Force And Laws Of Motion

The additional force is: The net centripetal force is: 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: