A Tunnel so Deep

Classical Mechanics Level 5

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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