Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distance r between sun and planet i.e. $$T2=Kr3$$ here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them $$isF=GMmr2$$, here G is gravitational constant. The relation between G and K is described as

Question

Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distance r between sun and planet i.e. $$T2=Kr3$$ here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them $$isF=GMmr2$$, here G is gravitational constant. The relation between G and K is described as

Infinity Learn · Accepted Answer

Gravitational force of attraction between sun and planet provides centripetal force for the orbit of planet.∴  $$GMmr2=mv2rv2=GMr$$                               . . . .(I)Time$$ period of the planet is given by $$T=2$$$$π$$rv,  $$T2=4$$$$π$$$$2r2v2T2=4$$$$π$$$$2r2GMr$$              $$(using$$ eqn. $$(i))T2=4$$$$π$$$$2r3GM$$                      . . . . $$(ii)According$$ to $$questionT2=Kr3$$                             . . . .$$(iii) Comparing equations (ii) and (iii), we get $$K=4$$$$π$$$$2GM$$  ∴  $$GMK=4$$$$π$$$$2$$

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