A satellite of mass m is in a circular orbit of radius R around a planet of mass M ( M ≫ m ). The satellite's orbital period is T 0 . Which of the following statements is/are correct?

Question

A satellite of mass m is in a circular orbit of radius R around a planet of mass M ( M ≫ m ). The satellite's orbital period is T 0 ​ . Which of the following statements is/are correct?

TestHub · Accepted Answer

A. For a circular orbit, the kinetic energy is $K = \frac{1}{2} m v^{2}$ . From centripetal force, $\frac{m v ^{2}}{R} = \frac{GM m}{R ^{2}} \Rightarrow m v^{2} = \frac{GM m}{R}$ . So $K = \frac{GM m}{2 R}$ . The potential energy is $U = - \frac{GM m}{R}$ . Total mechanical energy $E = K + U = \frac{GM m}{2 R} - \frac{GM m}{R} = - \frac{GM m}{2 R}$ . So A is correct.

B. The original orbital speed is $v_{0} = \frac{GM}{R}$ . The escape velocity from radius $R$ is $v_{esc} = \frac{2 GM}{R}$ . If the speed becomes $3 v_{0} = 3 \frac{GM}{R} = \frac{3 GM}{R}$ . Since $\frac{3 GM}{R} > \frac{2 GM}{R}$ , the satellite will escape. So B is correct. C. According to Kepler's Third Law, $T^{2} \propto R^{3}$ . So $(T_{n e w} / T_{0})^{2} = (4 R / R)^{3} = 4^{3} = 64$ . Therefore, $T_{n e w} = 64 T_{0} = 8 T_{0}$ . So C is correct.

D. The initial total mechanical energy is $E_{1} = - \frac{GM m}{2 R}$ . The final total mechanical energy in the new orbit of radius $2 R$ is $E_{2} = - \frac{GM m}{2 ( 2 R )} = - \frac{GM m}{4 R}$ . The total energy required (work done by external agent) to change the orbit is the difference in total mechanical energies: $W_{e x t} = E_{2} - E_{1} = - \frac{GM m}{4 R} - (- \frac{GM m}{2 R}) = \frac{GM m}{4 R}$ . So D is correct.

Physics - Gravitation Question with Solution | TestHub

Options:(select one or more)

Doubts & Discussion