Question

A satellite \(A\) of mass \(m\) is orbiting around the Earth in a stable circular orbit of radius \(r\) and another satellite \(B\) of mass \(2m\) is orbiting in a similar orbit of radius \(2r\). What is the ratio of the time periods of revolution of satellites \(A\) and \(B\)?

• A. \(1:2\sqrt2\)
• B. \(1:\sqrt2\)
• C. \(2\sqrt2:1\)
• D. \(1:2\)

Hint:

Kepler's third law: \[ T^2\propto r^3. \] Satellite mass does not affect the orbital time period.

Answer 1

The Correct Option is A

Concept: For a satellite in circular orbit, \[ T=2\pi\sqrt{\frac{r^3}{GM}}. \] Hence, \[ T\propto r^{3/2}. \] The time period is independent of the satellite mass.

Step 1: Write the ratio. \[ \frac{T_A}{T_B} = \left( \frac{r}{2r} \right)^{3/2}. \] \[ = \left(\frac12\right)^{3/2}. \] \[ = \frac{1}{2\sqrt2}. \] Therefore, \[ T_A:T_B = 1:2\sqrt2. \] \[\begin{aligned} \boxed{1:2\sqrt2} \end{aligned}\] Hence, option \(\mathbf{(A)}\) is correct.