The change in gravitational potential energy of a satellite of mass $m$ revolving around a planet of

Question

The change in gravitational potential energy of a satellite of mass $m$ revolving around a planet of mass $M$, when it is transferred from a circular orbit of radius $R_1$ to another of radius $R_2$ ($R_2 > R_1$), is

Professor PC Thomas Classes · Answer

(a) : The potential energy of a satellite in an orbit of radius $r$ is given by $U = -\frac{GMm}{r}$. When transferring from orbit $R_1$ to $R_2$, the initial potential energy is $U_1 = -\frac{GMm}{R_1}$ and the final potential energy is $U_2 = -\frac{GMm}{R_2}$. The change in potential energy is $\Delta U = U_2 - U_1 = \left(-\frac{GMm}{R_2}\right) - \left(-\frac{GMm}{R_1}\right) = \frac{GMm}{R_1} - \frac{GMm}{R_2} = GMm\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$.