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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 gravitational force exerted by the earth on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel.

Options

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

Correct answer

D. GMm R^3 x^2 + R^2 4

Step-by-step solution

Let O be the centre of the earth and C be the centre of the tunnel. The perpendicular distance from the centre of the earth to the tunnel is OC = R 2 . Let P be the position of the particle in the tunnel. The distance of the particle from the centre of the tunnel is CP = x . In the right-angled triangle OCP , the distance r of the particle from the centre of the earth is given by: r = OP = OC^2 + CP^2 = ( R 2 )^2 + x^2 = x^2 + R^2 4 The magnitude of the gravitational force exerted by the earth on a particle of mass

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