Two planets A and B having masses m₁ and m₂ move around the sun in circular orbits of r₁ and r₂ radii respectively. If angular momentum of A is L and that of B is 3L, the ratio of time period Tₐ/Tᵦ is:
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Question

Two planets A and B having masses m₁ and m₂ move around the sun in circular orbits of r₁ and r₂ radii respectively. If angular momentum of A is L and that of B is 3L, the ratio of time period Tₐ/Tᵦ is:

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G1 Physics · Accepted Answer

Angular momentum of a planet in circular orbit: L = mvr Also: v = 2πr/T Therefore: L = 2πmr²/T For planet A: L = 2πm₁r₁²/Tₐ For planet B: 3L = 2πm₂r₂²/Tᵦ Dividing: Tₐ/Tᵦ = 3(m₁/m₂)(r₁/r₂)² ...(1) From Kepler's third law for planets orbiting the same sun: T² ∝ r³ Therefore: (r₁/r₂)² = (Tₐ/Tᵦ)⁴ᐟ³ Substituting into (1): Tₐ/Tᵦ = 3(m₁/m₂)(Tₐ/Tᵦ)⁴ᐟ³ Hence: (Tₐ/Tᵦ)¹ᐟ³ = (1/3)(m₂/m₁) Therefore: Tₐ/Tᵦ = (1/27)(m₂/m₁)³