Question

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A tunnel is dug along the chord of the earth at a perpendicular distance r/2 from the earth's center. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the center of the tunnel​

Answer 1

Explanation:

Force exerted by the wall on a particle of mass m when it is at a distance x from the center of the tunnel is GMm/2R²

Given -

Wall of the tunnel is friction less.

Perpendicular distance = R/2

Let the radius of imaginary sphere is d.

So mass of imaginary sphere is (M') = 4/3 πd³ρ, where ρ is the density of earth

Mass of earth can be given by -

M = 4/3 πR³ρ

By dividing both the equations

M'/M = d³/R³ so M' = d³M/R³

Force between the particle and center of the earth is

F = GmM'/d²

By putting the values-

F = Gmd³M / d² R³

From the diagram we can see that F cos θ component gives the resulting force.

F cos θ = GMmd/R³ Cos θ

By putting the value of cos θ

F = GmM/2R²

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Answer 2

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