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Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel...
ask a mathematician dot com ^ | 4-2014 (orig 2012) | physicist

Posted on 04/19/2014 8:40:10 AM PDT by doug from upland

Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

Posted on August 3, 2012 by The Physicist

Physicist: This is a beautiful question, in a small part because it’s an interesting thought experiment with some clever math, but mostly because of all the reasons it couldn’t be done and wouldn’t work. Right off the bat; clearly a hole can’t be drilled through the Earth. By the time you’ve gotten no more than 30 miles down (less than 0.4% of the way through) you’ll find your tunnel filling will magma, which tends to gunk up drill bits (also melt everything).

Jumping into a hole drilled through the Earth. What’s the worst that could happen?

But! Assuming that wasn’t an issue, and you’ve got a tube through the Earth (made of unobtainium or something), you still have to contend with the air in the tube. In addition to air-resistance, which on its own would drag you to a stop near the core, just having air in the tube would be really really fatal. The lower you are, the more air is above you, and the higher the pressure. The highest air pressure we see on the surface of the Earth is a little under 16 psi. But keep in mind that we only have about 100 km of real atmosphere above us, and most of that is pretty thin. If the air in the tube were to increase in pressure and temperature the way the atmosphere does, then you’d only have to drop around 50 km before the pressure in the tube was as high as the bottom of the ocean.

Even worse, a big pile of air (like the atmosphere) is hotter at the bottom than at the top (hence all the snow on top of mountains). Temperature varies by about 10°C per km or 30 °F per mile. So, by the time you’ve fallen about 20 miles you’re really on fire a lot. After a few hundred miles (still a long way from the core) you can expect the air to be a ludicrously hot sorta-gas-sorta-fluid, eventually becoming a solid plug.

But! Assuming that there’s no air in the tube, you’re still in trouble. If the Earth is rotating, then in short order you’d be ground against the walls of the tunnel, and would either be pulverized or would slow down and slide to rest near the center of the Earth. This is an effect of “coriolis forces” which show up whenever you try to describe things moving around on spinning things (like planets). To describe it accurately requires the use of angular momentum, but you can picture it pretty well in terms of “higher things move faster”. Because the Earth is turning, how fast you’re moving is proportional to your altitude. Normally this isn’t noticeable. For example, the top of a ten story building is moving about 0.001 mph faster than the ground (ever notice that?), so an object nudged off of the roof can expect to land about 1 millimeter off-target. But over large changes in altitude (and falling through the Earth counts) the effect is very noticeable: about halfway to the center of the Earth you’ll find that you’re moving sideways about 1,500 mph faster than the walls of your tube, which is unhealthy.

The farther from the center you are, the faster you’re moving.

But! Assuming that you’ve got some kind of a super-tube, that the inside of that tube is a vacuum, and that the Earth isn’t turning (and that there’s nothing else to worry about, like building up static electricity or some other unforeseen problem), then you would be free to fall all the way to the far side of the Earth. Once you got there, you would fall right through the Earth again, oscillating back and forth sinusoidally exactly like a bouncing spring or a clock pendulum. It would take you about 42 minutes to make the trip from one side of the Earth to the other.

The clever math behind calculating how an object would fall through the Earth: As you fall all of the layers farther from the center than you cancel out, so you always seem to be falling as though you were on the the surface of a shrinking planet.

What follows is interesting mostly to physics/engineering majors and to almost no one else.

It turns out that spherically symmetric things, which includes things like the Earth, have a cute property: the gravity at any point only depends on the amount of matter below you, and not at all on the amount of matter above you. There are a couple of ways to show this, but since it was done before (with pictures!), take it as read. So, as you fall in all of the layers above you can be ignored (as far as gravity is concerned), and it “feels” as though you’re always falling right next to the surface of a progressively smaller and smaller planet. This, by the way, is just another reason why the exact center of the Earth is in free-fall.

The force of gravity is F = -\frac{GMm}{r^2}, where M is the big mass, and m is the smaller, falling mass. But, since you only have to consider the mass below you, then if the Earth has a fixed density (it doesn’t, but if it did) then you could say M = \rho \frac{4}{3}\pi r^3, where ρ is the density. So, as you’re falling F = -\left(\frac{Gm}{r^2}\right)\left(\rho \frac{4}{3}\pi r^3\right) = -\left(\frac{4}{3}G\rho \pi\right) mr.

Holy crap! This is the (in)famous spring equation, F = – kx! Physicists get very excited when they see this because it’s one of, like, 3 questions that can be exactly answered (seriously). In this case that answer is r(t) = R\cos{\left(t\sqrt{\frac{4}{3}G\rho \pi} \right)}, where R is the radius of the Earth, and t is how long you’ve been falling. Cosine, it’s worth pointing out, is sinusoidal.

