Wednesday, 22 April 2020

newtonian gravity - How was Newton able to guess that gravitational force is inversely proportional to distance squared?


This question is puzzling me since I learnt about the gravitation law in school. Why did Newton guess/assume that gravitational force is inversely proportional to the square of distance?


Did he verify that experimentally? (I remember hearing that the first experimental verification of the law of gravitation was after Newton's death.)



If the answer to the above question is no, is it for example more plausible to suppose that F1/r2 than to suppose that F1/r4? Did Newton carry out a thought experiment that makes F1/r2 a plausible guess?


So in summary: Why did Newton choose exponent of 2 instead of any other exponent? Was it a guess that depended on pure luck or an educated guess?



Answer



For a uniform circular orbit of radius r, the acceleration is


a = ω2r,ω = 2πT,


where T is the orbital period. Comparing eq. (A) with Kepler's third law


T2  r3,


we conclude that the gravitational acceleration


a  r2


is proportional to the inverse square distance r.



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