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