Gravitation 5 Question 11
12. A planet of mass $M$, has two natural satellites with masses $m_{1}$ and $m_{2}$. The radii of their circular orbits are $R_{1}$ and $R_{2}$, respectively. Ignore the gravitational force between the satellites. Define $v_{1}, L_{1}, K_{1}$ and $T_{1}$ to be respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and $v_{2}, L_{2}, K_{2}$ and $T_{2}$ to be the corresponding quantities of satellite 2. Given, $m_{1} / m_{2}=2$ and $R_{1} / R_{2}=1 / 4$, match the ratios in List-I to the numbers in List-II.
(2018 Adv.)
| List-I | List-II | ||
|---|---|---|---|
| P. | $v_{1} / v_{2}$ | 1. | $1 / 8$ |
| Q. | $L_{1} / L_{2}$ | 2. | 1 |
| R. | $K_{1} / K_{2}$ | 3. | 2 |
| S. | $T_{1} / T_{2}$ | 4. | 8 |
(a) $\mathrm{P} \rightarrow 4 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 3$
(b) $\mathrm{P} \rightarrow 3 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 1$
(c) $\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 4$
(d) $\mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 1$
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Answer:
Correct Answer: 12. (b)
Solution:
- As, $v=\sqrt{\frac{G M}{R}}$
Let $R_{1}=R$, then $R_{2}=4 R$
If $m_{2}=m$, then $m_{1}=2 m$
List-I
(P) $\frac{v_{1}}{v_{2}}=\sqrt{\frac{R_{2}}{R_{1}}}=\sqrt{\frac{4 R}{R}}=2: 1$
(Q) $L=m v R$
$$ \frac{L_{1}}{L_{2}}=\frac{R(2 m) v_{1}}{4 R(m) v_{2}}=\frac{1}{2}(2)=1: 1 $$
(R) $\frac{K_{1}}{K_{2}}=\frac{\frac{1}{2}(2 m) v_{1}^{2}}{\frac{1}{2}(m) v_{2}^{2}}=2(4)=8: 1$
(S) $\frac{T_{1}}{T_{2}}={\frac{R_{1}}{R_{2}}}^{3 / 2}=\frac{1}{4}^{3 / 2}=1: 8$
