The satellites S₁ and S₂ describe circular orbits of radii r and 2r respectively around a planet. If the orbital angular velocity of S₁ is ω , that of S₂ is

The correct option is ω/2√2.

To find the orbital angular velocity of the second satellite, we start by setting up the relation between a satellite's orbital radius and its angular velocity. For a satellite of mass $m$ orbiting a planet of mass $M$ in a circular orbit of radius $R$, the gravitational force provides the necessary centripetal force:

$\frac{GMm}{R^2}=m{\omega }^{2}R$

where $G$ is the universal gravitational constant and $\omega$ is the orbital angular velocity. Simplifying this expression by canceling $m$ and rearranging terms gives:

${\omega }^2=\frac{GM}{R^3}$

Taking the square root of both sides:

$\omega =\sqrt{\frac{GM}{R^3}}$

This shows that the orbital angular velocity is inversely proportional to $R^{3/2}$:

$\omega \propto \frac{1}{R^{3/2}}$

Now, let $r_1=r$ and ${\omega }_1=\omega$ for the first satellite S₁, and $r_2=2r$ and ${\omega }_2$ be the orbital radius and angular velocity of the second satellite S₂. Using the proportionality relation, we can write:

$\frac{{\omega }_2}{{\omega }_1}$

Substituting the given values into the equation:

$\frac{{\omega }_2}{\omega }$

$\frac{{\omega }_2}{\omega }$

$\frac{{\omega }_2}{\omega }$

Solving for ${\omega }_2$:

${\omega }_2=\frac{\omega }{2\sqrt{2}}$