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Demonstrating Inertia

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Date: 10/10/95 at 22:3:51
From: Pat S Pattillo

To introduce myself, I am an instructional assistant at Leander High
School in Leander, Texas.  I have two children, Jack, an eighth grader,
and Katie, a fifth grader.  Katie's fifth grade class was given this
hypothetical problem:

A hole is bored through the exact center of the earth, from the North
Pole to the South Pole, A mass is dropped into the hole in the exact
center.

WHAT HAPPENS NEXT?

I supposed the mass would oscillate.  I talked to physics teachers where
I work who agreed with me.  They, in turn, sent me to the calculus
teacher who said this is a pretty common hypothetical problem but not
especially for a class of fifth graders.  The calculus teacher said
there is a formula to figure this out.  Do you know this formula?  Can
you e-mail it to me? Please..

Pat Pattillo
Rt. 3 Box 166-A
Leander, TX 78641
(512) 258-5585
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Date: 10/11/95 at 13:6:13
From: Doctor Jonathan

Mr. Pattillo:

You are right: the object dropped will oscillate, and will eventually
come to rest at the center of the earth (assuming air friction). The
formula needed to figure this out is actually a differential equation,
relating the forces due to gravity, inertia and air friction in terms of
time and position.

As such, this probably isn't a problem that fifth graders could solve
analytically as opposed to just verbalizing an intuitive answer. I could
go through the theory and solution of the differential equation
mentioned above, or I could try to explain what would happen in
intuitive terms at a level appropriate to a fifth grader. Either way or
both is fine, and I'll be happy to provide you with as in depth an
answer as I can, but I just wanted to know what exactly you were looking
for first.

-Doctor Jonathan,  The Geometry Forum
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Date: 10/11/95 at 22:13:16
From: Pat S Pattillo

"Dr. Math"

Thank you very much for the prompt answer.

Now to the meat of the problem. I'll leave to you the way you
would like to explain the solution to this problem. Keeping in mind that
these are 10 and 11 year old students. I just wanted some example to
show to the kids to illustrate the complexity of the problem. It would
be interesting to have a chart or equation to show them that gives a
solution to the problem. I'm sure I couldn't solve a differential
equation if you were sitting beside me and guiding my pencil.

I'm indebted to you and anxiously awaiting your next message.

Mr Pat
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Date: 10/17/95 at 16:7:48
From: Doctor Jonathan

Mr. Pattillo:

Sorry it has taken me so long to get back to you--we just had midterms.

In order to understand this problem to any real degree, two concepts
must be first understood: Force and Inertia. In this problem, the force
is due to gravity, and the inertia is due to the mass of the object. The
force due to gravity can be thought of as a rubber band stretching from
the object to the center of the earth. Obviously, we're pretty far away
from the center of the earth so the rubber band is really trying hard to
pull the object toward the center of the earth. This feels like it's
pushing "down" to us.

So what would happen if we let go of the object over a hole that goes
all the way through the center of the planet and out the other side?
Well, the "rubber band" starts pulling the object "down" into the hole.
As the rubber band pulls on it, the object races faster and faster
toward the center.

When it finally reaches the center, our rubber band is no longer
stretched, and so it exerts no force on the object. However, the object
doesn't just stop, and for the same reason that a car doesn't stop when
you take your foot off the gas. This is called inertia. Inertia is the
tendency of objects that are moving to want to keep moving. The only
way to stop an object from moving is to push on it in the opposite
direction. If someone is ice skating and he can't stop, you have to grab
and pull him the opposite way so he doesn't crash into the wall.

Because of inertia, our object doesn't just stop at the center of the
planet, but instead keeps going "up" the other side of the earth. As it
does so, it starts to stretch the rubber band the opposite way. Like the
brakes on a car, the rubber band starts to slow the object down until it
stops. At this point, the object is just about on the surface of the
opposite side of the earth. Of course it then begins to "fall" back
towards us, and the process repeats itself forever.

If the kids are familiar with Sines and Cosines, the object's location
will be a sinusoidal function of time. An experiment you can do to
demonstrate this would be to suspect a heavy object between two
vertically oriented springs. Stretch them pretty far so that real
gravity doesn't really affect things and also so that they act linearly.
(One spring will work, as the mass will find an equilibrium point where
the spring and gravity cancel out, but it will be hard to explain this
to 5th graders, and will also ruin the visual symmetry of the problem.)
Where the object naturally rests can be thought of as the center of the
earth. If you lift the object up, it will be pushed down by the two
springs acting as gravity. Letting go, the mass will oscillate around
the equilibrium point (center of the earth) as the above discussion
predicted. To show that the motion is sinusoidal, you could attach a pen
to the mass and move a piece of paper behind it at a constant speed.
This might be awkward to do, but the resulting graph with be a pretty
good sinusoid.

Most likely the kids will figure out that eventually the mass will lose
energy and come to rest back at the center. This is because we have
neglected to mention the effects of air friction. Each oscillation, the
object is slowed down by air friction. However, unlike the spring, which
stored the objects energy and gave it back by sending the object
speeding in the opposite direction, energy lost to air friction is
lost forever and eventually takes away all of the object's energy,
leaving it motionless at the center.

Hopes this helps. If you feel any of this needs elucidation or would like
me to come up with a differential equation, feel free to write. I can't
really even give the equation for the solution of the differential
equation, because it involves Sines and Exponentials, which I assume

-Doctor Jonathan,  The Geometry Forum

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