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

Explanation

Newton considered the orbits of the planets as circular. For circular orbit of radius r, the centripetal acceleration towards the centre is
`a=-(v^2)/(r)" "...(1)`

Here v is the velocity and r, the distance of the planet from the centre of the orbit.
The velocity in terms of known quantities r and T, is
`v=(2pir)/(T)" "...(2)`
Here T is the time period of revolution of the planet. Substituting the value of 'v' in equation (1) we get ,
`a=-(((2pir)/(T))^(2))/(r)`
`=-(4pi^(2)r)/(T^2)" "...(3)`
Substituting the value of 'a' from (3) in Newton's second law, F = ma where 'm' is the mass of the planet.
`F=-(4pi^(2)mr)/(T^2)" "...(4)`
From Kepler's third law,
`(r^3)/(T^2)=k("constant")" "...(5)`
`(r)/(T^2)=(k)/(r^2)" "...(6)`
By equation (6) in the force expression, we can arrive at the law of gravitation.
`F=-(4pi^(2)mk)/(r^2)" "...(7)`
Here negative sign implies that the force is attractive and it acts towards the centre.He equated the constant `4pi^(2)k` to GM which turned out to be the law of gravitation.
`F=-(GMm)/(r^2)`