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 w, kinetic energy K, gravitational potential energy U, total energy E and angular momentum 1. As the radius r of the orbit increases, determine which of the above quantities increases and which ones decreases.

Answer 1

Consider a body of mass m is rotating around the star s in circular path of radius r.

(i) Orbital velocity –

vo = \(\sqrt{\frac{GM}{r}}\) or v ∝ \(\frac{1}{\sqrt r}\)

Orbital Velocity decreases

(ii) Angular velocity \(\frac{2\pi}{T}\)

By Kepler’s III law

T² \(\propto\) r³ or T² = Kr³

\(\omega\) = \(\frac{2\pi}{Kr^{\frac{3}{2}}}\) or \(\omega\) \(\propto\) \(\frac{1}{\sqrt{r^3}}\)

Hence, angular velocity decreases.

(iii) Kinetic Energy, K = \(\frac{1}{2}\)m\(\frac{GM}{r}\) or K \(\propto\) \(\frac{1}{r}\)

Hence K, decreases on increasing the radius.

(iv) Gravitational Potential Energy,

U = \(-\frac{GMm}{r}\)

or U \(\propto\) \(\frac{-1}{2}\)

So, on increasing radius of circular orbit the U increases.

(v) Total energy,

E = K + U = \(\frac{GMm}{2r}+\big(-\frac{Gmm}{r}\big)\)

E = −\(\frac{GMm}{2r}\)

So, increasing the radius, E will also be increased.

(vi) Angular momentum, L = mvr = mr\(\sqrt{\frac{GM}{r}}\)

L = m\(\sqrt{GMr}\) or L \(\propto\) \(\sqrt r\), increases