Motion in Straight Line

State and prove Kepler’s third law of planetary motion. - Wired Faculty

Question

State and prove Kepler’s third law of planetary motion.

Solution

Kepler’s third law states that the square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the elliptic orbit.
If orbit of the planet is circular, then square of the time period will be proportional to the cube of radius of the orbit.
Let a planet revolve around the sun in a circular orbit of radius r.
The necessary centripetal force required is provided by the gravitational pull,
i.e., $mrω squared equals GMm over straight r squared$
$rightwards double arrow$ $straight r fraction numerator 4 straight pi squared over denominator straight T squared end fraction equals GM over straight r squared$
$rightwards double arrow$ $space space straight T squared equals fraction numerator 4 straight pi squared over denominator GM end fraction straight r cubed$
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