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JEE MainPhysicsGravitation

Two identical stars, each of mass M , form a binary system moving in a circular orbit about their common centre of mass. The distance between the stars is d , and their time period of revolution is 4 days. Another binary system consists of two identical stars, each of mass 4M , separated by a distance 9d . The time period of revolution of this second binary system is _____ days.

Correct answer

54

Step-by-step solution

For a binary star system with two identical stars of mass M separated by a distance d , they revolve around their common centre of mass (which is at a distance d/2 from each star). The gravitational force provides the necessary centripetal force: G M^2 d^2 = M ^2 ( d 2 ) G M d^2 = ( 4 ^2 T^2 ) d 2 T^2 = 2 ^2 d^3 G M This gives the proportionality T d^3 M . For the first system: T₁ = 4 d^3 M For the second system: T₂ (9d)^3 4M = 729 d^3 4 M Taking the ratio: T₂ T₁ = 729 4 = 27 2 T₂ = 4 27 2 = 54 days. Answer: 54

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