Question

how did Newton guess the inverse Square rule​

Answer 1

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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”.

Answer 2

Answer

there was always been a great interest in the motion of planets . by the 16th century a lot of data on the motion of planets had been collected by many astronomers . based on this data johannes kepler derived 3 laws which govern the motion of planets .

the Kepler's law

• the orbit of a planet is an eclipse with the sun at one of the foei .
• the line joining the planet and the sun sweep equal area in equal interval of time . Thus, if the time of travel from A to B is the same that C to D then the area of AOB and OCD equal .
• the cube of the main distance of a planet from the sun is proportional to the square of its Orbit period T . or r^3/t^2 = constant

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 weekend by distance
• a simple argument goes like this we can assume that the planetary orbits are circular .

this is how Newton guess the Inverse Square rule .