Two planets A and B having masses m1 and m2 move around the sun in circular orbits of r1 and r2 radii respectively. ​

Question

Two planets \(\mathrm{A}\) and \(\mathrm{B}\) having masses \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\) move around the sun in circular orbits of \(r_{1}\) and \(r_{2}\) radii respectively. If angular momentum of \(\mathrm{A}\) is \(\mathrm{L}\) and that of B is 3L, the ratio of time period \(\left(\frac{T_{A}}{T_{B}}\right)\) is:

(1) \(\left(\frac{\mathrm{r}_{2}}{\mathrm{r}_{1}}\right)^{\frac{3}{2}}\)

(2) \(\left(\frac{r_{1}}{r_{2}}\right)^{3}\)

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

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

Answer 1

Correct option is : (3) \( \frac{1}{27}\left(\frac{m_{2}}{m_{1}}\right)^{3}\)  

\(\frac{\pi \mathrm{r}_{1}^{2}}{\mathrm{~T}_{\mathrm{A}}}=\frac{\mathrm{L}}{2 \mathrm{m}_{1}}\)   ..........(1)

\(\frac{\pi \mathrm{r}_{2}^{2}}{\mathrm{~T}_{\mathrm{B}}}=\frac{3 \mathrm{L}}{2 \mathrm{m}_{2}}\)   ...........(2)

\(\Rightarrow \frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}=3 \cdot \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}} \cdot\left(\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}\right)^{2}\)

\(\left(\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}\right)^{2}=\left(\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}\right)^{3} \Rightarrow\left(\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}\right)^{2}=\left(\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}\right)^{\frac{4}{3}}\)

\(\Rightarrow \frac{1}{27} \cdot\left(\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}}\right)^{3}=\left(\frac{\mathrm{T}_{\mathrm{A}}}{\mathrm{T}_{\mathrm{B}}}\right)\)