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Two spherical planets P and Q have the s...

Two spherical planets `P` and `Q` have the same uniform density `rho`, masses `M_(P)` and `M_(Q)` and surface areas `A` and `4A`, respectively. A spherical planet `R` also has uniform density `rho` and its mass is `(M_(P)+M_(Q))`. The escape velocities from the planets `P, Q` and `R`, are `V_(P), V_(Q)` and `V_(R)`, respectively.

A

`V_(Q)gtV_(R)gtV_(P)`

B

`V_(R)gtV_(Q)gtV_(P)`

C

`V_(R)//V_(P)=3`

D

`V_(P)//V_(Q)=1//2`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`V_(es)=sqrt((2GM)/R)=sqrt((2Grho.4/3piR^(3))/R)=sqrt((8piGrho)/3)R`
Surface area of `V_(es)propR`
`Q=4A=4piR_(Q)^(2)`
Surface area of `P=A=o4piR_(P)^(2)`
Mass `R` is `M_(R)=M_(P)+M_(Q)`
`implies R_(Q)=2R_/(P)`
`rho 4/3piR_(R)^(3)=rho 4/3piR_(P)^(3)+rho4/3piR_(Q)^(3)`
`implies R_(R)^(3)=R_(P)^(3)+R_(Q)^(3)=9R_(P)^(3)`
`R_(R)=9^(1/3)R_(P)implies R_(R)gtR_(Q)gtR_(P)`
Therefore `V_(R)gtV_(Q)gtV_(P)`
`(V_(R))/(V_(P))=9^(1//3)` and `(V_(P))/(V_(Q))=1/2`
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