Question

A particle of mass 𝑚 is under the influence of the gravitational field of a body of mass 𝑀 (≫ 𝑚). The particle is moving in a circular orbit of radius \(𝑟_0\) with time period \(𝑇_0\) around the mass 𝑀. Then, the particle is subjected to an additional central force, corresponding to the potential energy 𝑉c(𝑟) = 𝑚𝛼/𝑟 3 , where 𝛼 is a positive constant of suitable dimensions and 𝑟 is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius\( 𝑟_0\) in the combined gravitational potential due to 𝑀 and 𝑉c(𝑟), but with a new time period \(𝑇_1\), then\( (𝑇_1^2  − 𝑇_0^ 2 )/𝑇_1^ 2\) is given by [𝐺 is the gravitational constant.]

• A. \(\frac{3\alpha}{ 𝐺𝑀𝑟_0^2}\)
• B. \(\frac{\alpha}{ 2𝐺𝑀𝑟_0^2}\)
• C. \(\frac{\alpha}{ 𝐺𝑀𝑟_0^2}\)
• D. \(\frac{2\alpha}{𝐺𝑀𝑟_0^2}\)

Answer 1

The Correct Option is A

The correct option is (A):\(\frac{3\alpha}{ 𝐺𝑀𝑟_0^2}\)