Two planets A and B having masses and move around the sun in circular orbits of and radii respectively. If angular momentum of A is L and that of B is 3 L, the ratio of time period is

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Solution

Q.
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 L, the ratio of time period $(\frac{T_{A}}{T_{B}})$ is [2024]

1 $\frac{1}{27} {(\frac{m_{2}}{m_{1}})}^{3}$

2 ${(\frac{r_{1}}{r_{2}})}^{3}$

3 ${(\frac{r_{2}}{r_{1}})}^{\frac{3}{2}}$

4 $27 {(\frac{m_{1}}{m_{2}})}^{3}$

Ans.
(1)

We know, $T^{2} \propto r^{3} \Rightarrow {(\frac{T_{A}}{T_{B}})}^{2} = {(\frac{r_{1}}{r_{2}})}^{3}$

$\Rightarrow \frac{T_{A}}{T_{B}} = {(\frac{r_{1}}{r_{2}})}^{3 / 2}$ ...(1)

Given $L_{B} = 3 L_{A} \Rightarrow m_{2} v_{2} r_{2} = 3 (m_{1} v_{1} r_{1})$

$\Rightarrow \frac{v_{2}}{v_{1}} = 3 \frac{m_{1} r_{1}}{m_{2} r_{2}}$ ...(2)

Orbital velocity, $v = \sqrt{\frac{G M_{s}}{r}} \Rightarrow \frac{v_{2}}{v_{1}} = \sqrt{\frac{r_{1}}{r_{2}}}$ ...(3)

From eq. (2) and eq. (3)

$\sqrt{\frac{r_{1}}{r_{2}}} = \frac{3 m_{1}}{m_{2}} \frac{r_{1}}{r_{2}} \Rightarrow \frac{r_{1}}{r_{2}} = (\frac{m_{2}}{3 m_{1}})$ ...(4)

Using eq. (4) in eq. (1)

$\frac{T_{A}}{T_{B}} = {(\frac{m_{2}}{3 m_{1}})}^{3} = \frac{1}{27} {(\frac{m_{2}}{m_{1}})}^{3}$

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