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Gravitation

Section: Physics

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Question : 14 of 21

Marks: +1, -0

Two spherical planets P and Q have the same uniform density ρ, masses

M_P

M_Q

and surface areas A and 4A, respectively. A spherical planet R also has uniform density ρ and its mass is

(M_P+M_Q)

. The escape velocities from the planets P, Q and R are

V_P, V_Q

V_R

, respectively. Then

[JEE Adv 2012 P2]

$V_Q$ > $V_R$ > $V_P$
$V_R$ > $V_Q$ > $V_P$
$\frac{V_R}{V_P}$ = 3
$\frac{V_P}{V_Q}$ = $\frac{1}{2}$

Solution:

If the mass of the planet P is M and the radius of the planet P is R, then the mass of the planet Q is 8M; the radius of the planet Q is 2R; the mass of the planet R is 9M; and the radius of the planet R =

9^{1/3}R

.

V_P

=

\sqrt{\frac{2GM}{R}}

V_Q

=

\sqrt{\frac{2G8M}{2R}}

=

2V_P

V_R

=

\sqrt{\frac{2G8M}{9^{1/3}R}}

=

9^{1/3} V_P

Therefore,

V_R

>

V_Q

>

V_P

\frac{V_Q}{V_P}

= 2

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