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

Answer: (A)

$\frac{3\alpha}{ 𝐺𝑀𝑟_0^2}$

Answer 2

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