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A satellite is orbiting the earth in a circular orbit of radius $r.$ Its
kinetic energy varies as $r$
angular momentum varies as $\frac{1}{\sqrt r}$
linear momentum varies as $\frac{1}{r}$
frequency of revolution varies as $\frac{1}{r^{3/2}}$
Solution
$\because$ Satelite is orbiting
$\frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{\mathrm{r}^{2}}=\frac{\mathrm{mv}^{2}}{\mathrm{r}}$
$\Rightarrow \mathrm{mv}^{2}=\frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{\mathrm{r}}$
$\mathrm{K} \cdot \mathrm{E}=\frac{1}{2} \mathrm{mv}^{2}=\frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{2 \mathrm{r}}$
Angular momentum $=\mathrm{mvr}$
$=\mathrm{m} \sqrt{\frac{\mathrm{GM}}{\mathrm{r}}} \mathrm{r}$
$=\mathrm{L} \propto \mathrm{r}^{1 / 2}$
Momentum $\mathrm{P}=\mathrm{mv}=\mathrm{m} \sqrt{\frac{\mathrm{GM}}{\mathrm{r}}}$
$P$ oc $r^{-1 / 2}$
$\mathrm{T}^{2} \propto \mathrm{r}^{3} \Rightarrow \mathrm{T} \propto \mathrm{r}^{3 / 2} \quad \mathrm{f}=\frac{1}{\mathrm{T}}$
$\Rightarrow \mathrm{f} \propto \mathrm{r}^{-3 / 2}$
