The inverse-square rule

According to Newton’s law, the gravitational attraction force between two point masses is inversely proportional to the square of their separation distance and directly proportional to the product of their masses.

It is important to note that Kepler could not give a theory to explain the motion of planets. It was Newton who showed that the cause of the planetary motion is the gravitational force that the Sun exerts on them. Newton used the third law of Kepler to calculate the gravitational force of attraction. The gravitational force of the earth is weakened by distance. A simple argument goes like this. We can assume that the planetary orbits are circular. Suppose the orbital velocity is v and the radius of the orbit is r. Then the force acting on an orbiting planet is given by F ∝ v 2/r.
If T denotes the period, then v = 2πr / T,
so that v 2 ∝ r2 / T2
.We can rewrite this as v2 ∝ (1/r) × ( r3/T2 ). Since r3/T2 is constant by Kepler’s third law, we have v 2 ∝ 1/r. Combining
this with F ∝ v 2/ r, we get, F ∝ 1/ r2

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