A planet of mass M, has two natural satellites with masses m _ 1...

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}

and to be the corresponding quantities of satellite 2. Given

{m}_{1} / {m}_{2} = 2and{R}_{1} / {R}_{2} = \dfrac{1}{4}

, match the ratios in List-I to the numbers in List-II.

	LIST -I		LIST-II
A)	$\dfrac{{v}_{1}}{{v}_{2}}$	P)	$\dfrac{1}{8}$
B)	$\dfrac{{L}_{1}}{{L}_{2}}$	Q)	1
C)	$\dfrac{{K}_{1}}{{K}_{2}}$	R)	2
D)	$\dfrac{{T}_{1}}{{T}_{2}}$	S)	8

A planet of mass M, has two natural satellites with masses m _ 1 and m _ 2 . The radii of their circular orbits are R _

Options

Aa-p;b-q;c-s;d-r;
Ba-s;b-p;c-r;d-q;
Ca-s;b-q;c-r;d-p;
Da-r;b-q;c-s;d-p;

Correct answer

D. a-r;b-q;c-s;d-p;

Step-by-step solution

\left

\begin{array}{ll}\dfrac{{m}_{1}}{{m}_{2}} = 2 \\ \dfrac{{R}_{1}}{{R}_{2}} = \dfrac{1}{4}\end{array}

\right\}

given

\dfrac{GM{m}_{1}}{{R}_{1}^{2}} = \dfrac{{m}_{1}{V}_{1}^{1}}{{R}_{1}}

{v}_{1}^{2} = \dfrac{GM}{{R}_{1}}, {v}_{2}^{2} = \dfrac{GM}{{R}_{2}}

\dfrac{{v}_{1}^{2}}{{v}_{2}^{2}} = \dfrac{{R}_{2}}{{R}_{1}} = 4

\left(i\right)

\dfrac{{v}_{1}}{{v}_{2}} = 2

\left(ii\right)

L = mvR

\dfrac{{L}_{1}}{{L}_{2}} = \dfrac{{m}_{1}{v}_{1}{R}_{1}}{{m}_{2}{v}_{2}{R}_{2}} = 2 \times 2 \times \dfrac{1}{4} = 1

\left(iii\right)

K = \dfrac{1}{2}m{v}^{2}

\dfrac{{K}_{1}}{{K}_{2}} = \dfrac{{m}_{1}{v}_{1}^{2}}{{m}_{2}{v}_{2}^{2}} = 2 \times {\left(2\right)}^{2} = 8

( iv )

$T = 2\piR / V
\dfrac{{T}_{1}}{{T}_{2}} = \dfrac{{R}_{1}}{{v}_{1}} \times \dfrac{{v}_{2}}{{R}_{2}} = \dfrac{{R}_{1}}{{R}_{2}} \times \dfrac{{v}_{2}}{{v}_{1}} = \dfrac{1}{4} \times \dfrac{1}{2} = \dfrac{1}{8}

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