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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.

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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In equilibrium condition, the gravitational pull results in a centripetal force when a body moves around a star. Let us assume that a body of mass m is revolving around a star of mass M in a circle with radius r. Let us assume that a body of mass m is revolving around a star of mass M in a circle with radius r.

Orbital velocity

v_{0}=\sqrt{\frac{GM}{r}}\; or\; v_{0}\; \alpha\; \frac{1}{\sqrt{r}}

On the increasing radius of the circular path, orbital velocity decreases.

\omega =\frac{2\pi }{T}

T^{2}\; \alpha \; r^{3}  (by Kepler's third law)       

\omega =\frac{2\pi }{Kr^{\frac{3}{2}}}

\omega \; \alpha \; \frac{1}{\sqrt{r^{3}}}

On increasing the radius, angular velocity decreases.

E_{k}=\frac{1}{2}m\frac{GM}{r}

E_{k}\; \alpha \; \frac{1}{r}

On increasing the radius, the kinetic energy decreases.

Gravitation potential energy E_{p}=\frac{-GMm}{r}

E_{p}\; \alpha -\frac{1}{r}

On increasing the radius, P.E. increases.

Total Energy E=E_{k}+E_{p}=\frac{GMm}{2r}-\frac{GMm}{r}=-\frac{GMm}{2r}

Om increasing radius of the circular path the total energy also increases.

Angular momentum L=mvr=m\sqrt{\frac{GM}{r}}r

L=m\sqrt{GMr}

L\; \alpha \; \sqrt{r}

The increasing radius of the circular orbit increases angular momentum

 

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