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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. Determine the magnitude of the resultant force acting on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel.

Options

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

Correct answer

A. GMm R^3 x

Step-by-step solution

Let the centre of the earth be the origin O . The tunnel is excavated along a chord at a perpendicular distance R 2 from O . Let the centre of the tunnel be C . The distance OC = R 2 . When the particle is at a distance x from C , its distance r from the centre of the earth is given by: r = x^2 + ( R 2 )^2 The gravitational force exerted by the earth on the particle is directed towards the centre of the earth and its magnitude is: F_g = GMmr R^3 This force can be resolved into two components: 1. Along the tunnel (t

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