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16 July, 18:47

A planet's moon travels in an approximately circular orbit of radius 7.0 ✕ 10⁷ m with a period of 6 h 38 min. Calculate the mass of the planet from this information. ___ kg

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  1. 16 July, 21:21
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    3.56*10²⁶ Kg.

    Explanation:

    Note: The gravitational force is acting as the centripetal force.

    Fg = Fc ... Equation 1

    Where Fg = gravitational Force, Fc = centripetal force.

    Recall,

    Fg = GMm/r² ... Equation 2

    Fc = mv²/r ... Equation 3

    Where M = mass of the planet, m = mass of the moon, r = radius of the orbit and G = Universal gravitational constant.

    Substituting equation 2 and 3 into equation 1

    GMm/r² = mv²/r

    Simplifying the equation above,

    M = v²r/G ... Equation 4.

    The period of the moon in the orbit

    T = 2πr/v

    Making v the subject of the equation,

    v = 2πr/T ... Equation 5

    where r = 7.0*10⁷ m, T = 6 h 38 min = (6*3600 + 38*60) s = (21600+2280) s

    T = 23880 s, π = 3.14

    v = (2*3.14*7.0*10⁷) / 23880

    v = 18409 m/s

    Also Given: G = 6.67*10⁻¹¹ Nm²/kg²

    Also substituting into equation 4

    M = 18409²*7.0*10⁷ / (6.67*10⁻¹¹)

    M = 3.56*10²⁶ Kg.

    Thus the mass of the planet = 3.56*10²⁶ Kg.
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