Question

Two satellites $A$ and $B$ of masses $200\text{kg}$ and $400\text{kg}$ are revolving round the earth at height of $600\text{km}$ and $1600\text{km}$ 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:
$[$Given : Radius of earth$=6400\text{km}$, Mass of earth$=6\times {10}^{24}\text{kg}]$

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

Answer 1

The correct option is C $1.33\times {10}^3\text{s}$

We know that, time period,
$T=2\pi \sqrt{\frac{r^3}{G{M}_e}}$

$T_B-T_A=2\pi \sqrt{\frac{r_B^3}{G{M}_e}}-2\pi \sqrt{\frac{r_A^3}{G{M}_e}}$

$\Rightarrow T_B-T_A=2\pi \times \frac{1}{\sqrt{G{M}_e}}(\sqrt{r_B^3}-\sqrt{r_A^3})$

$\Rightarrow T_B-T_A=2\pi \times \frac{1}{\sqrt{6.67\times {10}^{-11}\times 6\times {10}^{24}}}(\sqrt{(8000\times {10}^3{)}^3}-\sqrt{(7000\times {10}^3{)}^3})$

$\Rightarrow T_B-T_A\approx 1.33\times {10}^3\text{s}$