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Given the Mass & Orbital Period, Find the Gravitational Force

  1. Oct 26, 2014 #1
    1. The problem statement, all variables and given/known data
    A large telescope of mass 8410 kg is in a circular orbit around the earth, making one revolution every 927 minutes. What is the magnitude of the gravitational force exerted on the satellite by the earth?
    [itex]M_E = 6.0x10^{24} kg[/itex]
    [itex]m_s = 8410 kg[/itex]
    [itex]T_s = 927 min = 55,620 s[/itex]
    [itex]G = 6.67x10^{-11} Nm^2/kg^2[/itex]

    2. Relevant equations
    [itex]T^2 = \frac {4{\pi}^2 r^3}{GM_E} \Rightarrow r = \sqrt[3]{\frac {T^2 GM_E}{4\pi^2}}[/itex]
    [itex]F_G = \frac {GM_E m_s}{r^2}[/itex]

    3. The attempt at a solution
    [itex]r = \sqrt[3]{\frac {T^2 GM_E}{4\pi^2}} = \sqrt[3]{\frac {(55,620 s)^2 (6.67x10^{-11} Nm^2/kg^2)(6.0x10^{24} kg)}{4\pi^2}} = 3.1532x10^7 m[/itex]

    [itex]F_G = \frac {GM_E m_s}{r^2} = \frac {(6.67x10^{-11} Nm^2/kg^2)(6.0x10^{24} kg)(8410 kg)}{(3.1532x10^7 m)^2} = 3385 N[/itex]

    Is there a simpler equation to get the radius? Am I doing it right?

    Thank-you
     
    Last edited: Oct 26, 2014
  2. jcsd
  3. Oct 26, 2014 #2

    gneill

    User Avatar

    Staff: Mentor

    Your results look good. I don't know if there's a simpler way to get to the radius unless you memorize some fact about another radius and period. For example, if an object could orbit at the Earth's surface it would have a period of 84.5 minutes. Then knowing the Earth's radius you could set up the ratio:
    $$\left( \frac{927}{84.5} \right)^2 = \left( \frac{r}{6378 km} \right)^3$$
    and solve for r.
     
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