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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 plantes P,Q and R are `V_P V_Q and V_R` respectively. Then

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`
`V_(es)propR`
Surface area of `P=A=4piR_(P)^(2)`
Surface area of `Q=4A=4piR_(Q)^(2)`
`impliesR_(Q)=2R_(P)`
Mass `R` is `M_(R)=M_(P)+M_(Q)`
`rho 4/3 piR_(R)^(3)=rho 4/3 piR_(P)^(3)+rho 4/3 piR_(Q)^(3)`
`impliesR_(R)^(3)=R_(P)^(3)+R_(Q)^(3)=9R_R_(P)^(3)`
`=R_(R)=9^(1//3)R_(P)impliesR_(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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