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Explain how Newton derived his law of gravitation from Kepler's third law.

Answer» <html><body><p></p>Solution :Newton considered the orbits of the planets as circular. For circular orbit of radius r, the centripetal acceleration towards the centre is <br/>`a=-(v^2)/(r)""...(<a href="https://interviewquestions.tuteehub.com/tag/1-256655" style="font-weight:bold;" target="_blank" title="Click to know more about 1">1</a>)` <br/><img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/PRE_GRG_PHY_XI_V02_C06_E01_032_S01.png" width="80%"/> <br/>Here v is the velocity and r, the distance of the planet from the centre of the orbit.<br/>The velocity in <a href="https://interviewquestions.tuteehub.com/tag/terms-1242559" style="font-weight:bold;" target="_blank" title="Click to know more about TERMS">TERMS</a> of known quantities r and T, is <br/>`v=(2pir)/(T)""...(2)` <br/>Here T is the time period of <a href="https://interviewquestions.tuteehub.com/tag/revolution-623158" style="font-weight:bold;" target="_blank" title="Click to know more about REVOLUTION">REVOLUTION</a> of the planet. Substituting the value of 'v' in equation (1) we get ,<br/>`a=-(((2pir)/(T))^(2))/(r)` <br/>`=-(<a href="https://interviewquestions.tuteehub.com/tag/4pi-1882352" style="font-weight:bold;" target="_blank" title="Click to know more about 4PI">4PI</a>^(2)r)/(T^2)""...(3)` <br/>Substituting the value of 'a' from (3) in Newton's second law, F = ma where 'm' is the mass of the planet.<br/>`F=-(4pi^(2)mr)/(T^2)""...(4)` <br/>From Kepler's third law,<br/>`(r^3)/(T^2)=k("constant")""...(5)` <br/>`(r)/(T^2)=(k)/(r^2)""...(6)` <br/>By equation (6) in the force expression, we can arrive at the law of gravitation.<br/>`F=-(4pi^(2)mk)/(r^2)""...(7)` <br/>Here negative sign implies that the force is <a href="https://interviewquestions.tuteehub.com/tag/attractive-887662" style="font-weight:bold;" target="_blank" title="Click to know more about ATTRACTIVE">ATTRACTIVE</a> and it acts towards the centre.He equated the constant `4pi^(2)k` to GM which turned out to be the law of gravitation.<br/>`F=-(GMm)/(r^2)`</body></html>


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