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NTA Abhyas JEE Main2020PhysicsGravitationPractice

A hypothetical planet in the shape of a sphere is completely made of an incompressible fluid and has a mass M and radius R . If the pressure at the surface of the planet is zero, then the pressure at the centre of the planet is [ G = universal constant of gravitation]

Options

  1. AP = 3 G M 2 8 π R 4
  2. BP = 3 G M 2 4 π R 4
  3. CP = 3 G M 2 8 π 2 R 4
  4. DP = 3 G M 2 4 π 2 R 4

Correct answer

A. P = 3 G M 2 8 π R 4

Step-by-step solution

The pressure gradient at a distance r from the centre of the spherical planet is d P d r = - ρ g , where g is the acceleration due to gravity at that point d P d r = - 3 M 4 π R 3 G M R 3 r d P = - 3 G M 2 4 π R 6 r d r ∫ P 0 d P = - 3 G M 2 4 π R 6 ∫ 0 R r d r P = 3 G M 2 8 π R 4

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