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Question

# A planet moves around sun in nearly circular orbit period of revolution 't', radius of orbit r mass of sun m. Time period if direcyly proportional to mass of sun, distance between planet and sun and universal gravitational constant. Prove T​​​​​​2 is directly proportional to r​​​​​​3

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Solution

## YOUR ANSWER GOES IN THE WAY OF LAW STATED BY KEPLER. 3rd LAW OF KEPLER: T² ∝R³ PROOF : It is known as Law of periods.. Let us consider a planet P of mass m moving with a velocity v around the sun of mass M in a circular orbit of radius r. The gravitational force of attraction of the sun on the planet is, F=GMm/r². The centripetal force is,F = mv²/r. equating the two forces, mv²/r=GMm/r². v²=GM/r -----›(i) If T be the period of revolution of the planet around the sun, then v=2πr/T-------›(ii) Substituting (ii) in (i) 4π²r²/T²=GM/r r³/T²=GM/4π² GM is a constant for any planet. •°• T²∝R³.

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