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

[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!

[boop]

Omen 2 See if this works.

[Lonesome in Massachussets]

I am not sure about the millimeter, or the construction of a skyscraper. Assume that a skyscraper is a six sided solid, with four identical rectangular sides and a square base and top, constructed in the mid latitudes, with the sides of its base aligned to North-South and East-West and the vertical sides perpendicular to the ellipsoid at the bottom,as near as possible. If you release a coin (in a vacuum) from the top, it would land slightly to the North of where it was released. From the north side, slightly further than the release point, from the south side, slightly closer. From the east or west side same distance away, but always slightly north.

[IronJack]

But you wouldn’t fall out of the hole on the other side. In fact, you wouldn’t make it to the opposite surface

You can’t gain more energy than you started with.

[Mad Dawgg]

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

You wouldn't make it all the way because about two kilometers down you would burn to a crisp and I know this because Al Gore told me "the interior of the earth is extremely hot, several million degrees..."

[Bobalu]

The closer you get to the center the less you weigh.

At the center there is weightlessness...this is because the mass of the earth is equal in all directions. You are pulled up as much as down, to one side as much as the other..weightless.

A 150 lb man standing on the surface weighs almost the same regardless of where on the surface he stands (bit less at equator) This is because he has the entire mass of Earth beneath him. But if he is at the center of the earth the mass is roughly equal in all directions...so he weighs
0 lb.

As you fall in, the maximum speed you can attain is limited by friction caused by atmosphere....this effect becomes greater as you approach the center because of your declining weight and makes you move slower and slower. Remember that objects of differing mass fall at the same speed only in a vacuum. Air pressure is zero at the center because the air molecules are at zero g....but the friction of air against the body is still present.

Objects always weigh most at the surface of a planet. At any location inside the planet objects weigh less. It would be a very small difference but if you could measure it a man would weigh less in a coal mine than he would standing on the ground. Of course this is due to the weak pull of gravity from the mass of the earth that was above him when he is deep in the mine....

[USFRIENDINVICTORIA]

That was like the cheese slicer from H*ll.

I never even considered the cables. Thanks for the link.

[Lonesome in Massachussets]

Steve,

If it helps you visualize, look at his diagram again:

The building (I assume you are assuming) would follow the line called "perpendicular to the ellipsoid", i.e., "level". The diagram illustrates where the lines cross the surface of the ellipsoid. Move the "direction of gravity line to the left, without changing its direction, until it intersects the top of the "normal to the ellipsoid" line. While not perfectly accurate technically (it's complicated) that should illustrate the idea - the penny falls to the north in the northern hemisphere, to the south in southern hemisphere. (Flip the illustration upside down, "prove" it.)

[steve86]

I understand the diagram — it is pointing out a straightforward consequence of the earth being an ellipsoid — not a sphere — meaning things on the surface that are “vertical” do not point exactly away from the center of the earth, except at the poles, etc. Not a consequence of gravity or the density of the earth, just the shape. Thanks for the diagrams.

[Lonesome in Massachussets]

The ellipsoidal shape is the shape that would be assumed by a rotating liquid held together by gravity and bulging in the middle due to centrifugal (not centripetal) force. The surface gravity arises from the shape - a sphere surrounded by the toroidal bulge. The gravity due to the (homogeneous) sphere would be in the direction of the center of the sphere. The gravity due to the toroid would point towards the part of the toroid that happened to be closer, it would tilt the gravity vector towards the equator, unless you were at the poles or equator, where tilting towards the equator is no tilt at all.

As you get further from the center of the earth, the differences in the tugs from the left and right side become smaller relative to the total gravitational vector, and the field looks more spherical, i.e., the gravity vector points more nearly towards the center of the earth, so the higher you go, the more nearly gravity points towards the center.

[Former Fetus]

I know! I had hoped that some of my students had a good enough understanding of acceleration to come up with that anwer... It was disappointing and humbling (I wasn’t as good a Physics teacher as I thought I was) when nobody came up with the right answer!

[OldNewYork]

I enjoyed this.

[SgtHooper]

Good grief! No wonder you are Lonesome in Mass! /s

[Lonesome in Massachussets]

No, it’s fairly basic navigation knowledge.

[SgtHooper]

Seeeeeeee!

[doug from upland]

Thanks. That looks interesting.

[zeugma]

Some of them are great. The guy has a really twisted sense of humor.

[Vermont Lt]

I can nearly say it once. And I don’t know why I would say it all.

[daniel1212]

One would wonder why it would not be feasible to drill to intersecting holes to that depth, connect piping to them, and pump fluid to serve as geothermal energy steam or other heat.

Reykjavik, Iceland, which heats 95 percent of its buildings using geothermal energy, is considered one of the cleanest cities in the world [source: International Geological Congress Oslo].

Geothermal plants are also considered to be more reliable than coal or nuclear plants because they can run consistently, 24 hours a day, 365 days a year. ..

a home geothermal energy pump can cut energy bills by 30 to 40 percent and will pay for itself within 5 to 10 years [source: Consumer Energy Center]. http://science.howstuffworks.com/environmental/energy/geothermal-energy2.htm

But as in eternity, location is key. Here are some European projects: http://sciencenordic.com/danes-drill-deep-geothermal-energy

Geothermal energy is considered renewable because the heat is continually replaced. The water that is removed is put right back into the ground after its heat is used.

[Future Snake Eater]

The biggest danger would come from venturing too close to the central sun.

Don't forget having to contend with the Mole People

[Oshkalaboomboom]

But you wouldn’t fall out of the hole on the other side. In fact, you wouldn’t make it to the opposite surface

That is what is meant by the Chinaman's point of view. People don't fall up out of holes they fall down into them.