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How to prove that the sum of two rotating vectors in an ellipse?

  1. Aug 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Within the xy-plane, two vectors having lengths P and Q rotate around the z-axis with angular velocities ω and –ω. At t = 0,these vectors have orientations with respect to the x axis specified by θ1 and θ2. How do I find the orientation of the major axis of the resulting ellipse relative to the x-axis.


    3. The attempt at a solution

    P=|p|cos(θ1+ωt) x^+ |p|sin(θ1+ωt) y^
    Q=|q|cos(θ1+ωt) x^+ |q|sin(θ1+ωt) y^

    x^- x hat
    y^-y hat

    How do I solve this after these 2 equations?
    I tried to group the x and y vectors separately. But i could not figure out anything after that.

    X= |p|cos(θ1+ωt)+|q|cos(θ1+ωt)
    Y=|p|sin(θ1+ωt)+ |q|sin(θ1+ωt)

    Without the θ terms I could have just squared both sides and added it. But now I am stuck. Thank you for the help :)
     
  2. jcsd
  3. Aug 30, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    The expressions are not correct.
    The vectors have length P and Q, use them instead of |p| and |q|.
    The vectors rotate in opposite directions (the angular velocities are ω and -ω).
    One vector encloses θ1 angle with the x axis at t=0, the other one encloses θ2.

    Expand the cosine and sine terms in the expression for X and Y, collect the terms with cos(ωt) and sin(ωt).
    There is an easy method to find the angle of the principal axis: Just think that the vectors rotate in opposite directions, and there is a time instant when they are on the same line, so the resultant has the longest length.
    ehild
     
    Last edited: Aug 30, 2012
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