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Differential Equations

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Moderate

A Mass in a Channel Through the Earth

DIFFEQ-LL13EC

Consider that a channel had been dug straightly from the north pole to the south pole of the earth. There was no air in the channel. A mass $m$ was released at the north pole. Then, under the gravitational force, the mass would bounce back and forth in the channel between the north pole and the south pole.

Assume that the earth is a perfect ball with uniform density and that the gravity of the mass at the surface is $F=GMm/R^2$ where $M$ is the mass of the earth, $R$ is the radius of the earth and $G$ is a constant.

One can then show that the gravity of the mass is $Fr/R$ where $r$ is the distance to the center of the earth ($r\le R$).

Create an axis such that the center of the earth is the origin and the north pole is on the positive side. Let:

$$\omega=\sqrt{F/mR}$$

At time $t$, the mass is at position:

A

$R\cos(\omega t)$

B

$R-R\sin(\omega t)$

C

$\cfrac{R}{2}(e^{\omega t}+e^{-\omega t})$

D

This is a poor problem since we can't dig the earth like so!