Explain how Newton derived his law of gravitation from Kepler's third law.

Explanation

Newton's inverse square Law:
Newton considered the orbits of the planets as circular. For circular orbit of radius r, the centripetal accelartion towards the centre is
`a = (v^2)/(r)" "........(1)`
Here v is the velocity and r is the distance of the planet from the center of the orbit.

The velocity in terms of known quantities r and T, is
`v = (2pi r)/T" ".......(2)`
Here T is the time period of revoluation of the planet. Substituting this value of v is equation (1) we get,
`a = (((2pi r)/(T))^(2))/(r) = - (4pi^(2)r)/(T^2)" "........(3)`
Substituting the value of 'a' from in Newtons second law, F = ma, where 'm' is the mass of the planet.
`F = -(4pi^(2) mr)/(T^2)" ".........(4)`
From Keplers third law,
`(r^3)/(T^2) = k` (Constant)`" "........(5)`
`r/(T^2) = k/(r^2)" "........(6)`
By substituting equation (6) in the force expression, we can arrive at the law of gravitation.
`F = -(4 pi^2 mk)/(r^2)" "........(7)`
Here negative sign implies that the force is attractive and it acts towards the centre. In equation (7), mass of the planet 'm' comes explicity. But Newton strongly felt that according to his third law, if Earth is attracted by the Sun, then the Sun must also be attracted by the Earth with the same magitude of force. So he felt that the Sun's mass (M) should also occur explicityly in the expression for force. From this insight, he equated to constant `4pi^(2)k` to GM which turned out to be the law of gravitation
`F = -(GMm)/(r^2)`
Again the negative sign in the above equation implies that the gravitational force is attractive.