A Tunnel so Deep

A tunnel is dug in the Earth's crust joining any 2 points on surface (not diametric end points). Now a long spring of length l(l < 1/4th length of chord ) is attached to the mid point of chord .To the other point of the spring , mass of \(m\) is attached.

Find the Time-period of SHM , if a constant force of 100 N is applied in the direction so as to extend spring.

Your answer can be represented as \( 2 \pi \sqrt{a}\) sec.

Enter \(a\) as your answer.

Details and Assumptions:

  • Radius of Earth \(R = \)\( 6.4 \times {10}^{6} m\), \(K = 10 N/m\), \(m = \)\(6.4 \times {10}^{6}Kg\), \(g = 10m/s^2\)
  • Neglect Frictional force
  • If you feel it would not perform SHM then put 0 as your answer.
  • No motion occurs in the other half of chord.
  • Assume during its complete motion block never goes out of earth's inner body. i.e. max displacement of block from earth's centre is less than half the length of chord.

This is a part of my set Aniket's Mechanics Challenges.
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