Using Kepler’s 3rd law and Newton’s law of universal gravitation, find the period of revolution P of the planet as it moves around the sun.

Question

Using Kepler’s 3rd law and Newton’s law of universal gravitation, find the period of revolution P of the planet as it moves around the sun. Assume that the mass of the planet is much smaller than the mass of the sun. Use G for the gravitational constant.

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Neala 5 years 2021-08-06T03:56:09+00:00 1 Answers 187 views 0

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  1. thachthao
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    2021-08-06T03:58:08+00:00
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    Answer:

    Explanation:

    According to the Kepler’s third law, the square of the time period of a planet is directly proportional to the length of cube of semi major axis.

    Let M is the mass of sun, m is the mass of planet and r is the radius of orbit and T is the time period of the planet around the sun. Here ω is the angular velocity of the planet around the sun.

    G is the universal gravitational constant.

    The centripetal force is balanced by the gravitational force between the planet and the sun.

    mr\omega^{2}=\frac{GMm}{r^{2}}

    r\times \frac{4 \pi^{2}}{T^{2}}=\frac{GM}{r^{2}}

    T^{2}=\frac{4 \pi^{2}r^{3}}{GM}

    T=\sqrt \frac{4 \pi^{2}r^{3}}{GM}

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