Question

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 $T_B - T_A$ is : [$R_e = 6400$ km, $M_e = 6 \times 10^{24}$ kg]
 

• A. $1.33 \times 10^3$ s
• B. $4.24 \times 10^3$ s
• C. $4.24 \times 10^2$ s
• D. $3.33 \times 10^2$ s

Hint:

The time period of a satellite depends only on the orbital radius $r$ and the mass of the planet. It is independent of the satellite's own mass.

Answer 1

The Correct Option is A

Step 1: Orbital radii: $r_A = 6400 + 600 = 7000$ km; $r_B = 6400 + 1600 = 8000$ km.
Step 2: Time period formula $T = 2\pi\sqrt{\frac{r^3}{GM}}$.
Step 3: $GM = (6.67 \times 10^{-11}) \times (6 \times 10^{24}) \approx 4 \times 10^{14}$ m$^3$/s$^2$.
Step 4: $T_A = 2\pi\sqrt{\frac{(7 \times 10^6)^3}{4 \times 10^{14}}} \approx 5840$ s.
Step 5: $T_B = 2\pi\sqrt{\frac{(8 \times 10^6)^3}{4 \times 10^{14}}} \approx 7140$ s.
Step 6: $T_B - T_A \approx 1300$ s $= 1.3 \times 10^3$ s. Closest option is (A).