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) $P \rightarrow 4 ; Q \rightarrow 2 ; R \rightarrow 1 ; S \rightarrow 3$
(b) $P \rightarrow 3 ; Q \rightarrow 2 ; R \rightarrow 4 ; S \rightarrow 1$
(c) $P \rightarrow 2 ; Q \rightarrow 3 ; R \rightarrow 1 ; S \rightarrow 4$
(d) $P \rightarrow 2 ; Q \rightarrow 3 ; R \rightarrow 4 ; S \rightarrow 1$
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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) $\frac{T _1}{T _2}={\frac{R _1}{R _2}}^{3 / 2}=\frac{1}{4}^{3 / 2}=1: 8$
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