Two satellites $A$ and $B$ of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively. If $T_A$ and $T_B$ are the time periods of $A$ and $B$ respectively then the value of $T_B - T_A$:
[Given: radius of earth = 6400 km, mass of earth = $6\times10^{24}$ kg]
Answer: (C) $1.33\times10^3$ s
$T = 2\pi\sqrt{\dfrac{r^3}{GM}}$, independent of the satellite's mass. $GM = 6.67\times10^{-11}\times6\times10^{24} \approx 4.0\times10^{14}\ \text{m}^3\text{s}^{-2}$.
$r_A = 7000$ km: $T_A = 2\pi\sqrt{\dfrac{(7\times10^6)^3}{4.0\times10^{14}}} \approx 5.82\times10^3$ s
$r_B = 8000$ km: $T_B = 2\pi\sqrt{\dfrac{(8\times10^6)^3}{4.0\times10^{14}}} \approx 7.11\times10^3$ s
$T_B - T_A \approx 1.3\times10^3$ s, which matches $1.33\times10^3$ s.
Solution by Sreeraj P, M.Sc Physics