Question 79: Gravitation

Question

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:

Accepted Answer

$$\begin{aligned}
& \frac{v_1}{v_2}=\sqrt{\frac{r_2}{r_1}} \quad 	ext{.... (i)}\
& m_1 v_1 r_1=h \
& m_2 v_2 r_2=32 \
& \Rightarrow \frac{v_1}{v_2}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \quad 	ext{.... (ii)}
\end{aligned}$$
From (i) & (ii)
$$\begin{aligned}
& \sqrt{\frac{r_2}{r_1}}=\frac{1}{3} \frac{m_2}{m_1} \frac{r_2}{r_1} \
& \frac{3 m_1}{m_2}=\sqrt{\frac{r_2}{r_1}} \
& \frac{T_1}{T_2}=\left(\frac{r_1}{r_2}ight)^{3 / 2}=\left(\frac{m_2}{3 m_1}ight)=\frac{1}{27}\left(\frac{m_2}{m_1}ight)^3
\end{aligned}$$

Answer

$$\left(\frac{r_2}{r_1}\right)^{\frac{3}{2}}$$

Answer

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

Answer

$$\left(\frac{r_1}{r_2}\right)^3$$

Answer

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

Gravitation

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:

Choose the correct answer: