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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 $v_{1}, L_{1}, K_{1} a n d T_{1}$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite1; and $v_{2}, L_{2}, K_{2} a n d T_{2}$ to be the corresponding quantities of satellite 2.Given $\frac{m_{1}}{m_{2}} = 2$ and $\frac{R_{1}}{R_{2}} = \frac{1}{4}$ , match the ratios in List-I to the numbers in List-II.
LIST–I LIST–II
P. $\frac{v_{1}}{v_{2}}$ 1. $\frac{1}{8}$
Q. $\frac{L_{1}}{L_{2}}$ 2. 1
R. $\frac{K_{1}}{K_{2}}$ 3. 2
S. $\frac{T_{1}}{T_{2}}$ 4. 8

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P → 4; Q → 2; R → 1; S → 3

P → 2; Q → 3; R → 4; S → 1

P → 2; Q → 3; R → 1; S → 4

P → 3; Q → 2; R → 4; S → 1

answer is A.

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Detailed Solution

$\frac{G M m_{1}}{R_{1}^{2}} = \frac{m_{1} V_{1}^{2}}{R_{1}}$

$\Rightarrow V_{1}^{2} = \frac{G M}{R_{1}}$

$\frac{G M m_{2}}{R_{2}^{2}} = \frac{m_{2} V_{2}^{2}}{R_{2}}$

$\Rightarrow V_{2}^{2} = \frac{G M}{R_{2}}$

$\frac{V_{1}^{2}}{V_{2}^{2}} = \frac{R_{2}}{R_{1}} = 4$

$(P) \frac{V_{1}}{V_{2}} = 2$

$(Q) L = m v R$

$\frac{L_{1}}{L_{2}} = \frac{m_{1} v_{1} R_{1}}{m_{2} v_{2} R_{2}} = 2 \times 2 \times \frac{1}{4} = 1$

$(R) K = \frac{1}{2} m v^{2}$

$\frac{K_{1}}{K_{2}} = \frac{m_{1} v_{1}^{2}}{m_{2} v_{2}^{2}} = 2 \times (2)^{2} = 8$

$(S) T = 2 π R / V$

$\frac{T_{1}}{T_{2}} = \frac{R_{1}}{v_{1}} \times \frac{v_{2}}{R_{2}} = \frac{R_{1}}{R_{2}} \times \frac{v_{2}}{v_{1}} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$

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