Planet P₁ revolves in a circular orbit of radius R around a star of mass 2M. Planet P₂ revolves in a circular orbit of radius 2R around another star of mass 4M. What is the ratio of their orbital periods T(P₂) / T(P₁)?
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Question

Planet P₁ revolves in a circular orbit of radius R around a star of mass 2M. Planet P₂ revolves in a circular orbit of radius 2R around another star of mass 4M. What is the ratio of their orbital periods T(P₂) / T(P₁)?

Options

G1 Physics · Accepted Answer

For a circular planetary orbit, T = 2π√[r³ ÷ (GM)]. Therefore, T(P₁) = 2π√[R³ ÷ (2GM)]. For P₂, T(P₂) = 2π√[(2R)³ ÷ (4GM)] = 2π√[8R³ ÷ (4GM)] = 2π√[2R³ ÷ (GM)]. Now, T(P₂) ÷ T(P₁) = √{[8R³ ÷ (4GM)] ÷ [R³ ÷ (2GM)]} = √4 = 2. Correct option: B.