A planet of mass has two natural satellites with masses and . The radii of their circular orbits are and , respectively. Ignore the gravitational force between the satellites. [2018]

Question

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 $\frac{m_{1}}{m_{2}} = 2$ and $\frac{R_{1}}{R_{2}} = \frac{1}{4}$ , match the ratios in List-I to the numbers in List-II. [2018]

	LIST-I		LIST-II
P.	$\frac{v_{1}}{v_{2}}$	1.	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

	LIST-I		LIST-II
P.	$\frac{v_{1}}{v_{2}}$	1.	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

Smart Achievers · Accepted Answer

(2)

$∵$ Orbital velocity,

$V = \sqrt{\frac{G M}{R}}, or, V \propto \frac{1}{\sqrt{R}} \Rightarrow \frac{V_{1}}{V_{2}} = \sqrt{\frac{R_{2}}{R_{1}}} = \frac{2}{1}$

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

$Kinetic energy, K = \frac{G M m}{2 R}$

$∴$ $\frac{K_{1}}{K_{2}} = \frac{m_{1}}{m_{2}} \times \frac{R_{2}}{R_{1}} = \frac{2 \times 4}{1 \times 1} = \frac{8}{1}$

From Kepler's law of planetary motion

$T^{2} \propto R^{3} \Rightarrow \frac{T_{1}}{T_{2}} = {(\frac{R_{1}}{R_{2}})}^{3 / 2} = \frac{1}{8}$

Solution

1 P → 4; Q → 2; R → 1; S → 3

2 P → 3; Q → 2; R → 4; S → 1

3 P → 2; Q → 3; R → 1; S → 4

4 P → 2; Q → 3; R → 4; S → 1

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