Two planets, $A$ and $B$ are orbiting a common star in circular orbits of radii $R_A$ and $R_B$,…
Two planets, $A$ and $B$ are orbiting a common star in circular orbits of radii $R_A$ and $R_B$, respectively, with $R_B=2 R_A$. The planet $B$ is $4 \sqrt{2}$ times more massive than planet $A$. The ratio $\left(\frac{L_B}{L_A}\right)$ of angular momentum $\left(L_B\right)$ of planet $B$ to that of planet $A\left(L_A\right)$ is closest to integer ________.
Solution
$\mathrm{L}=\mathrm{mv}_0 \mathrm{R}=\mathrm{m} \sqrt{\frac{\mathrm{GM}}{\mathrm{R}}} \mathrm{R}=\mathrm{m} \sqrt{\mathrm{GMR}}$ here M is mass of star $\begin{aligned} & \frac{\mathrm{L}_{\mathrm{B}}}{\mathrm{~L}_{\mathrm{A}}}=\frac{\mathrm{m}_{\mathrm{B}}}{\mathrm{~m}_{\mathrm{A}}} \sqrt{\frac{\mathrm{R}_{\mathrm{B}}}{\mathrm{R}_{\mathrm{A}}}} \\
& =4 \sqrt{2} \sqrt{\frac{2}{1}} \\
& \frac{\mathrm{~L}_{\mathrm{B}}}{\mathrm{~L}_{\mathrm{A}}}=8 \end{aligned}$