A particle of mass is under the influence of the gravitational field of a body of mass . The particle is moving in a circular orbit of radius with time period around the mass . Then, the particle is subjected to an additional central force [2024]

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 $V_{c} (r) = \frac{m α}{r^{3}},$ 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 $V_{c} (r)$ , but with a new time period $T_{1}$ , then $\frac{T_{1}^{2} - T_{0}^{2}}{T_{1}^{2}}$ is given by [G is the gravitational constant.] [2024]

Smart Achievers · Accepted Answer

(1)

Particle of mass $(M ≫ m)$ is moving in a circular orbit of radius $r_{0}$ with time period $T_{0}$ around the mass $M$ provides centripetal force

$F_{c} = \frac{m v^{2}}{r_{0}}$

$\frac{G M m}{r_{0}^{2}} - \frac{3 m α}{r_{0}^{4}} = \frac{m v^{2}}{r_{0}}$

$or, \frac{ω_{1}^{2}}{ω_{0}^{2}} = \frac{\frac{G M}{r_{0}^{2}} - \frac{3 α}{r_{0}^{4}}}{\frac{G M}{r_{0}^{2}}}$

$or, \frac{T_{0}^{2}}{T_{1}^{2}} = 1 - \frac{3 α}{G M r_{0}^{2}} \Rightarrow \frac{T_{1}^{2} - T_{0}^{2}}{T_{1}^{2}} = \frac{3 α}{G M r_{0}^{2}}$

Solution

1 $\frac{3 α}{G M r_{0}^{2}}$

2 $\frac{α}{2 G M r_{0}^{2}}$

3 $\frac{α}{G M r_{0}^{2}}$

4 $\frac{2 α}{G M r_{0}^{2}}$

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Solution

1 3αGMr02

2 α2GMr02

3 αGMr02

4 2αGMr02

Ans. (1) Particle of mass m is moving in a circular orbit of radius r0 with time period T0 around the mass M provides centripetal force Fc=mv2r0 GMmr02-3mαr04=mv2r0 or, ω12ω02=GMr02-3αr04GMr02 or, T02T12=1-3αGMr02⇒T12-T02T12=3αGMr02

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1 $\frac{3 α}{G M r_{0}^{2}}$

2 $\frac{α}{2 G M r_{0}^{2}}$

3 $\frac{α}{G M r_{0}^{2}}$

4 $\frac{2 α}{G M r_{0}^{2}}$