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:
Solution
<p>$$\begin{aligned}
& \frac{v_1}{v_2}=\sqrt{\frac{r_2}{r_1}} \quad \text{.... (i)}\\
& m_1 v_1 r_1=h \\
& m_2 v_2 r_2=32 \\
& \Rightarrow \frac{v_1}{v_2}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \quad \text{.... (ii)}
\end{aligned}$$</p>
<p>From (i) & (ii)</p>
<p>$$\begin{aligned}
& \sqrt{\frac{r_2}{r_1}}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \\
& \frac{3 m_1}{m_2}=\sqrt{\frac{r_2}{r_1}} \\
& \frac{T_1}{T_2}=\left(\frac{r_1}{r_2}\right)^{3 / 2}=\left(\frac{m_2}{3 m_1}\right)=\frac{1}{27}\left(\frac{m_2}{m_1}\right)^3
\end{aligned}$$</p>
About this question
Subject: Physics · Chapter: Gravitation · Topic: Satellites and Orbital Motion
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