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Period of Planets orbiting a Star

  1. May 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Planet 1 orbits Star 1 and Planet 2 orbits Star 2 in circular orbits of the same radius. However, the orbital period of Planet 1 is longer than the orbital period of Planet 2. What could explain this?

    A) Star 1 has less mass than Star2.
    B) Star 1 has more mass than Star 2
    C) Planet 1 has less mass than Planet 2
    D )Planet 1 has more mass than Planet 2.
    E) The masses of the planet are much less than the masses of the stars.

    2. Relevant equations

    F=(G m1 x m2 ) / (r2)
    ac = mv2 / r
    (2π x r ) / T = V

    3. The attempt at a solution
    I think it is C.
    I used F=(G m1 x m2 ) / (r2) and set it equal to mv2 / r
    I didn't see anything related to Period so I remembered that circumference divided by period equals V.
    I solved for V in the first equation and got : v = √[(Gm)/r]. Mass is the only variable that could cause the speed and therefore the Period to change. So I thought that increasing the mass of Planet would increase the speed and make the Period longer.
    I probably messed up .....
     
  2. jcsd
  3. May 4, 2015 #2

    BvU

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    You sure the mass in your expression is the planet mass and not the star mass ? In short: sort out which is which in m1, m2 and m and which one divides out !
     
  4. May 4, 2015 #3
    Ohhhhh, dumb me...
    I assigned m1 as the Planet and m2 as the Star.
    Since the m in mv2 / r is referring to the orbiting mass, the mass of the planet cancels out and leaves me with the star so it should be B right?
     
  5. May 4, 2015 #4

    BvU

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    How do you deduce that from your equations ?
     
  6. May 4, 2015 #5
    I cancelled m1 out.
     
  7. May 4, 2015 #6

    BvU

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    I mean how do you deduce that it's B and not A
     
  8. May 4, 2015 #7
    So if V increases that means the period would increase. Oh wait. T is in the denominator of 2πr / T.. so that means if I increase V that would mean that Period would go down. I assume that's my mistake. So it has to be that T increases when V decreases meaning that the mass of the star has to be less. So A.
     
  9. May 4, 2015 #8

    BvU

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    Something like that. If you work out T2 you get something with .../(GM..)
     
  10. May 4, 2015 #9
    How would your approach look like? I am wondering how I would arrive at T2
     
  11. May 5, 2015 #10

    BvU

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    1. F=(G m1 x m2 ) / r2
    2. not ac but Fc= m2v2 / r
    3. (2π x r ) / T = v
    3: ##\ \ \displaystyle {1\over T^2} = \left ({v\over 2\pi r} \right )^2\ \ ##. Now equate 1 and 2:
    $${v^2\over r} = {Gm_1 \over r^2} \quad \Rightarrow \quad 1/T^2 = \left ({v\over 2\pi r} \right )^2 = {1\over (2\pi)^2} {Gm_1 \over r^3} \quad \Rightarrow \\ T = 2\pi\sqrt {r^3\over Gm_1} $$

    as in wikipedia (here you can ignore the planet mass m wrt the M of the star)
     
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