Question

How did Newton guess the inverse-square rule?

Answer 1

Hint: Newton had an intuitive idea of gravity that any object whether small or large with a definite amount of mass, should experience the same acceleration at the same point in space. He had also the help of Kepler’s third law from which the inverse square relationship can be derived.

Complete step by step answer:
Newton’s Law of Gravitation states that, each particle in the universe attracts every other particle in the universe with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. So we can write Newton’s law as,
$F=-G\dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$
Where, G is the proportionality constant also known as the gravitational constant.
${{\text{m}}_{\text{1}}}\text{,}{{\text{m}}_{\text{2}}}\text{ }$are the masses of the two bodies.
r is the distance between the two bodies.
The negative sign indicates that the force is attractive.
Newton actually derived his inverse square law from Kepler’s third law of planetary motion which states that, “The square of time period of a planet around the sun is directly proportional to the cube of the semi-major axis of the orbit”. Which can be stated as, ${{\text{T}}^{\text{2}}}\propto {{\text{r}}^{\text{3}}}$.
We know that a particle executing a circular motion in a path of radius R experiences an acceleration known as the centripetal acceleration. So suppose a planet of mass m revolves around the sun in a perfect orbit whose radius is r with an angular velocity $\text{ }\\!\\!\omega\\!\\!\text{ }$. The centripetal acceleration of the planet is given by the formula,
$F=m{{\omega }^{2}}r$
${{\text{ }\\!\\!\omega\\!\\!\text{ }}^{\text{2}}}$ can be written as ${{\left( \dfrac{2\pi }{\text{T}} \right)}^{2}}$
$\therefore \text{F}=\text{mr}{{\left( \dfrac{\text{2 }\\!\\!\pi\\!\\!\text{ }}{\text{T}} \right)}^{2}}$
$\therefore \text{F}=\text{mr}\left( \dfrac{\text{4}{{\text{ }\\!\\!\pi\\!\\!\text{ }}^{2}}}{{{\text{T}}^{2}}} \right)$
From Kepler’s law, ${{\text{T}}^{\text{2}}}\text{=k}{{\text{r}}^{\text{3}}}$, k is proportionality constant, so we can write
$F=\dfrac{4{{\pi }^{2}}mr}{K{{r}^{3}}}$
$\Rightarrow F\propto \dfrac{1}{{{r}^{2}}}$
Since $\dfrac{\text{4}{{\text{ }\\!\\!\pi\\!\\!\text{ }}^{\text{2}}}\text{m}}{\text{K}}$ is a constant.
So from this we can see that how the gravitational force is inversely proportional to the square of the distance between the masses.

Note: Newton’s law is accurate in predicting the motions of masses in weak gravitational fields. While in regions of high gravitational field, the general relativity by Einstein is applicable and newton’s is not applicable in these areas.
Newton published his Law of Gravitation in his famous work Philosophiæ Naturalis Principia Mathematica , otherwise known as the “The Principia Mathematica” on July 5, 1687. It is regarded as one of the greatest works in science.