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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