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A satellite of $10^3 \mathrm{~kg}$ mass is revolving in circular orbit of radius $2 \mathrm{R}$. If $\frac{10^4 \mathrm{R}}{6} \mathrm{~J}$ energy is supplied to the satellite, it would revolve in a new circular orbit of radius:
(use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2, \mathrm{R}=$ radius of earth)
$2.5 \mathrm{R}$
$3 \mathrm{R}$
$4 \mathrm{R}$
$4 \mathrm{R}$
Solution
Total energy $=\frac{-\mathrm{GMm}}{2(2 \mathrm{R})}$
if energy $=\frac{10^4 \mathrm{R}}{6}$ is added then
$\frac{-\mathrm{GMm}}{4 \mathrm{R}}+\frac{10^4 \mathrm{R}}{6}=\frac{-\mathrm{GMm}}{2 \mathrm{r}}$
where $r$ is new radius of revolving and $g=\frac{G M}{R^2}$
$-\frac{\mathrm{mgR}}{4}+\frac{10^4 \mathrm{R}}{6}=-\frac{\mathrm{mgR}^2}{2 \mathrm{r}} \quad\left(\mathrm{m}=10^3 \mathrm{~kg}\right)$
$-\frac{10^3 \times 10 \times \mathrm{R}}{4}+\frac{10^4 \mathrm{R}}{6}=-\frac{10^3 \times 10 \times \mathrm{R}^2}{2 \mathrm{r}}$
$-\frac{1}{4}+\frac{1}{6}=-\frac{\mathrm{R}}{2 \mathrm{r}}$
$\mathrm{r}=6 \mathrm{R}$
