Gravitation

JEE Main level, single correct, medium

211 of 223

Question 211 of 223

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

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Answer: (C) rA3=4rB3{r}_{A}^{3}=4{r}_{B}^{3}

From Kepler's law of periods,
T2∝R3{T}^{2}\propto {R}^{3}
Therefore,
⇒(TATB)2=(rArB)3\Rightarrow {\left(\dfrac{{T}_{A}}{{T}_{B}}\right)}^{2}={\left(\dfrac{{r}_{A}}{{r}_{B}}\right)}^{3}
⇒22=rA3rB3\Rightarrow {2}^{2}=\dfrac{{r}_{A}^{3}}{{r}_{B}^{3}}, ⇒rA3=4rB3\Rightarrow {r}_{A}^{3}=4{r}_{B}^{3}

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