# Two satellites A and B of masses 200 kg and 400 kg

Question:

Two satellites A and B of masses $200 \mathrm{~kg}$ and $400 \mathrm{~kg}$ are revolving round the earth at height of $600 \mathrm{~km}$ and $1600 \mathrm{~km}$ respectively. If $T_{A}$ and $T_{B}$ are the time periods of $A$ and $B$ respectively then the value of $T_{B}-T_{A}$ :

$\left[\right.$ Given : radius of earth $=6400 \mathrm{~km}$, mass of earth $\left.=6 \times 10^{24} \mathrm{~kg}\right]$

1. (1) $4.24 \times 10^{2} \mathrm{~s}$

2. (2) $3.33 \times 10^{2} \mathrm{~s}$

3. (3) $1.33 \times 10^{3} \mathrm{~s}$

4. (4) $4.24 \times 10^{3} \mathrm{~s}$

Correct Option: , 3

Solution:

(3)

$\mathrm{V}=\sqrt{\frac{\mathrm{GM}_{\mathrm{e}}}{\mathrm{r}}}$

$\mathrm{T}=\frac{2 \pi_{\mathrm{r}}}{\sqrt{\frac{\mathrm{GM}_{\mathrm{e}}}{\mathrm{r}}}}=2 \pi \mathrm{r} \sqrt{\frac{\mathrm{r}}{\mathrm{GM}_{\mathrm{e}}}}$

$\mathrm{T}=\sqrt{\frac{4 \pi^{2} \mathrm{r}^{3}}{\mathrm{GM}_{\mathrm{e}}}}=\sqrt{\frac{\overline{4 \pi^{2} \mathrm{r}^{3}}}{\mathrm{GM}_{e}}}$

$\mathrm{T}_{2}-\mathrm{T}_{1}=\sqrt{\frac{4 \pi^{2}\left(8000 \times 10^{3}\right)^{3}}{\mathrm{G} \times 6 \times 10^{24}}}-\sqrt{\frac{4 \pi^{2}\left(7000 \times 10^{3}\right)^{3}}{G \times 6 \times 10^{24}}}$

$\cong 1.33 \times 10^{3} \mathrm{~s}$