Gravitation 2 Question 6

9. Two spherical planets P and Q have the same uniform density ρ, masses MP and MQ, and surface areas A and 4A, respectively. A spherical planet R also has uniform density ρ and its mass is (MP+MQ). The escape velocities from the planets P,Q and R, are vP,vQ and vR, respectively. Then

(a) vQ>vR>vP

(b) vR>vQ>vP

(c) vR/vP=3

(d) vP/vQ=1/2

(2012)

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Answer:

Correct Answer: 9. (a)

Solution:

  1. Surface area of Q is four times. Therefore, radius of Q is two times. Volume is eight times. Therefore, mass of Q is also eight times.

So, let

MP=M and RP=r

Then,

MQ=8M and RQ=2r

Now, mass of R is (MP+MQ) or 9M. Therefore, radius of R is (9)1/3r. Now, escape velocity from the surface of a planet is given by

v=2GMr(r= radius of that planet )vP=2GMrvQ=2G(8M)(2r)vR=2G(9M)(9)1/3r

From here we can see that,

vPvQ=12 and vR>vQ>vP



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