Question

Two planets $A$ and $B$ of equal mass are having their period of revolutions $T _{ A }$ and $T _{ B }$ such that $T _{ A }=2 T _{ B }$. These planets are revolving in the circular orbits of radii $r_{A}$ and $r_{B}$ respectively. Which out of the following would be the correct relationship of their orbits?

• A. $2 r_{ A }^{2}= r _{ B }^{2}$
• B. $r _{ A }^{3}=2 r _{ B }^{3}$
• C. $r _{ A }^{3}=4 r _{ B }^{3}$
• D. $T _{ A }^{2}- T _{ B }^{2}=\frac{\pi^{2}}{ GM }\left( r _{ B }^{3}-4 r _{ A }^{3}\right)$

Answer 1

Answer: (C)

$\begin{array}{c} T ^2 \propto r ^3 \\ \frac{ T _{ A }^2}{ r _{ A }^3}=\frac{ T _{ B }{ }^2}{ r _{ B }^3} \\ r _{ A }{ }^3=4 r _{ B }^3 \end{array}$