Tube through the Earth

I see Cecil neglected to take account of the tidal forces affecting bodily fluids, which would decelerate the participant at some point before forever. What an amateur.

Never doubt the wisdom of Cecil. His discussion was about the uniform density model, in which the force is a linear function of distance; thus the tidal force does not change as your body oscillates back and forth. If the tidal force is constant, there is no dissipation of energy.

You’re forgetting the bunny rabbit. And the tidal forces lead to the dissipation of energy as heat. How are you suggesting this heat gets turned back into kinetic energy?

ETA: Not to mention that Cecil doesn’t state he’s dealing with uniform density, it’s just been assumed in this thread for ease of calculation.

Thanks for the confirmations, I was just making sure there wasn’t a glitch in my model of reality. I assumed the force was insignificant, my main worry was all the rethinking I was in for if it didn’t exist at all. I’ve had educated people argue there’s a perfect zero gravity at the center even after I’ve increased the hole to 5,000 miles and remade the Earth from neutronium.

rickbdotcom gave a fairly decent estimate of how long it would take to get from the North Pole to the South Pole, but the lack of accuracy clouds a lot of the history associated with this problem. The letter at the beginning of Cecil’s article alludes to why you need a slightly more accurate answer:

"You may recall that Alice wondered about how deep the tube was while she was falling down the rabbit hole into Wonderland. "

Lewis Carroll took this idea one step further to calculate how long it would take to go from one end of a straight tube to the other no matter where the tube were located (obviously, this would only work if the Earth didn’t rotate, but …). If the tube is straight, then a traveler would be falling downhill at some angle for half the trip and their momentum would carry them back up the hill on the other half the trip.

It turns out that, no matter where the ends of the tube are located, the duration of the trip is the same. The time to go from the North Pole to the South Pole by gravity train? 42 minutes. The time to go from New York to London by gravity train? 42 minutes. The time to go from London to Sydney by gravity train? 42 minutes. Etc.

Lewis Carroll was fascinated by this idea. It’s the reason that the number 42 constantly pops up in his Alice books - including one of my favorite puns: Why was the mad hatter mad? He planned tea for two, but Alice showed up for tea, too.

I imagine other readers might think of at least one other author that seemed to like the number 42.

I assumed you were talking about the tidal force from the Earth’s gravity causing the blood to slosh back and forth with each oscillation. My point is that no sloshing would occur and no heat would be generated since the tidal force in the uniform density model is constant. If you were talking about the tidal force of the moon, yes that does oscillate every twelve hours but that is much much smaller than the already negligible tidal force from the Earth. The only reason the Moon has observable tidal effect on the Earth is because the Earth is so much bigger than a person.

You’re absolutely right! My bad.

Just to be a really annoying pendant, if we pretend that Cecil had been talking conditions of uniform density, and ignore non terrestrial gravitational sources, wouldn’t there be a differential in gravity across an individual’s height? Obviously this is minuscule for Earth, but presumably non-zero, hence your blood would imperceptibly slosh towards the extremity closest to the Earth’s center of mass. I’ll reserve the Moon as a tidal back-up to help speed things along, not that the energy dissipation would be speedy in any event.

Yes, this is exactly my point. There would be a differential in gravity, which is the tidal effect I’ve been talking about. Under the standard assumption, the gravitational force is exactly proportional to the distance along the tube. Thus, the difference in force between your head and your toes is a constant, a few milligrams, it doesn’t matter whether you are at the center of the Earth, the surface, or somewhere in between. The dissipation due to the tides is caused by the time variation of the tidal force. If you have a constant tide, it just stretches things out. It is the act of stretching that requires an input of energy.

Incidentally, the magnitude of this constant differential force is exactly the same as the tidal force you “feel” at the surface of the Earth. Inside the tube it is constant (first derivative of a linear function), outside the tube it goes down by the cube of the distance from the center of the Earth (first derivative of the inverse square law). The Moon’s effect on you is vastly smaller, since the the Earth is much more massive than the Moon, and more importantly, the cube of the distance from the center of the Moon is many orders of magnitude larger than the cube of the radius of the Earth.

But presuming you don’t rotate at the centre of the Earth, wouldn’t the blood slosh back and forth relatively speaking?

Why would it do this?

I’m not sure I understand the question. When you jump in the hole, your body is already stretched by the Earth’s tidal force. When you reach the center of the Earth it is still stretched by the same amount. When you reach the other side of the Earth, same story. The stretch doesn’t change, so no sloshing.

Going upright into the hole, your feet experience a greater amount of gravity than your head, so blood pools there. However past the Earth’s centre of mass the effect is reversed, so it would begin to pool in your head causing tidal heating. The differential force is constant in size, but would still lead to the dissipation of energy, unless I’ve misunderstood.

Captain,

First, let me apologize for making an error. I’ll explain it in the third paragraph, but it does not change the conclusion.

The tidal effect is a differential force, represented by a compression or tension, not by a direction toward or away from the Earth’s center. After you jump into the hole feet first, your feet feel a weaker pull than your head. The effect is to compress your body, as if your body was in a vise exerting a few milligrams of force* from head to foot. At the center of the earth, there is no net force, but there is still a tidal effect squeezing your body the same amount. When you reach the other side of the Earth, there is still a few milligrams of squeezing force. Since there is no change in the amount of squeezing, there is no dissipation of energy.

In my earlier posts, my mistake was to say that the effect was to stretch the body, rather than squeeze it, and to say that the tidal force was identical to that at the surface of the Earth when completely out of the hole. In fact, the two effects are opposite in sign (since gravity gets weaker with distance in the inverse square law and stronger with distance within the hole). There is also a factor of two difference in magnitude. Here are the formulae for the tidal effects:

Outside the hole: -2GM/x^3 (where x is the distance from the center of the Earth)
Inside the hole: GM/R^3 (where R is the radius of the Earth, notice no x dependence)

To get the squeezing or stretching force, you need to multiply these terms by the mass of your head and feet and by the distance between your head and feet . Depending on the mass distribution in your body, the squeezing or stretching force at the surface is smaller than your weight by a factor of roughly your height divided by the radius of the Earth, which accounts for my earlier estimate of a few milligrams*.

  • Among friends, please forgive my use of mass and weight interchangeably. Physicists do this all the time. It is only physics teachers who scold their students for doing so.

Just to add one clarification:

It might be helpful to think about riding inside an elevator falling through the tube (a good idea if you hope to breath). Since you and the elevator are in freefall, you will be weightless for the entire journey and feel nothing different going past the center of the Earth or turning around near the surface. In principle, very careful measurements would show the constant tidal effect, but no pooling of blood in either your head or feet.

Thanks for your explainable, I think I’ve finally grasped it. Presumably going from non-linear compression jumping from the outside, to linear compression inside, would have a non-zero, but immeasurable tidal decay effect.

BobG_80918 said:

Except that it is inaccurate. There were at least 4 “people” at the tea besides Alice.

Mad Hatter, March Hare, Doormouse, White Rabbit.

Carry on with the physics.

Hatters of the day were mad because they suffered from mercury poisoning, which was used somehow in the process of making hats.

I’m fairly certain that you would just die from the heat of the earth’s core or the rapid change in atmosphere- but it might be a fun ride if mother nature didn’t shank you at the end…

This topic/cecil response was originally op’d in 2003. Since that thread has seniority, I have placed a note there.Falling trough the center of the earth? - #104 by Leo_Bloom