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Elliptical motion about the origin

  1. Mar 8, 2012 #1
    1. The problem statement, all variables and given/known data

    A ball of mass m fastened to a long rubber band is spun around so that the ball follows an elliptical path about the origin given by:

    r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y)

    b, ω constants
    bold type indicates vectors

    Find the period of the balls motion.

    2. Relevant equations

    r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y)

    3. The attempt at a solution
    I think the period is, T=2Pi/ω because the motion is harmonic but I'm not sure if this applies for elliptical motion..?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 8, 2012 #2

    gneill

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    Staff: Mentor

    Hi Identify, Welcome to Physics Forums.

    All you have to do is establish after what time period the function r(t) repeats. What do you know about finding the overall period of a function that is comprised of other functions with their own periods?
     
  4. Mar 8, 2012 #3
    Thanks gneill.
    I think when 2 periodic functions are added their periods are Pi(the lowest common multiple of the two periods). In this case the answer would be T=2Pi/w, since the period of the cos and sin functions are both 2Pi.
     
  5. Mar 8, 2012 #4

    gneill

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    Staff: Mentor

    Your result is fine.
     
  6. Mar 8, 2012 #5
    To find the distance from the origin I take,

    |r(t)|=((bcos(ωt))^2 + (2bsin(ωt))^2))^1/2

    Is there a way I can use the sin^2(u) + cos^2(u) = 1 identity to simplify this any further? Or is this the simplified form? If the identity can be used here Im having trouble with the b and 2b coefficients.
     
  7. Mar 8, 2012 #6

    gneill

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    Staff: Mentor

    Well, you can pull the b out for starters. You could also convert either cos2 or sin2 via your identity to get everything in terms of just sin2 or just cos2.
     
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