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Finding the period of rotation of this system...

  1. Jan 12, 2017 #1
    1. The problem statement, all variables and given/known data
    Three identical stars, each with mass m, form the verticies of an equilateral triangle with side length d and rotate in a circular orbit due to their mutual gravitation. What is the period τ of their rotation?

    I set up the FBD for each star and am now trying to figure out what to do from there. I know how to get period into the equation from circular motion, but is that all there is to it? The FBD gives different equations for each mass so when I solve I get different periods for each star. What should I be doing here?

    Force equations

    Star at bottom left point:

    x: Fcosθ+F = max
    y: Fsinθ = may

    Star at bottom right:

    x: -(Fcosθ+F) = max
    y: Fsinθ = may

    Star at top:

    y: -2F = may (or should this be -2Fcos(θ/2)?)
    x: forces cancel to 0

    If you draw an equilateral triangle you will see what I'm talking about and then how I oriented my axes around the stars.
     
  2. jcsd
  3. Jan 12, 2017 #2
    When I solve for period my units are seconds - so I know my method is correct. Anyway I would solve each of the force equations above for period and compare because they should be the same (or at least what I have been thinking) and they are off by a factor due to the cosθ or sinθ
     
  4. Jan 12, 2017 #3

    TSny

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    Instead of working with x and y components of force, think about centripetal components of forces. Use what you know about circular motion.
     
  5. Jan 12, 2017 #4
    I should neglect the force that each is exerting on each other in x or y and only focus on the forces going towards the center of the system (triangle)? I thought about that but I was afraid to try it.
     
  6. Jan 12, 2017 #5
    Okay I think this might be right. Using the centripetal components I was able to solve for period using the star at the top which my answer was T = (2πd3/2)/(√(Gm))(31/4). Units check out.

    (2Gm2cos(θ/2))/d2 = mv2/d this was my equation from F = ma.
     
  7. Jan 12, 2017 #6

    TSny

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    Is the orbital radius equal to d?
     
  8. Jan 12, 2017 #7

    haruspex

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    I get a different constant factor. Please post all your working.
     
  9. Jan 12, 2017 #8
    dsinθ-dsinθ/2 is the distance? Here is the work. Sorry for the blur.
     

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  10. Jan 12, 2017 #9
    OH MY GOD I DID IT THANK YOU haruspex as usual and Tsny . No dsinθ -dsinθ/2 was definitely not it
     
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