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A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let $r$ be the distance of the body from the centre of the star and let its linear velocity be $v$, angular velocity $\omega $, kinetic energy $K $, gravitational potential energy $U$, total energy $E$ and angular momentum $l$. As the radius $r$ of the orbit increases, determine which of the abovequantities increase and which ones decrease.
Solution
As shown in figure, where a body of mass $m$ is revolving around a star of mass $\mathrm{M}$.
Linear velocity of the body,
$v=\sqrt{\frac{\mathrm{GM}}{r}}$
$\therefore v \propto \frac{1}{\sqrt{r}}$
Therefore, when $r$ increases, $v$ decreases.
Angular velocity of the body $\omega=\frac{2 \pi}{\mathrm{T}}$
According to Kepler's $3^{\text {rd }}$ law,
$\mathrm{T}^{2} \propto r^{3}$
$\therefore \mathrm{T}=k r^{\overline{2}}$
$\therefore \omega=\frac{2 \pi}{k r^{\frac{3}{2}}}\left(\because \omega=\frac{2 \pi}{\mathrm{T}}\right)$
$\therefore \omega \propto \frac{1}{r^{\frac{3}{2}}}$
Kinetic energy of the body,
$\mathrm{K}=\frac{1}{2} m v^{2}=\frac{1}{2} m \times \frac{\mathrm{GM}}{r}=\frac{\mathrm{GM} m}{2 r}\left(\because v=\sqrt{\frac{\mathrm{GM}}{r}}\right)$
$\therefore \mathrm{K} \propto \frac{1}{r}$
