Question

A particle of mass \( m \) is under the influence of the gravitational field of a body of mass \( M \) (\( M \gg m \)). 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 \( V(r) = \frac{\alpha m{r^3} \), 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 } \[ \frac{T_1^2 - T_0^2}{T_1^2} \] 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. \( \frac{2\alpha}{GMr_0^2} \)

Hint:

Analyze forces and potentials to find the net effect on orbital time periods.

Answer 1

The Correct Option is A

The additional force is: \[ F = -\frac{dV}{dr} = -\frac{3\alpha m}{r^4}. \] The net centripetal force is: \[ F_{\text{net}} = \frac{GMm}{r_0^2} + \frac{3\alpha m}{r_0^4}. \] The time period is: \begin{align*} T_1 &= \frac{2\pi t_0}{v} = \sqrt{\frac{2\pi t_0^2}{Gmr_0^3 - 3\alpha}}
\text{Also,} \quad T_0^2 &= \frac{4\pi^2 r_0^3}{Gm}
\frac{T_1^2 - T_0^2}{T_1^2} &= -1 - \frac{T_0^2}{T_1^2} = -1 - \frac{4\pi^2 r_0^3}{Gm} \frac{Gmr_0^3 - 3\alpha}{2\pi t_0^2 r_0^3}
&= 1 - 1 + \frac{3\alpha}{Gmr_0 t_0^2} } \end{align*} Expanding: \[ = \frac{3\alpha}{GMr_0^2} \]