Explain how Newton verified his law of gravitation.

Explanation

Newton inverse square law: 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`
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 this value of v in equation (1) we get,
` a = (((2pir)/T)^(2))/r = (4pi^(2)r)/(T^(2))`
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))`
From Kepler.s third law`(r^(3))/(T^(2))= k ` (constant ) ,
`r/(T^(2)) = k/(r^(2))`
By substituting equation (6) in the force expression, we can arrive at the law of gravitation.
`F = -(4pi^(2)mk)/(r^(2))`
Here negative sign implies that the force is attractive and it acts towards the center. 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 magnitude of force. So he felt that the Sun.s mass (M) constant`4pi^(2)k` to GM which turned out be the law of gravitation.
`F = - (GMm)/(r^(2))`
Again the negative sign in the above equation implies that the gravitational force is attractive.