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  4. Consider the earth as a uniform sphere of mass M and radius R . Imagine a straight smooth tunnel made through the earth which connects any two points on its surface. Determine the time that a particle would take to go from one end to the other through the tunnel.

Q. Consider the earth as a uniform sphere of mass and radius . Imagine a straight smooth tunnel made through the earth which connects any two points on its surface. Determine the time that a particle would take to go from one end to the other through the tunnel.

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Solution:

Suppose at some instant the particle is at radial distance r from centre of earth O. Since, the particle is constrained to move along the tunnel, we define its position as distance x from C. Hence, equation of motion of the particle is,
max = Fx
The gravitational force on mass m at distance r is,

(towards O)
Therefore,

Since, Fx ∝ - x, motion is simple harmonic in nature. Further,

∴ Time period of oscillation is,

The time taken by particle to go from one end to the other is

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