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Question

Density of a spherical planet of radius R is given by ρ=ρ0(1+rR) where r is distance from center of planet, then
  1. Mass of planet is 73πρ0R3
  2. Acceleration due to gravity at surface of planet is 53Gρ0πR
  3. Acceleration due to gravity at center of planet is zero
  4. Potential at surface of planet is 73Gπρ0R2


Solution

The correct options are
A Mass of planet is 73πρ0R3
C Acceleration due to gravity at center of planet is zero

dm=ρdv=ρ0(1+rR)4πr2dr

M=dm=4πρ0R0(r2+r3R)dr

M=4πρ0[R33+R44R]=4πρ0R3712

g=E=GMR2=73Gρ0πR

at center E = 0 so g = 0

Coming to potential,
dV=GdmRV=GMR=73Gπρ0R2

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