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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 and released. The particle exhibits simple harmonic motion. Find the time period of this motion.

Options

  1. A2 R^2 GM
  2. B2 R^3 GM
  3. C2 2R^3 GM
  4. D2 R^3 2GM

Correct answer

B. 2 R^3 GM

Step-by-step solution

Let the centre of the earth be O and the centre of the tunnel be C . The distance OC = R/2 . When the particle is at a distance x from C , its distance from the centre of the earth is r = x^2 + (R/2)^2 . The gravitational force acting on the particle of mass m at a distance r from the centre of the earth is directed towards O and is given by: F = GMmr R^3 The component of this force along the tunnel acts as the restoring force. From the geometry of the right-angled triangle formed by O, C , and the particle, the co

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