IIT JEE , AIEEE Forum - Physics , Mechanics - gravitation [admin]:derive keplar's laws of planetary motion

cvramana

Let Us get Kepler's Law from Newton's Law of Gravitation.

Let two masses m and M be revolving around their C.M. such that the Gravitational Force of attraction be providing the Centripetal Force. Let R₁ and R₂ be the respective distances of m and M from their common C.M. and let R be the distance between the two masses.

Therefore mR₁ = MR₂ --------------------(1)

And R₁ + R₂ = R -----------------------(2)

Solving (1) and (2) we get R₁ = M R / (m + M) ----------------------(3)

Also

GMm / R² = m w² R₁ --------------------------(4)

From (3) and (4) we get

G(m + M) = w² R³ ------------------------- (5)

Therefore T² is proportional to R³ . Which is nothing but Kepler?s Third Law.

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