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7.Gravitation
medium
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 $I_{A}$ and $I_{B}$ 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
$T_{A}^{2}-T_{B}^{2}=\frac{\pi^{2}}{G M}\left(r_{B}^{3}-4 r_{A}^{3}\right)$
(JEE MAIN-2022)
Solution
$T =\frac{2 \pi}{\sqrt{ Gm _{ A }}} r ^{\frac{3}{2}}$
$T ^{2} \propto r ^{3}$
$\left(\frac{ T _{ A }}{ T _{ B }}\right)^{2}=\left(\frac{ r _{ A }}{ r _{ B }}\right)^{3}$
$\Rightarrow\left(\frac{2}{1}\right)^{2}=\left(\frac{ r _{ A }}{ r _{ B }}\right)^{3} \Rightarrow r _{ A }^{3}=4 r _{ B }^{3}$
Standard 11
Physics
