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JEE MainPhysicsGravitation

Two satellites of masses m₁ and m₂ are orbiting a planet in circular orbits of radii R₁ and R₂ respectively. If the ratio of their time periods is T₁ : T₂ = 8 : 1 and the ratio of their masses is m₁ : m₂ = 3 : 1 , then the ratio of their orbital angular momenta L₁ : L₂ is

Options

  1. A6 : 1
  2. B12 : 1
  3. C3 : 4
  4. D6 2 : 1

Correct answer

A. 6 : 1

Step-by-step solution

According to Kepler's Third Law, the time period T and orbital radius R are related by: T^2 R^3 R T^ 2/3 Given the ratio of time periods T₁ T₂ = 8 , the ratio of their orbital radii is: R₁ R₂ = ( T₁ T₂ )^ 2/3 = (8)^ 2/3 = 4 The orbital velocity of a satellite is given by v = GM R . The orbital angular momentum L is: L = mvr = m GM R R = m GMR Thus, L m R . The ratio of their orbital angular momenta is: L₁ L₂ = ( m₁ m₂ ) R₁ R₂ Substitute the given mass ratio m₁ m₂ = 3 and the calculated radius ratio R₁ R₂ = 4 : L₁ L

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