A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M$ ($\gg m$). 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 $V_c(r) = \frac{m\alpha}{r^{3}}$, where $\alpha$ 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 $V_c(r)$, 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.]

1. A $\frac{3\alpha}{GM r_0^{2}}$
2. B $\frac{\alpha}{2GM r_0^{2}}$
3. C $\frac{\alpha}{GM r_0^{2}}$
4. D $\frac{2\alpha}{GM r_0^{2}}$

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Answer: A. $\frac{3\alpha}{GM r_0^{2}}$

Gravitation A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M$ ($\gg m$). 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 $V_c(r) = \frac{m\alpha}{r^{3}}$, where $\alpha$ 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 $V_c(r)$, 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.] Watch this solved on the board The full walkthrough, written out a line at a time with narration explaining each step as it appears. Pause anywhere and ask why a line is there. Free to watch

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