Falling Through

Cecil's Columns/Staff Reports
1

Regarding:
http://www.straightdope.com/mailbag/mdoughnut.html

I have long imagined having a tunnel that went clean through the Earth. This tunnel would be about 20ft wide, open to land at both ends (which would take some scouting), and be miraculously shielded by such pesky planetary annoyances as pressure, heat, continental drift, and currents of liquid hot magma.

Jumping into our magnificent feat of engineering, you would fall and reach your terminal velocity. The terminal velocity of a skydiver is about 200 kph, depending on the position of the body, and the diameter of the Earth is the equator is 12,750 km, giving you a 32-hour trip to the center of mass (better bring a book).

The first impulse may be to expect shooting past the center of gravity like a ball rolling from the lip of a bowl shoots past the bottom of the bowl, rolls up the other side, and eventually settles at the bottom. But this doesn’t take into account the amount of mass you’re falling through, leaving behind you, also exerting the force of gravity on you in an upward direction. Your terminal velocity would drop steadily.

What then? Would you find equilibrium as you reached the center of mass, or would there be momentum enough to cause you to pass through even marginally before falling back the other way to find rest floating in negated gravity at the center of the planet?

What about the atmosphere? Not long after your entrance into the tunnel, you’d notice a sharp pain in your ears. That would be the atmosphere crushing down on you. But what becomes of atmospheric pressure in an open-ended tunnel, especially once you’ve reached the center where gravity is negated on you?

If there were a ladder running through our tunnel, it would be an easy climb at first. 2 million rungs further, and you’d drag yourself out at the top. If you kept up a brisk pace of one rung per second, you be home for supper in 33 days (23 days of straight climbing, plus 8 days worth of sleeping, plus 2 days worth of lunches and bathroom breaks – please bring a portable sewage container, lest you pollute the center of mass for anyone else falling past you on your way up).

Also, on the way down, mind the sides of the tunnel. Scraping your knuckles at 250 kph stings a bit.

–Chad-Michael–

2

And, of course, there’s Cecil’s take on What if you fell into a tube through the earth?

3

Note that you will hit the sides unless you start at one of the poles, due to the rotation of the Earth.

4

I had a wonderufl converstaion with my sophomore year physics teacher on this very idea. We came up with this

The original idea - that you will go through the center at high speed, not reach the other side before getting pulled back, and go back and forth a few times before settling down in the center. This is partially due to the fact of air resistance, and also due (BUM BUM BUM BUUUUM!) to the changing gravity conditions.

When you are first falling, the gravity acting on you seems to lessen and lessen (in reality, it’s increasing upward and to the sides, but those forces are small). Assuming that air remains at the same viscosity (it wouldn’t really but whatever), you no longer have enough acceleration acting on you (gravity downward) to keep you at 200 kph. You will slow.

You will still pass the Center and do the old back-and-forth, but you won’t pass by far.

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The gravity cancels itself out, for example if you were anywhere inside a huge massive hollow sphere you would be weightless and the sphere would exert no net gravitational forces on you.

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<< and the sphere would exert no net gravitational forces on you. >>

In short, you’d have nothing to sphere but the sphere itself.

7

Bad Dexter! No biscuit.

But settling out in the center would be entirely due to air resistance (or volcanic fume resistance, as the case may be), not to changing gravity conditions. Neglecting air resistance and treating the Earth as having uniform density, you get a perfect simple harmonic oscillator (a rather rare beast, incidentally: Approximate SHOs abound, but perfect ones are very difficult to find), so you would come out exactly the same height on the other side. Now, the Earth doesn’t have constant density, but even so, gravity is a conservative force, so you have to have just the same amount of mechanical energy at all points on your trip. In particular, at the points when you’re stopped, you need to have the same potential energy, which means the same height.

8

Actually, this was once addressed in a short story by Larry Niven, in which a quantum black hole is clumsily dropped into Mars by an angry scientist.

A quantum black hole is a black hole of an extremely tiny size. The one in the story was microscopic… but still had a truly insane mass (naturally), and was falling through Mars, absorbing matter into itself as it fell… making a tiny hole, straight through the planet.

The other members of the expedition naturally asked the scientist what was likely to happen.

He replied, much as Cecil did, that theoretically, at least, the hole would yo-yo back and forth through the planet’s center of mass until it finally reached equilibrium in the planet’s core somewhere. The difference was that, instead of having a tunnel to bob up and down in, the QBH was making its OWN tunnel, and likely whipping around in big loose curves like a Spirograph, since the planet was rotating as the QBH ripped back and forth through it. It would continue to absorb mass from Mars even after achieving equilibrium, and would eventually consume the entire planet, incidentally becoming about large enough to see with the naked eye by that time.

At this point, someone pointed out that they couldn’t leave Mars until perihelion… some six months away. How long would it take the thing to eat Mars?

…at which point the scientist gets a weird look on his face, and begins totting up numbers in his head…

9

Ah, but what side do you hit?

Imagine a hole through the Equator. You jump in. What side of the hole do you hit? I am asking this as a riddle; I know the answer. I feel evil today.

As it happens, it is difficult to find a good thorough essay on this matter. I have found several pieces, but none covers the whole story. I have written one, but it’s over 3000 words long. :slight_smile: I need to edit it and punch it up a bit, and then (sigh) find a place to publish it.

Oh: the atmosphere problem is interesting. Gravity gets smaller as you go down, so finding the air pressure at a given height I believe would be… challenging. This is one idea I haven’t tackled yet. Anyone have any ideas?

10

I will make a guess that you would hit the east side, since your angular velocity (is this a correct term) would be higher on the surface than 50/100/1000 miles down.

I do have a question though, even if gravity lessens as you head toward the center, would the air pressure lessen? given that there is no place for it to escape as it does on the outside of an object like the moon, wouldn’t the air molecules fill up the “vacuum” and equalize the pressure?

11

I’ll pass on doing the analysis – differential equations were never my strong suit. But a nonquantitative thought experiment seems to suggest you’d be hit from the west.

12

I am not much of a mathematician, but from my guesswork, assuming that I understand the concept of the conservation of angular momentum… here is some “proof” of my original guess

at the earth’s surface the diameter is roughly 12,756 kms.
which would give a radius of 2030 kms and a spin rate of 0.14764 km/s

dropping only 30 miles the spin rate becomes 0.14544 km/s so by this time, you are now travelling 2 m/s faster then the earth is 30 km down from the surface.

Seeing as it takes some time to fall 30 km, and the tunnel is only 5 - 6 metres wide, I would assume that the person falling will have long been road rashed out of existance.

If I am wrong, please correct me…

I also wonder would the fact that the earth is travelling 28.x km/s around the sun affect how long it would take to hit the side… since the jumper is no longer being affected by friction contact with the ground… and falling so far, would there be the chance of hitting the nbrth, or south, prior to hitting the east side?

13

You both are correct. It’s kinda funny to think you’d hit the forward wall! Really, this is an example of the Coriolis effect, just in a radial direction. Weird!