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26 June, 02:24

Show that the radius r of the orbit of a moon of a given planet can be determined from the radius R of the planet, the acceleration of gravity at the surface of the planet, and the time τ required by the moon to complete one full revolution about the planet. Determine the acceleration of gravity at the surface of the planet Jupiter knowing that R = 71 492 km and that τ = 3.551 days and r = 670.9 x 10³ km for its moon Europa.

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  1. 26 June, 02:50
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    g = 24.78 m/s^2

    Explanation:

    Given:

    - R = 71492 km

    - T = 3.551 days

    - r = 670.9*10^3 km

    Find:

    - acceleration due to gravity at jupiter's surface g:

    Solution:

    - The time period T of a satellite in orbit is:

    T = 2*pi*sqrt (r^3/GM)

    GM=4*pi^2*r^3/T^2

    - The local gravitational acceleration of planet is give by:

    g = GM / R^2

    - Combining the two expressions:

    g = 4*pi^2*r^3/T^2*R^2

    - Plug in values:

    g = 4*pi^2*670900^3/306806^2*71492000^2

    g = 24.78 m/s^2
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