Question

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

• A. Kinetic energy varies as $\frac{1}{R}$
• B. Angular momentum varies as $\frac{1}{\sqrt{R}}$
• C. Linear momentum varies as $\sqrt{R}$
• D. Frequency of revolution varies as $\frac{1}{R^{3/2}}$
• E. Potential energy varies as R

Answer 1

Answer: (D)

Orbital velocity of the satellite is $v=\sqrt{\frac{G_E}{r}}$ 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, $L=mvr$
$=m\sqrt{\frac{G{M}_E}{r}r}$
$=m\sqrt{G_{E}r}\therefore L\propto \sqrt{r}$
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, $v=\frac{1}{T}=\frac{1}{2\pi }\sqrt{\frac{G{M}_E}{r^3}}$ $\therefore v\propto \frac{1}{r^{3/2}}$