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Explain how Newton derived his law of gravitation from Kepler's third law. |
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Answer» Solution :Newton considered the orbits of the planets as circular. For circular orbit of radius r, the centripetal acceleration towards the centre is `a=-(v^2)/(r)""...(1)`  Here v is the velocity and r, the distance of the planet from the centre of the orbit. The velocity in TERMS of known quantities r and T, is `v=(2pir)/(T)""...(2)` Here T is the time period of REVOLUTION of the planet. Substituting the value of 'v' in equation (1) we get , `a=-(((2pir)/(T))^(2))/(r)` `=-(4PI^(2)r)/(T^2)""...(3)` Substituting the value of 'a' from (3) in Newton's second law, F = ma where 'm' is the mass of the planet. `F=-(4pi^(2)mr)/(T^2)""...(4)` From Kepler's third law, `(r^3)/(T^2)=k("constant")""...(5)` `(r)/(T^2)=(k)/(r^2)""...(6)` By equation (6) in the force expression, we can arrive at the law of gravitation. `F=-(4pi^(2)mk)/(r^2)""...(7)` Here negative sign implies that the force is ATTRACTIVE and it acts towards the centre.He equated the constant `4pi^(2)k` to GM which turned out to be the law of gravitation. `F=-(GMm)/(r^2)` |
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