A satellite moves in a circular orbit of radius $r$ around the earth. Match List-I with List-II, which shows how each quantity depends on $r$.
$$\begin{array}{|l|l|l|l|}\hline & \text{List-I} & & \text{List-II} \\ \hline \text{A.} & \text{Orbital speed} & \text{I.} & \propto r^{-1/2} \\ \hline \text{B.} & \text{Time period} & \text{II.} & \propto r^{3/2} \\ \hline \text{C.} & \text{Angular momentum about the earth's centre} & \text{III.} & \propto r^{1/2} \\ \hline \text{D.} & \text{Magnitude of total energy} & \text{IV.} & \propto r^{-1} \\ \hline \end{array}$$
Choose the correct answer from the options given below:
Answer: (C) A-I, B-II, C-III, D-IV
Concept. Gravity provides the centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$.
Step 1. Speed. $v = \sqrt{\dfrac{GM}{r}} \propto r^{-1/2}$, so A-I.
Step 2. Period. $T = \dfrac{2\pi r}{v} = 2\pi\sqrt{\dfrac{r^3}{GM}} \propto r^{3/2}$, so B-II.
Step 3. Angular momentum. $L = mvr = m\sqrt{GMr} \propto r^{1/2}$, so C-III.
Step 4. Total energy. $E = -\dfrac{GMm}{2r}$, so $|E| \propto r^{-1}$, D-IV.
Answer. A-I, B-II, C-III, D-IV (Option C).
Solution by Sreeraj P, M.Sc Physics