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Do you know how Newton guessed the inverse-square rule?

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Newton used the third law of Kepler to calculate the gravitational force of attraction. So, what are Kepler’s laws?
Johannes Kepler derived three laws, which govern the motion of planets, called Kepler's laws. These are:

  1. The orbit of a planet is an ellipse with the Sun at one of the foci. This law is also known as the 'law of orbits'.
  2. The line joining the planet and the Sun sweep equal areas in equal intervals of time. Thus, if the time of travel from A to B is the same as that from C to D then the areas OAB and OCD are equal. This law is also called "law of areas"
  3. The cube of the mean distance of a planet from the Sun is proportional to the square of its orbital period T,  Or, r^{3} / T^{2}=constant.

To derive inverse square law, Newton made some assumptions in planetary motion that planetary orbits can be considered as circular orbits.

Then he suggested that if planets are revolving around the sun, then there must be a centripetal force acting on them which is given by, F=m \frac{v^{2}}{r}

where, m is the mass of the planet moving around the sun in a circular path and v is the speed of the planet and r is its distance from the sun.

Then the force acting on an orbiting planet is given by F \propto \frac{v^{2}} { r}--------------(1)

If the time period of the planet is T, v=\frac{2 \pi r}{T}, So, we can write v \propto\frac{r}{T} Or, v^{2} \propto \frac{r^{2}} {T^{2}}

So, we can write v^{2} \propto\frac{1}{r} \times\frac{r^3}{T^2}

Since r^{3} / T^{2}  is constant by Kepler's third law, we get v^{2} \propto \frac{1}{r}------(2) 

From equation (1) and (2) we can say F \propto \frac{1}{r^2}

 

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shubham.krishnan

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