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

Explanation

Newton law of gravitation states that a particle of mass `M_(1)` attracts any other particle of mass `M_(2)` in the universe with an attractive force. The strength of this force of attraction was found to be directly proportional to the product of their masses and is inversely proportional to the square to the distance between them. In mathematical from, it can be written as:
`vec(F) = (GM_(1)M_(2))/(r^(2)) hat(r )`
where`hat(r )` is the unit vector from `M_(1)` towards`M_(2)` as shown in given figure, and G is the Gravitational constant that has the value of `6.67 xx10^(-11) " N m"^(2) kg^(-2)` and r is the distance between the two masses `M_(1) and M_(2)` and In given figure the vector denotes the gravitational force experienced by `M_(2)` due to`M_(1)` . Here the negative sign indicates that the gravitational force is always attractive in nature and the direction of the force is along the line joining the two masses. In Cartesian coordinates, the square of the distance is expressed as `r^(2) = (x^(2) +y^(2) +z^(2))`