Question

Using newton's law of gravitation prove keplar's third law of planetary motion for circular orbits?

Answer 1

From Newton's law of Gravitation we can say,

`(mv^2)/r = (GMm)/r^2` ( As,the force of attraction between two objects of mass `M` and `m` will be supplying the necessary centripetal force)

Again, `T= (2pi)/omega = (2pir)/v`

So,putting `V= (2pir)/T` in the 1st equation we get,

`(m4(pi)^2r)/T^2 = (GMm)/r^2`

or, `T^2 = (4pi^2)/(GM)*r^3`

That is `T^2 prop r^3` ...which is the Kepler's 3rd law