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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)` 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
The two are spherical planets of same density. Hence `M prop R^(3)` and `A prop R^(2)`.
From ratio of the surface area given, we can write the following :
`(R_(Q))/(R_(P)) = sqrt((4)/(1)) = 2`
Now we can write the ratio of masses
`(M_(Q))/(M_(P)) = 8`
`M_(R ) = M_(P) + M_(Q) = 9 M_(P)`
`R_(R ) = 3 R_(P)`
Escape velocity can be written as
`v = sqrt(2gR) prop R`
Hence we can write the following :
`V_(R ) gt V_(Q) gt V_(P)` and `V_(P)//V_(Q) = (1)/(2)`
hence options (b) and (d) are correct.
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