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Physics and Geometry

  1. Jul 20, 2013 #1
    1. The problem statement, all variables and given/known data
    A certain quaternary star system consists of three stars, each of mass m, moving in the same circular orbit of radius r about a central star of mass M. The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. Show that the period of each of the three stars is given by


    2. Relevant equations



    3. The attempt at a solution
    What I am having difficulty is with the geometry of this problem. I attached a diagram that the answer key provides. How am I to know that the three planets form a equilateral triangle, what betokens this. Likewise, why is the angle between two sides of the triangle 60 degrees?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Jul 20, 2013 #2

    phinds

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    is that not clear?
     
  4. Jul 20, 2013 #3

    462chevelle

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    think about it.
    360/3=120
    so if everything is equal. wherever each point is has to equal said number
    and a triangle =180 and divided by 3 equals......
     
  5. Jul 20, 2013 #4

    tiny-tim

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    Hi Bashyboy! :smile:
    If they move uniformly in a circle, the total force on one of of the outer planets from the other two must point towards the centre …

    isn't it obvious then that the position must be symmetic? :wink:

    (and the angles of an equilateral triangle must be 60° because the angles of any triangle must add to 180°)
     
  6. Jul 20, 2013 #5
    Well, I had already determined that a 120 degree angle was was maintained between adjacent planets. However, I was not sure if that related to anything.

    Why must the angles of any triangle sum to 180 degrees, does this follow from some definition?
     
  7. Jul 20, 2013 #6

    tiny-tim

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    erm :redface: … you should be able to prove this in about 17 different ways!

    (eg divide the triangle into two right-angled triangles)

    you need to study an elementary geometry book!!
     
  8. Jul 20, 2013 #7
    I have another question, would the force of one planet on another produce some tangential acceleration?
     
  9. Jul 20, 2013 #8

    haruspex

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    Yes, but by symmetry the tangential affects of each pair on the third cancel.
     
  10. Jul 20, 2013 #9
    Oh, I see. There are two tangential forces acting on each planet, each of which is equal and opposite to each other, is this correct?
     
  11. Jul 20, 2013 #10

    tiny-tim

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    yes, but you're analysing this too much …

    isn't it obvious that, if you have two planets of the same mass at the same distance, then the total force will be toward their midpoint? :smile:
     
  12. Jul 20, 2013 #11
    No, it is not immediately evident; however, after having analyzed the problem, I can see that. I don't think I am analyzing the problem too much, I want to understand every detail of every problem I solve.
     
  13. Jul 20, 2013 #12
    Tiny Tim, is what you say always true?
     
  14. Jul 21, 2013 #13

    tiny-tim

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    yes, because of symmetry

    if you reflect it in a mirror (through the midpoint), you'll have exactly the same …

    so the force in the reflected situation must be same as the original force, in other words it must be its own reflection, in other words it must be in the mirror itself, ie towards (or away from) the midpoint

    it's this concept of symmetry that you're missing …

    many physics exam problems are deliberately constructed with a symmetry in, to help you and to save you time

    it is perfectly acceptable in an exam to say "from symmetry, it is obvious that …"​

    you need to think about symmetry a lot (sorry, but it isn't really a subject you can look up in books), until you're used to spotting it, and using it! :smile:
     
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