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JEE Main 20248 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 8 April
Shift: Morning Shift
Subject: Physics
Chapter: Gravitation

JEE Main 2024 · 8 April · Morning Shift · Physics · Gravitation

Two planets A and B having masses m_1 and m_2 move around the sun...

Two planets

A and B

having masses

m_1 and m_2

move around the sun in circular orbits of

r_1 and r_2

radii respectively. If angular momentum of

A

is

L

and that of

B

is

3 \mathrm{~L}

, the ratio of time period

\left(\frac{T_A}{T_B}\right)

is:

Options

A $\left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}$
B $\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3$
C $27\left(\frac{m_1}{m_2}\right)^3$
D $\left(\frac{r_1}{r_2}\right)^3$

Correct answer

B. $\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3$

Step-by-step solution

\begin{aligned} & \frac{\pi \mathrm{r}_1^2}{\mathrm{~T}_{\mathrm{A}}}=\frac{\mathrm{L}}{2 \mathrm{~m}_1} \ldots \ldots .(1) \\ & \frac{\pi \mathrm{r}_2^2}{\mathrm{~T}_{\mathrm{B}}}=\frac{3 \mathrm{~L}}{2 \mathrm{~m}_2} \ldots \ldots .(2) \\ & \Rightarrow \frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}=3 \cdot \frac{\mathrm{m}_1}{\mathrm{~m}_2} \cdot\left(\frac{\mathrm{r}_1}{\mathrm{r}_2}\right)^2 \\ & \left(\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}\right)^2=\left(\frac{\mathrm{r}_1}{\mathrm{r}_2}\right)^3 \Rightarrow\left(\frac{\mathrm{r}_1}{\mathrm{r}_2}\right)^2=\left(\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}\right)^{\frac{4}{3}} \\ & \Rightarrow \frac{1}{27} \cdot\left(\frac{\mathrm{m}_2}{\mathrm{~m}_1}\right)^3=\left(\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}\right)\end{aligned}

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