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Quantization of Earth's angular momentum

  1. Apr 7, 2015 #1
    1. The problem statement, all variables and given/known data
    If the angular momentum of Earth in its motion around the Sun were quantized like a hydrogen electron, what would Earth's quantum number be? How much energy would be released in a transition to the next lowest level? Would that energy release be detectable? What would be the radius of that orbit?

    2. Relevant equations
    L = mvr = nħ
    F = GMm/r2 = mv2/r
    En = -E0/n2

    3. The attempt at a solution
    I found Earth's quantum number by solving mvr = nħ for n, with m = 5.972 x 1024 kg, r = 149.7 x 109 m and v = 29.78 x 103 m/s.

    n = 2.523 x 1074

    The second part is what gets me.
    In order to express the energy of a energy level in terms of a planet, I used
    F = GMm/r2 = mv2/r
    and E = KE + PEg = mv2/2 - GMm/r
    to get:

    E = -GMm/2r

    I then plugged in v = (GM/r)1/2 to r = nħ/mv getting r = n2ħ2/m2GM

    And then plugged this into E getting:

    E = -G2M2m3/2n2ħ2

    The energy of a transition would equal: En+1 - En = -E0/(n+1)2 + E0/n2

    Which I could solve with a super huge/super small online calculator. I already assume that the energy will be super small and the orbit won't noticeably change. My issue is...what do I plug in for m? Or should I do it a completely different way? Please help!
     
    Last edited: Apr 7, 2015
  2. jcsd
  3. Apr 7, 2015 #2
    Oh duh, I went through all that work and didn't realize I need the mass of the sun as well as the Earth.
     
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