JEE Advanced · 2012 · 08 Apr 2012 · Paper 2 · Gravitation

Two spherical planets

P

and

Q

have the same uniform density

ρ

, masses

M_{P}

and

M_{Q}

, and surface areas

A

and

4 A

, 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}

and

V_{R}

, respectively. Then

Answer: (B), (D)

Solution

Surface area ratio

4 : 1

implies radius ratio

2 : 1

, so if

R_{P} = R

, then

R_{Q} = 2 R

. With same density, mass ratio is

M_{Q} / M_{P} = (R_{Q} / R_{P})^{3} = 8

. So if

M_{P} = M

, then

M_{Q} = 8 M

and

M_{R} = 9 M

. Radius of

R

R_{R} = 9^{1/3} R

. Escape velocity

V = \frac{2 GM}{R}

. Therefore:

V_{P} = \frac{2 GM}{R}

V_{Q} = \frac{2 G ( 8 M )}{2 R} = 2 V_{P}

V_{R} = \frac{2 G ( 9 M )}{9 ^{1/3} R} = 9^{1/3} V_{P} \approx 2.08 V_{P}

. Thus

V_{R} > V_{Q} > V_{P}

and

V_{P} / V_{Q} = 1/2

JEE Advanced · 2012 · 08 Apr 2012 · Paper 2 · One or more correct options