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- \rm I to the numbers in List - \rm II. .tg border-collapse:collapse;border-spacing:0;border-color:#999; .tg tdfont-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#999;color:#444;background-color:#F7FDFA; .tg thfont-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#999;color:#fff;background-color:#26ADE4; .tg .tg-wv9zfont-size:22px;text-align:center .tg .tg-de48font-weight:bold;font-size:22px;text-align:center 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

P \to 3;Q \to 2;R \to 4;S \to 1

P \to 4;Q \to 2;R \to 1;S \to 3

P \to 2;Q \to 3;R \to 1;S \to 4

P \to 2;Q \to 3;R \to 4;S \to 1

Gravitation PYQ — JEE Advanced 2018 · Shift 2 · Q50 with Solution

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-

{\rm I}

to the numbers in List-

{\rm II}.

View written solutionFree

Correct answer: B

Use standard results for circular orbit around a planet of mass $M$ :

$v = \sqrt{\frac{GM}{R}}$ $L = m v R = m\sqrt{GMR}$ $K = \frac{1}{2}mv^2 = \frac{GMm}{2R}$ $T = 2\pi\sqrt{\frac{R^3}{GM}}$

Given: $\frac{m_1}{m_2}=2, \qquad \frac{R_1}{R_2}=\frac14$
Find $\dfrac{v_1}{v_2}$

Since $v \propto \dfrac{1}{\sqrt R}$ , $\frac{v_1}{v_2} = \sqrt{\frac{R_2}{R_1}} = \sqrt{4}=2$

So, $P \to 3$
Find $\dfrac{L_1}{L_2}$

Since $L \propto m\sqrt R$ , $\frac{L_1}{L_2} = \frac{m_1}{m_2}\sqrt{\frac{R_1}{R_2}}$ $= 2\times \sqrt{\frac14} = 2\times \frac12 = 1$

So, $Q \to 2$
Find $\dfrac{K_1}{K_2}$

Since $K \propto \frac{m}{R}$ , $\frac{K_1}{K_2} = \frac{m_1}{m_2}\cdot \frac{R_2}{R_1}$ $= 2\times 4 = 8$

So, $R \to 4$
Find $\dfrac{T_1}{T_2}$

Since $T \propto R^{3/2}$ , $\frac{T_1}{T_2} = \left(\frac{R_1}{R_2}\right)^{3/2} = \left(\frac14\right)^{3/2} = \frac{1}{8}$

So, $S \to 1$
Final matching

$P \to 3, \quad Q \to 2, \quad R \to 4, \quad S \to 1$

This corresponds to Option B.