# Imagine a straight tunnel passing through the center of the Earth, which is regarded as a 2 0f 2 sphere of radius R and uniform mass density. A particle is dropped into the tunnel from the surface.

Imagine a straight tunnel passing through the center of the Earth, which is regarded as a 2 0f 2 sphere of radius R and uniform mass density. A particle is dropped into the tunnel from the

surface. Show that the particle undergoes simple harmonic motion at the angular frequency w = m, where g is the gravitational acceleration at Earth’s surface. (Hint: The gravi— tational acceleration at a point inside a spherically symmetric mass distribution is the same as if all of the mass interior to the point were concentrated at the center, and all of the mass

exterior to the point were neglected.) Estimate how long it takes the particle to reach the

other end of the tunnel. Finally, assuming that the tunnel is smooth (i.e., ignoring friction),

show that motion is simple harmonic even if the tunnel does not pass through the center of

the Earth, and that the period is the same as before. [From Ingard 1988.]

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