JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Gravitation

The distance between two stars of masses

3 M_{S}

and

6 M_{S}

9 R

. Here

R

is the mean distance between the centers of the Earth and the Sun, and

M_{S}

is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period

n T

, where

T

is the period of Earth's revolution around the Sun. The value of

n

is ____.

Answer: 9

Solution

For Earth orbiting the Sun:

T = 2 π \frac{R ^{3}}{G M _{S}}

For the binary star system, both stars orbit their common center of mass. The gravitational force provides the centripetal force.

The period of a binary system is:

T_{binary} = 2 π \frac{d ^{3}}{G ( M _{1} + M _{2} )}

where

d = 9 R

is the separation and

M_{1} + M_{2} = 3 M_{S} + 6 M_{S} = 9 M_{S}

n T = 2 π \frac{( 9 R ) ^{3}}{G \cdot 9 M _{S}}

n \cdot 2 π \frac{R ^{3}}{G M _{S}} = 2 π \frac{729 R ^{3}}{9 G M _{S}}

n \frac{R ^{3}}{G M _{S}} = \frac{81 R ^{3}}{G M _{S}}

n = 81 = 9

Therefore,

n = 9

JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Integer answer

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JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Gravitation

Solution

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