Question

A satellite of mass $m$ revolves around earth of mass $M$ in a circular orbit of radius $r$ with angular velocity $\omega$. If radius of the orbit becomes $9r$, then angular velocity of the satellite in the new orbit will be

• A. $9\omega$
• B. $\frac{\omega }{9}$
• C. $27\omega$
• D. $\frac{\omega }{27}$

Answer 1

The correct option is D $\frac{\omega }{27}$

Orbital velocity $v_0$ of the satellite is given by

$v_0=\sqrt{\frac{GM}{r}}$

Angular velocity will be

$\omega =\frac{v}{r}=\left(\frac{\sqrt{GM/r}}{r}\right)==\sqrt{\frac{GM}{r^3}}$

So, $\omega \propto (\frac{1}{r}{)}^{3/2}$

$\therefore \frac{{\omega }_2}{{\omega }_1}={\left(\frac{r_1}{r_2}\right)}^{3/2}={\left(\frac{r}{9r}\right)}^{3/2}$

$\Rightarrow \frac{{\omega }_2}{\omega }={\left(\frac{1}{9}\right)}^{3/2}=\frac{1}{27}(\because {\omega }_1=\omega )$

$\Rightarrow {\omega }_2=\frac{\omega }{27}$

Hence, option $\left(d\right)$ is correct.