JEE MAIN - Physics (2024 - 8th April Morning Shift - No. 3)
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:
$$\left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}$$
$$27\left(\frac{m_1}{m_2}\right)^3$$
$$\left(\frac{r_1}{r_2}\right)^3$$
$$\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3$$
Explanation
$$\begin{aligned} & \frac{v_1}{v_2}=\sqrt{\frac{r_2}{r_1}} \quad \text{.... (i)}\\ & m_1 v_1 r_1=h \\ & m_2 v_2 r_2=32 \\ & \Rightarrow \frac{v_1}{v_2}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \quad \text{.... (ii)} \end{aligned}$$
From (i) & (ii)
$$\begin{aligned} & \sqrt{\frac{r_2}{r_1}}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \\ & \frac{3 m_1}{m_2}=\sqrt{\frac{r_2}{r_1}} \\ & \frac{T_1}{T_2}=\left(\frac{r_1}{r_2}\right)^{3 / 2}=\left(\frac{m_2}{3 m_1}\right)=\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3 \end{aligned}$$
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