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JEE Main202325 Jan 2023Morning ShiftPhysicsGravitationActual

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g . The time period of the motion of the particle will be (approximately) (take g = 10 ms - 2 , radius of earth = 6400 km )

Options

  1. A24 hours
  2. B1 hour 24 minutes
  3. C1 hour 40 minutes
  4. D12 hours

Correct answer

B. 1 hour 24 minutes

Step-by-step solution

The gravitational force acting on a particle of mass m at a distance   r   ( < R ) from the centre of Earth is F = - G M ' m r 2 ....(1) where M ' is the mass of portion of Earth enclosed by a sphere of radius r concentric with Earth. Here - sign indicates radially inward direction of force. Taking the density of Earth to be uniform M ' 4 πr 3 = M 4 πR 3 ⇒ M ' = M r 3 R 3 ....(2) Substituting M ' from equation 2 into equation 1: F = - G M m R 3   r Which is sim

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