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Orbital Period

  1. Nov 2, 2013 #1
    1. The problem statement, all variables and given/known data

    hl0OrQX.jpg

    The three planets (v1, v2 and v3) in the diagram all have similar mass and are in a line equally spaced so that v1 and v3 are orbiting around v2 synchronously. If the mass of each of the planets are M and the radius of the orbit is R, what is the orbital period?

    2. Relevant equations

    T=[itex]\frac{2*pi*r}{v}[/itex]

    v=[itex]\sqrt{GM/r}[/itex]

    Fg=GMm/r^2

    3. The attempt at a solution

    I am pretty sure you just need to set two equations equal to each other and solve for the variable T, but I am unsure which two equations this is. It would be appreciated if somebody could explain all this to me. Thank you.
     
  2. jcsd
  3. Nov 2, 2013 #2

    rude man

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    One equation.
    Equate the centripetal force of v1 to the gravitational pull exerted on v1 by v2 and v3.
     
  4. Nov 2, 2013 #3
    [itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{2*G*M*M}{R^2}[/itex]

    v=[itex]\sqrt{2GM}[/itex]

    T=[itex]\frac{2*pi*R}{v}[/itex]=[itex]\frac{2*pi*R}{sqrt(2GM)}[/itex]
     
  5. Nov 2, 2013 #4

    rude man

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    You don't have the correct expression for the total gravitationl attraction of v2 and v3 on v1.

    I also suggest changing mv^2/R to m(w^2)R.
     
  6. Nov 2, 2013 #5
    How would I find the total gravitational attraction? Since they both have the same radius and masses, I assume that the gravitational forces would be the same and therefore have two times that.
     
  7. Nov 2, 2013 #6

    rude man

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    Look at the picture. Is the distance from v1 to v2 the same as the distance of v1 to v3?
     
  8. Nov 2, 2013 #7

    [itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{G*M*M}{R^2}[/itex]+[itex]\frac{G*M*M}{2R^2}[/itex]

    v=[itex]\sqrt{3GM}[/itex]

    T=[itex]\frac{2*pi*r}{\sqrt{3GM}}[/itex]
     
  9. Nov 2, 2013 #8

    rude man

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    Still not right. re-examine the second term on the right.
     
  10. Nov 2, 2013 #9
    [itex]\frac{Mv^2}{R}[/itex]=[itex]\frac{G*M*M}{R^2}[/itex]+[itex]\frac{G*M*M}{2R^2}[/itex]

    v=[itex]\sqrt{5/4*GM}[/itex]

    T=[itex]\frac{2*pi*r}{\sqrt{5/4*GM}}[/itex]
     
  11. Nov 2, 2013 #10

    rude man

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    What is the distance between v1 and v3? Ergo, what is the grav. attraction between them?
     
  12. Nov 2, 2013 #11
    Isn't the distance 2R?

    Alright, I figured out I simplified to v wrong resulting in a wrong answer.
     
    Last edited: Nov 2, 2013
  13. Nov 2, 2013 #12

    tms

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    Yes, but when you write it, you should write [itex](2R)^2[/itex], not [itex]2R^2[/itex].
     
  14. Nov 2, 2013 #13
    Yes, so it's 4R^2
     
  15. Nov 3, 2013 #14

    rude man

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    So it is.
     
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