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Gravitation — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsGravitation5 Marks
Question
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 above quantities increase and which ones decrease.
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Answer

In equilibrium condition, when a body moves around a star, the gravitational pull result in a centripetal force. Let us consider a body of mass m is rotating around the star S of mass M in circular path of radius r.
$\text{F}_\text{c}=\frac{\text{mv}^2_0}{\text{r}}=\frac{\text{GMm}}{\text{r}^2}$
  1. Then orbital velocity $\text{v}_0=\sqrt{\frac{\text{GM}}{\text{r}}}\text{or}\text{ v}_0\propto\frac{1}{\sqrt{\text{r}}}$
Hence, on increasing radius of circular path orbital velocity decreases.
  1. Angular velocity $\omega=\frac{2\pi}{\text{T}}$ and $\text{T}^2\propto\text{r}^3$ by Kepler’s third law
$\therefore\ \omega=\frac{2\pi}{\text{Kr}^{\frac{3}{2}}} \text{ or }\omega\propto\frac{1}{\sqrt{\text{r}^3}}$

Hence, on increasing the radius of circular orbit the angular velocity decreased.
  1. Kinetic energy $\text{E}_\text{k}=\frac{1}{2}\text{ m }\frac{\text{GM}}{\text{r}}$
Or $\text{E}_\text{k}\propto\frac{1}{\text{r}}.$ Hence on increasing the radius of circular path the kinetic energy decreased.
  1. Gravitation potential energy $\text{E}_\text{p}=\frac{-\text{GMm}}{\text{r}}$ or $\text{E}_\text{p}\propto-\Big(\frac{1}{\text{r}}\Big)$ so.
On increasing radius of circular orbit the P.E. (Ep) increases.
  1. Total energy $\text{E}=\text{E}_\text{k}+\text{E}_\text{p}=\frac{\text{GMm}}{2\text{r}}+\Big(\frac{-\text{GMm}}{\text{r}}\Big)$
$\text{E}=\frac{-\text{GMm}}{2\text{r}}$

Hence, on increasing the radius of circular orbit the total energy E will also be increased.
  1. Angular momentum $=\text{L}=\text{mvr}=\text{m}\sqrt{\frac{\text{GM}}{\text{r}}}\text{r}$
$\text{L}=\text{m}\sqrt{\text{GMr}}$ or $\text{L}\propto\sqrt{\text{r}}$

Hence, the increasing radius r of circular orbit increases the angular momentum.

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