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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 gt V_R gt V_P`

B

`V_R gt V_Q gt V_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((2Gr.4/3pR^3)/R) =sqrt((8pGr)/3)R rArr V_"es" prop R`
Surface area of `Q=4A = 4pR_Q^2`
Surface area of P=A= `4pR_P^2 " " rArr R_Q=2R_P`
Mass of R is `M_R = M_P + M_Q`
`r 4/3pR_R^3= r 4/3pR_P^3 + r4/3pR_Q^3 rArr R_R^3 = R_P^3 +R_Q^3 =9R_P^3`
`R_R =9^(1//3) R_P rArr R_R gt R_Q gt R_P " " Therefore V_R gt V_Q gt V_P,V_R/V_P=9^(1//3)` and `V_P/V_Q=1/2`
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