Two satellites of masses 3M and M orbit the earth in the circular orbits of radii r and 3r respectively. The ratio of their orbital velocities is

Concept:

• Orbital Velocity: Orbital velocity is the velocity with which a body or a satellite revolves along a circular orbit around the earth. The expression for orbital velocity is,

\(\Rightarrow {\rm{v}} = \sqrt {\frac{{{\rm{GM}}}}{{\rm{r}}}} \)

From the expression, since G, gravitational constant, and M, the mass of the earth, are constants,

Calculation:

Given orbital radius is 3r and r for the mass of planets, m, and 3m

\(\Rightarrow {\rm{v}} \propto \sqrt {\frac{1}{{\rm{r}}}} \)

• This means the orbital velocity depends on the orbital radius of the satellite but not on the mass of the satellite. From the relation, we thus form an equation,

\(\Rightarrow \frac{{{{\rm{v}}_1}}}{{{{\rm{v}}_2}}} = \sqrt {\frac{{{{\rm{r}}_2}}}{{{{\rm{r}}_1}}}} \)

In the question,

\(\Rightarrow \frac{{{{\rm{r}}_1}}}{{{{\rm{r}}_2}}} = \frac{1}{3}\)

Then,

\(\Rightarrow \frac{{{{\bf{v}}_1}}}{{{{\bf{v}}_2}}} = \frac{{\sqrt 3 }}{1}\)