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JEE MainPhysicsGravitation

Two satellites S₁ and S₂ are revolving around a planet in circular orbits of radii R₁ and R₂ respectively. If the radius of the orbit of S₁ is 9 times the radius of the orbit of S₂ ( R₁ = 9 R₂ ), what is the correct relationship between their respective time periods of revolution T₁ and T₂ ?

Options

  1. AT₁ = 9 T₂
  2. BT₁ = 27 T₂
  3. CT₁ = 3 T₂
  4. DT₁ = 81 T₂

Correct answer

B. T₁ = 27 T₂

Step-by-step solution

According to Kepler's third law of planetary motion, the square of the time period of revolution is directly proportional to the cube of the radius of the circular orbit. T^2 R^3 Taking the ratio for the two satellites: ( T₁ T₂ )^2 = ( R₁ R₂ )^3 Given that R₁ = 9 R₂ , we have R₁ R₂ = 9 . Substituting this value into the equation: ( T₁ T₂ )^2 = (9)^3 = 729 Taking the square root on both sides: T₁ T₂ = 729 = 27 Therefore, T₁ = 27 T₂ . Answer: T₁ = 27 T₂

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