Interesting fun-fact: the time it takes to oscillate back-and-forth through a planet is dependent only on the density of that planet and not on the size!


TOPICS: Science; Weird Stuff
KEYWORDS: science; throughtheearth; tunnel
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1 posted on 04/19/2014 8:40:10 AM PDT by doug from upland
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To: doug from upland
mostly because of all the reasons it couldn’t be done and wouldn’t work.

What? Is he going make fun of Einstein for imagining riding along with a light beam? That couldn't be done either.

2 posted on 04/19/2014 8:42:42 AM PDT by DManA
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To: doug from upland
Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

A: With my eyes closed!!!

3 posted on 04/19/2014 8:42:56 AM PDT by varon (Para bellum)
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To: doug from upland
Cosine, it’s worth pointing out, is sinusoidal.

It can't be said often enough, if you ask me.

4 posted on 04/19/2014 8:43:35 AM PDT by ClearCase_guy
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To: doug from upland

Clever the way the writer keeps bringing back in the “problems” of air resistance, sideways motions, etc.


5 posted on 04/19/2014 8:44:06 AM PDT by Robert A. Cook, PE (I can only donate monthly, but socialists' ABBCNNBCBS continue to lie every day!)
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To: doug from upland

Okay then...I’ll just scratch that one off the list ;-)


6 posted on 04/19/2014 8:45:13 AM PDT by Ouchthatonehurt ("When you're going through hell, keep going." - Sir Winston Churchill)
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To: doug from upland

“how would you fall? “

Get pushed?


7 posted on 04/19/2014 8:46:06 AM PDT by CodeToad (Arm Up! They Are!)
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To: doug from upland

I asked this question years ago, as an extra credit question, in a Physics test. I asked the students to consider only the effect of the gravity. More than half answered that “you would fall out the bottom of the hole”, while other answers ranged from “you would not fall into the hole if the tunnel was dug parallel to the equator” to “who cares?” As the Physics teacher, it was a very humbling moment.


8 posted on 04/19/2014 8:46:17 AM PDT by Former Fetus (Saved by grace through faith)
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To: doug from upland
Once you got there, you would fall right through the Earth again, oscillating back and forth sinusoidally exactly like a bouncing spring or a clock pendulum. It would take you about 42 minutes to make the trip from one side of the Earth to the other.

Cool. I'd always assumed you'd slow down and rest at the center.

9 posted on 04/19/2014 8:46:35 AM PDT by Drew68
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To: doug from upland

No two vertical lines are parallel as a plumb line always points to the center of the earth.


10 posted on 04/19/2014 8:49:13 AM PDT by gorush (History repeats itself because human nature is static)
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To: doug from upland

Well one reason it won’t work, is because when your Dad comes home, the neighborhood kids scatter and your left to watch his head turn bright red when he realizes you’ve dug a 8 foot hole in the back yard. Then you have to fill it in without the help of the neighborhood kids who were helping you dig to China in the first place.

But that was a fun childhood memory.


11 posted on 04/19/2014 8:49:41 AM PDT by DannyTN
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To: ClearCase_guy
Cosine, it’s worth pointing out, is sinusoidal.
I wonder if seeing a psychiatrist would help?
12 posted on 04/19/2014 8:52:08 AM PDT by oh8eleven (RVN '67-'68)
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To: doug from upland

Didn’t even answer the question straight. Guess the answer requires government funds for a research study.


13 posted on 04/19/2014 8:52:35 AM PDT by sagar
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To: All

Didn’t AlGore say that it was ten million degrees just a mile deep?

What sort of drill could withstand those temperatures?!?!?!?

Be hard enough to find material that would withstand the temp of the ‘run of the mill’ magma that we foolishly thought was there before Professor Gore informed us of the truth.


14 posted on 04/19/2014 8:52:36 AM PDT by LegendHasIt
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To: LegendHasIt

Plas-magma.


15 posted on 04/19/2014 8:53:58 AM PDT by cripplecreek (REMEMBER THE RIVER RAISIN!)
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To: sagar
Guess the answer requires government funds for a research study.

Step 1 -- drill the hole.
Step 2 -- drop Obama down the hole.
Step 3 -- observe and report.

16 posted on 04/19/2014 8:56:06 AM PDT by ClearCase_guy
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To: doug from upland
Interesting thought exercise. I read that Earth's gravity is strongest about its rotational axis, meaning that gravity is stronger at the poles than the equator. Gravity is a phenomenon solely associated with mass, and unlike electromagnetism, it seems to be unidirectional, but that doesn't explain centrifugal force. If you punched a hole through a rotating ball, in a vacuum, and allowed an object to pass through the hole, how would that object react?
17 posted on 04/19/2014 8:57:05 AM PDT by factoryrat (We are the producers, the creators. Grow it, mine it, build it.)
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To: Drew68

You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side.

