Discussion
Two satellites A and B are orbiting the earth at a height of 2.5R and 7.5R respectively from the centre of the earth. The ratio of time periods of A and B is
Answer: B
The ratio of time periods is \(1:3\sqrt{3}\).
By Kepler's third law \(T^{2} \propto r^{3}\), where r is measured from the centre of the earth. The question gives the distances from the centre directly: 2.5R and 7.5R, in the ratio 1 : 3.
\[ \begin{aligned} \frac{T_{A}}{T_{B}} &= \left(\frac{r_{A}}{r_{B}}\right)^{3/2} \\ &= \left(\frac{1}{3}\right)^{3/2} = \frac{1}{3\sqrt{3}} \end{aligned} \]
\(3\sqrt{3}:1\) inverts the ratio; the nearer satellite has the shorter period. \(1:\sqrt{3}\) uses a power of \(\frac{1}{2}\) instead of \(\frac{3}{2}\).
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