Question

A satellite is orbiting the earth in circular orbit of radius r. Its

• A. kinetic energy varies as r
• B. angular momentum varies as $\frac{1}{\sqrt{r}}$
• C. linear momentum varies as $\frac{1}{r}$
• D. frequency of revolution varies as $\frac{1}{r^{3/2}}$

Answer 1

The correct option is D frequency of revolution varies as $\frac{1}{r^{3/2}}$

Orbital velocity of the satellite is $v=\sqrt{\frac{G{M}_E}{r}}$ where $M_E$ is the mass of the earth

Kinetic energy, $K=\frac{1}{2}m{v}^2=\frac{G{M}_{E}m}{2r}$

where m is the mass of the satellite. $K\propto \frac{1}{r}$

Hence, option (a) is incorrect. Angular momentum, $L=mvr$

$=m\sqrt{\frac{G{M}_E}{r}}r$

$=m\sqrt{G{M}_{E}r}\therefore L\propto \sqrt{r}$

Hence, option (b) is incorrect.

Linear momentum, $p=mv=m\sqrt{\frac{G{M}_E}{r}}\therefore P\propto \frac{1}{\sqrt{r}}$

Hence, option (c) is incorrect.

Frequency of revolution, $v=\frac{1}{T}=\frac{1}{2\pi }\sqrt{\frac{G{M}_E}{r^3}}$

$\therefore v\propto \frac{1}{r^{3/2}}$

Hence, option (d) is correct.