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JEE Main 202228 JuneMorning Shift

Exam: JEE Main
Year: 2022
Date: 28 June
Shift: Morning Shift
Subject: Physics
Chapter: Gravitation

JEE Main 2022 · 28 June · Morning Shift · Physics · Gravitation

Two planets A and B of equal mass are having their period of...

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?

Options

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

Correct answer

C. ${r}_{A}^{3} = 4{r}_{B}^{3}$

Step-by-step solution

From Kepler's law of periods,

${T}^{2} ∝ {R}^{3}$

Therefore,

$⇒ {\left(\dfrac{{T}_{A}}{{T}_{B}}\right)}^{2} = {\left(\dfrac{{r}_{A}}{{r}_{B}}\right)}^{3}$

$⇒ {2}^{2} = {r}_{A}^{3}{r}_{B}^{3}$ , $⇒ {r}_{A}^{3} = 4{r}_{B}^{3}$

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