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Suppose the gravititional force varies i...

Suppose the gravititional force varies inversely as the `n^(th)` power of distance then, find the expression for the time period of a planet in a circular orbit of radius r around the sun.

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As the gravitational force varies inversely as the `n^(th)` power of the distance, so the gravitaitonal force on the planet is given by
`F = (GMm)/(r^n)`
This force provides the centripetal force `mrw^(2)` to the planet
`(GMm)/(r^n) = mr omega^(2)`
`= mr ((2pi)/(T))^(2)`
or `T^(2) = (r xx 4 pi^2 xx r^N)/(GM) = (4pi^2r^(n+1))/(GM)`
`T = (2pi)/(sqrt(GM)) cdot r^((n+1)/2)`
`:. T prop r^((n+1)/2)`.
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