Aside from the impossibility of it, it would be a great transportation method. Maybe it would work on geologically inactive moons.


18 posted on 04/19/2014 9:00:00 AM PDT by cripplecreek (REMEMBER THE RIVER RAISIN!)
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To: doug from upland
You would cheat the undertaker out of his cremation cost!
19 posted on 04/19/2014 9:00:01 AM PDT by dalereed
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To: doug from upland

Can we throw Harry Reid down this hole?


20 posted on 04/19/2014 9:00:25 AM PDT by GrandJediMasterYoda (Hitlery: Incarnation of evil.)
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To: LegendHasIt

With Al Gore in the equation nothing is possible.


21 posted on 04/19/2014 9:00:49 AM PDT by taterjay
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To: GrandJediMasterYoda
Can we throw Harry Reid down this hole?

Nah, he'd just return in a little under two hours.
22 posted on 04/19/2014 9:03:32 AM PDT by cripplecreek (REMEMBER THE RIVER RAISIN!)
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To: doug from upland
it’s an interesting thought experiment

Damn. There go my weekend plans.

23 posted on 04/19/2014 9:05:24 AM PDT by martin_fierro (< |:)~)
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To: doug from upland

Thank you Doug for posting this. Another item crossed off the bucket list.


24 posted on 04/19/2014 9:06:45 AM PDT by upchuck (Support ABLE, the Anybody But Lindsey Effort. Yes, we are the ABLE!!)
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To: doug from upland

That was very interesting. It also explains why I got out of engineering. I’m not good at it.


25 posted on 04/19/2014 9:06:53 AM PDT by St_Thomas_Aquinas ( Isaiah 22:22, Matthew 16:19, Revelation 3:7)
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To: doug from upland

This guy is a real Debbie Downer!


26 posted on 04/19/2014 9:08:26 AM PDT by MNDude
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To: doug from upland

D=25000/3.14


27 posted on 04/19/2014 9:09:45 AM PDT by fedupjohn (America...Designed by Geniuses...Now inhabited by Idiots..Palin 2016...)
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To: DannyTN

There was a guy in my neighborhood that tried to literally, build a stairway to heaven.


28 posted on 04/19/2014 9:10:24 AM PDT by Zeneta (Thoughts in time and out of season.)
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To: CodeToad; All
"how would you fall?
“Get pushed?"
finally..the question; had been answered:
"Who pushed Huppity-Dumpity?"

Ans: the "Collisis (sp?) effect" and gravity!


29 posted on 04/19/2014 9:10:44 AM PDT by skinkinthegrass (The end move in politics is always to pick up a gun. Cattlegate..0'Caligula / 0'Reid? ;-)
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To: ClearCase_guy

The ultimate yo-yo.


30 posted on 04/19/2014 9:13:51 AM PDT by reg45 (Barack 0bama: Implementing class warfare by having no class.)
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To: doug from upland
It would take you about 42 minutes to make the trip from one side of the Earth to the other.

42 minutes? The diameter of the earth is 7,918 miles. That is an average speed of over 11,000 mph.

31 posted on 04/19/2014 9:13:59 AM PDT by FatherofFive (Islam is evil and must be eradicated)
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To: taterjay
With Al Gore in the equation nothing is possible.

And he would remind us that it is a million degrees down there.

32 posted on 04/19/2014 9:14:15 AM PDT by TangoLimaSierra (To win the country back, we need to be as mean as the libs say we are.)
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To: doug from upland

33 posted on 04/19/2014 9:15:18 AM PDT by Domandred (Fdisk, format, and reinstall the entire .gov system.)
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To: cripplecreek
"You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side."

As a retired skydiver, I must question that assertion. A skydiver accelerates at first, but then acceleration slows as one approaches terminal velocity. Terminal velocity is dependent upon altitude (actually the thickness of the air mass at any given altitude) Terminal velocity is at its slowest the closer you get to the earth and presumably slower yet at the earth's center. (assuming our magical tunnel has air but no magma)

34 posted on 04/19/2014 9:15:39 AM PDT by gorush (History repeats itself because human nature is static)
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To: doug from upland
Impossible? It isn't impossible at all! All you need is an 8000-mile-long perfectly straight tube filled with perfect vacuum and with frictionless sides, and the ability to stop the Earth from rotating for only 42 minutes. The rest (as an infuriating old math professor of mine used to say) is merely engineering.

Well, it sure used to spin up the engineering students.

Actually the entire article is a fake. The Nazis will do nearly anything to keep us from discovering that the Earth is hollow and filled with their UFOs.

