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Concepts Of Physics MCQ Edition [Volume 1]PhysicsSimple Harmonic Motion

Suppose a tunnel is excavated along a chord of the earth at a perpendicular distance R/2 from the earth's centre, where R is the radius of the earth. The wall of the tunnel is frictionless. A particle of mass m is placed in the tunnel at a distance x from the centre of the tunnel. Find the magnitudes of the components of the earth's gravitational force on the particle along the tunnel and perpendicular to the tunnel,

Options

  1. AGMm R^3 x , GMm R^3 x^2 + R^2 4
  2. BGMm R^2 x , GMm 2R^3
  3. CGMm R^3 x , GMm 2R^2
  4. DGMm 2R^2 , GMm R^3 x

Correct answer

C. GMm R^3 x , GMm 2R^2

Step-by-step solution

Let the centre of the earth be the origin (0,0) . Let the tunnel be parallel to the x-axis at a perpendicular distance y = R 2 . The position vector of the particle at a distance x from the centre of the tunnel is given by r = x i + R 2 j . The gravitational force on a particle of mass m inside the earth at a position vector r from the centre is directed towards the centre and is given by: F = - GMm R^3 r Substituting the position vector r into the force equation, we get: F = - GMm R^3 ( x i + R 2 j ) F = - GMm R^3

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