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A planet of mass M, has two natural satellites with masses m_{

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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 $v_{1}, L_{1}$ , $K_{1}$ 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.

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Book:

Mechanics Volume 2 (DC Pandey)

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Benjamin Discussed

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.

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Text solutionVerified

(b)

v = \frac{GM}{R}

Let

R_{1} = R

, then

R_{2} = 4 R

m_{2} = m

, then

m_{1} = 2 m

List-I
(P)

\frac{V _{1}}{V _{2}} = \frac{R _{2}}{R _{1}} = \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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Question Text	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.
Topic	Gravitation
Subject	Physics
Class	Class 11
Answer Type	Text solution:1
Upvotes	112

Text solutionVerified

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