35 posted on 04/19/2014 9:16:20 AM PDT by Billthedrill
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To: CodeToad; doug from upland; All
lookie here:
http://stratus.ssec.wisc.edu/courses/gg101/coriolis/coriolis.html

36 posted on 04/19/2014 9:16:51 AM PDT by skinkinthegrass (The end move in politics is always to pick up a gun. Cattlegate..0'Caligula / 0'Reid? ;-)
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To: doug from upland
This has been done before when I was a kid I saw a documentary on this subject:


37 posted on 04/19/2014 9:19:30 AM PDT by central_va (I won't be reconstructed and I do not give a damn.)
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To: gorush

Physicists usually ignore friction and drag when coming up with such hypotheticals. This problem is a matter of simple harmonic oscillation. As a pendulum has zero acceleration at its lowest point, the same applies to any object at the earth’s center.


38 posted on 04/19/2014 9:20:08 AM PDT by Hoodat (Democrats - Opposing Equal Protection since 1828)
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To: doug from upland

Damm!

I SO wanted to try this!


39 posted on 04/19/2014 9:20:12 AM PDT by left that other site (You shall know the Truth, and The Truth Shall Set You Free.)
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To: Former Fetus

How do we know that atmosphere plays no part in falling, only mass below?


40 posted on 04/19/2014 9:20:20 AM PDT by amihow
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To: Hoodat

I concur.


41 posted on 04/19/2014 9:22:02 AM PDT by gorush (History repeats itself because human nature is static)
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To: cripplecreek
You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side.

That was my thought also.
Kinda like stepping on an elevator on the ground floor and stepping off on the 26th floor.

42 posted on 04/19/2014 9:22:40 AM PDT by Vinnie
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To: doug from upland
Holy crap! This is the (in)famous spring equation, F = – kx!

Robert Hooke is one of the greatest scientific minds ever.

43 posted on 04/19/2014 9:24:08 AM PDT by Hoodat (Democrats - Opposing Equal Protection since 1828)
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To: doug from upland

If you are in a free falling elevator would jumping up at the very end, just before impact, save your life?


44 posted on 04/19/2014 9:25:45 AM PDT by central_va (I won't be reconstructed and I do not give a damn.)
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To: gorush
No two vertical lines are parallel as a plumb line always points to the center of the earth.

Laplace would be surprised to hear that. That is true on a spherical planet, and on an ellipsoid of rotation, only on the equator and at the poles. It's complicated.


45 posted on 04/19/2014 9:26:17 AM PDT by Lonesome in Massachussets (This is known as "bad luck". - Robert A. Heinlein)
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To: DannyTN; All
"diggin' to China"
or just tell him, "It's a fallout shelter, Dad..just thinkin' of the
future w/Jimmah "Malaise" Carter"
..(now w/0'Muslim)

46 posted on 04/19/2014 9:27:25 AM PDT by skinkinthegrass (The end move in politics is always to pick up a gun. Cattlegate..0'Caligula / 0'Reid? ;-)
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To: doug from upland

They can’t get permits to build the keystone pipeline.


47 posted on 04/19/2014 9:27:34 AM PDT by ThomasThomas (Some learn from others... The rest of them have to pee on the electric fence for themselves.)
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To: FatherofFive; gorush

He’s assuming there is a vacuum in the hole. You would not reach terminal velocity because there would be no air resistance.

You would accelerate all the way to the center, so you’d get moving pretty fast.

However, acceleration due to gravity would steadily decrease as you approach the center. Your momentum would carry you through the center, then you begin decelerating.


48 posted on 04/19/2014 9:28:41 AM PDT by MV=PY (The Magic Question: Who's paying for it?)
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To: doug from upland
When I was about 8 years old, my friends and I decided that we would dig a hole to China. We were serious. We actually thought that if we were determined enough, and gave up watching Speed Racer and Gilligan's Island on the TV, that our perseverance would pay off and our shovels would eventually strike daylight and we'd emerge into a world of smiling Chinese people with fortune cookies, rice paddies and beef teriyaki on a stick (our favorite).

With great enthusiasm, we proceeded to start the hole - in my backyard, because other parents wouldn't let their kids dig in theirs - and after the first afternoon of digging, we actually had a pretty decent sized hole with mounds of dirt and rocks all around.

The next morning, I was woken up to the sounds of my father shouting for me. I came downstairs and my mother grabbed me by the hair and shoved me to the floor in front of my glowering father whose face was rapidly turning a deep shade of red. He simply pointed to the backyard and said "fill it" and walked away.

None of my friends came over to help me fill it. It took pretty much all morning and sure enough, there was not enough dirt to completely fill it and even to this day, some 40 years later, there is still a slight depression in the yard where the aborted hole to China lies unfinished.

49 posted on 04/19/2014 9:30:27 AM PDT by SamAdams76
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To: FatherofFive

http://en.wikipedia.org/wiki/Schuler_tuning

It’s the same period as the period of a satellite orbiting the earth just above the surface.


50 posted on 04/19/2014 9:30:45 AM PDT by Lonesome in Massachussets (This is known as "bad luck". - Robert A. Heinlein)
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