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.]

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.]

cdquestions Admin · Accepted Answer

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

Answer

$\frac{3\alpha}{ 𝐺𝑀𝑟_0^2}$

Answer

$\frac{\alpha}{ 2𝐺𝑀𝑟_0^2}$

Answer

$\frac{\alpha}{ 𝐺𝑀𝑟_0^2}$

Answer

$\frac{2\alpha}{𝐺𝑀𝑟_0^2}$

The Correct Option is A

Solution and Explanation

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