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The masses of two spherical planets A an...

The masses of two spherical planets A and B are `M_(A)` and `M_(B)`, surface areas are S and 9 S respectively. The mass of a spherical planet C is `(M_(A) + M_(B))`. If the densities of all the planets A, B, C are same, then the escape velocities `V_(A)`, `V_(B)`, `V_(C)` from the surface of these planets follow:

A

`V_(C)` gt `V_(B)` gt `V_(A)`

B

`V_(B)` = 3`V_(A)`

C

`V_(C)` lt `V_(B)` lt `V_(A)`

D

`V_(C)` gt `V_(B)` lt `V_(A)`

Text Solution

Verified by Experts

The correct Answer is:
C, D
As area of B is 9 times area of A
Densities of all planets is same, then
`r_(B) = 3r_(A), M_(B) = 27 M_(A)`
`r_(C ) = 3sqrt(28)r_(A) = 3.03 r_(A)`
As `V_(A) = sqrt((2GM_(A))/(r_(A))) , V_(B) = sqrt((2GM_(B))/(r_(B))) , V_(A) = sqrt((2GM_(C ))/(r_(C )))`
`rArr V_(A) = sqrt((2GM_(A))/(r_(A))), V_(B) = sqrt((2G xx 27 M_(A))/(3r_(A))) = sqrt((18 GM_(A))/(r_(A)))`
`V_(C ) = sqrt((2GM_(C ))/(r_(C )))`
`= sqrt((2G xx 28 xx M_(A))/(3.03 r_(A)))`
`= sqrt((18.48 GM_(A))/(r_(A)))`
`V_(C ) gt V_(B) gt V_(A)`
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