Motion in a Tunnel through the Earth

Suppose a mass moves without friction through a linear tunnel dug between two points on the surface of the Earth. Assume that the Earth has uniform density. The force of gravity acts toward the center of the Earth. As this force is proportional to the displacement, it can be shown that the mass experiences simple harmonic motion with a period of oscillation of 84.3 minutes, independent of the position of the tunnel.

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This idea was apparently first mentioned in a letter from Robert Hooke to Isaac Newton in the 17th century. Hooke derived a formula describing how objects might fall if dropped through hypothetical tunnels through the Earth at varying angles.
See the Wikipedia entry for "Gravity train".
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