Two planets $A$ and $B$ having masses $m_1$ and $m_2$ move around the sun in circular orbits of $r_1$ and…

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:
  1. $\left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}$
  2. $\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3$
  3. $27\left(\frac{m_1}{m_2}\right)^3$
  4. $\left(\frac{r_1}{r_2}\right)^3$

Solution

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

Asked in: JEE Main 2024 (08 Apr Shift 1)

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