If a ball were dropped in a hole bored through the center of the earth, it would

Question

If a ball were dropped in a hole bored through the center of the earth, it would be attracted toward the center with a force directly proportional to the distance of the body from the center. Find: the amplitude of the motion

Note:

when the body (mass = m) is on the surface of the earth,

the attraction force is F = GMm/R2, where M is the mass of the earth, and R is the radius of the earth.

When the body is "below" the surface, say, with distance y from the center.

You may think the body is on the surface of the "sub-earth" whose radius is y.

And the mass of the "sub-earth" is Ms = M (y/R)3.

So the attraction force now is F = GMsm/y2 = (GMm/R3)y which is proportional to y.

To do this problem, you will need to use numbers R=4000 miles and GM/R2 = g = 32 ft/sec2

Explanation / Answer

As the force is directly proportional to the displacement

F = -ky

Then this is just like a spring.

For a spring, the amplitude of the motion is the initial compression.

Here, you start at the surface of the earth, y = 6371 km or 4000 mi.

Thus, the amplitude of the motion is 4000 mi. It will actually go back to you if only there is a holeconnecting the opposite parts of the earth.

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