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Question about Gravity

Vanyel

The Imperious Leader
Premium Member
Gravity pulls you down, simple idea, the hows and whys are complicated.

My question is:

Imagine you could drill a hole straight through the earth and ignore the increasing heat and molten rock as you get deeper. Now if you were to drop a rock down that hole, gravity would pull it down, but what happens when the rock reaches the center of the Earth? Does its momentum keep it going until it reaches the other side? Or does it stop at the center since gravity is pulling in from all directions? Or does it do both? Meaning that its momentum pushes it past the center, but gravity pulls it back, and it keeps going back and forth like a Yo-yo until it runs out of momentum.
 
Actually the answer is relatively straight forward...

The thing to understand is that there is no gravitational pull inside a hollow spherical mass. This means that as the rock falls deeper and deeper into the Earth, only the mass of the Earth inside of the remaining sphere about the Earth's center is actually still exerting a pull on it. As it reaches the center there isn't any inner spherical mass left to pull on it so there is no longer any gravitational force present.

As for the effect, if the hole has no air (and the hole is running from pole to pole), then the rock would enter a pendulum motion that would last forever. Without something like air friction, the system follows conservation of energy (reaching full kinetic energy as it passes the center of the Earth and full potential energy as it approaches either side).
 
Isn't that how they had the Jupiter 2 escape the pull of gravity in the Lost in Space Movie that came out a while back. Because they didn't have the necessary thrust to go up they decided that due to the planets crust breaking up they decided it better to go through the center of the planet and out through the other side, builidng up enough thrust to break out of the planet there.

I always thought it looked pretty cool.
 
Interesting question...and what would happen to that rock when it reached the center? My answer is that the rock would experience a yo yo effect and slowly settle down suspended over absolute dead center.
 
Anybody familiar with the Keno Don Rosa Unca Scrooge story where our anatidoid heroes accidentally create a hole all the way into Earth's core by spilling some very powerful solvent, and have to go after that solvent before it eats away all matter and condenses it into superdense dust?

Don Rosa got much of the gravity issues right, but he thought that there would be some negative pull towards the end of the journey. And actually he might have had that right, as Earth wouldn't be a hollow sphere; there could be higher-order effects from the basically cylindrar mass surrounding the narrow hole. Or then the Junior Woodchucks got it wrong, and the "negative gravity" they observed was just irregularities in the acceleration of the cart they used.

As the ducks reach the dead center, where the solvent has eaten a little cavity for itself due to yo-yoing for a while, they correctly observe zero gravity - although the solvent itself seems to have great mass from all the superdense waste it has accumulated, an issue that is conveniently forgotten after Unca Scrooge bags the stuff in a solventproof jar.

Other issues not evident in the comic, but not necessarily incorrectly portrayed, either, include the fact that the hole wouldn't necessarily be straight: the solvent would encounter coriolis forces on its way down, possibly influencing its path (even though its progress was necessarily slow and friction-limited as it ate through the rock). But the ducks rode down in a cart hugging the walls, not in a freefall capsule, so the curvature wouldn't affect the adventure.

There's also the classic Jeff Hawke adventure where aliens have pulled a cable through their smallish planet to construct a powerful magnet to regulate their planet's climate, and invite/abduct a few human engineers to help them out after a civil war severs the cable and savages the planet. That one gets the gravity part right, too - and it, too, involves a cart riding along the walls for the most part, leaving the freefall stuff for a single character only.

The Jeff Hawke tunnel was in a vacuum, another thing correctly and convincingly portrayed. The fatigued heroes just didn't think it through, though, so one of them got sucked into the shaft when the lid was forced open. He fell all the way through, using a bottle of compressed gas to compensate for any momentum lost, and emerged at the other pole. (The pole-to-pole ride also eliminated coriolis forces and thus fatal collisions with the walls.)

The Unca Scrooge one was also in a vacuum, with the Junior Woodchucks (or was it Gyro Gearloose?) correctly describing how the presence of 6,700 km of air in the hole would have made the rescue operation impossible. (A quantity of air did enter with the solvent initially, before the shaft was air-sealed, and there was enough of it left unconsumed by the solvent for our ducks to shed their spacesuits at the core.)

Dontcha love it when even Disney gets the science right for once? I wonder how Trek would have handled the concept... They did grasp things like space elevators, even when they fumbled easy ones like black holes or exploding stars.

Timo Saloniemi
 

Not long ago the History Channel's show The Universe had an ep focusing on gravity and they talked about this gravity train concept. Including how it would always take 42 minutes to go between any two points on the Earth's surface. Fascinating.

I can't help thinking of the line from the Pink Floyd song "Have A Cigar".

We're going riding the gravity traaaaiiiiiin.

Robert
 
Actually if there were somehow to be a hollow chamber at the core of the Earth, wouldn't any object at the center of it be pulled AWAY equally in all directions (ignoring, more or less, mass differences in the material surrouding the hollow space) back TOWARD the surface?
 
Isn't that how they had the Jupiter 2 escape the pull of gravity in the Lost in Space Movie that came out a while back. Because they didn't have the necessary thrust to go up they decided that due to the planets crust breaking up they decided it better to go through the center of the planet and out through the other side, builidng up enough thrust to break out of the planet there.

I always thought it looked pretty cool.

Seemed silly to me since after it passes the center of the core it has to fight gravity again.

As for the OQ. My understanding is that, say you had a hollow tube extending from pole to pole and you vaccumed all the air out of this tube to make it virtually frictionless. You've got a ball being suspended at the end of the tube with a magnet (to keep the tube sealed) and you then release your end of the magnet (releasing the ball.)

The ball would begin to fall towards the center of the Earth gaining more and more speed (without air there'd be no terminal velocity) and reaches, I believe, a speed of about 1600 miles an hour until it reaches the core about 80-some minutes later. It rockets past the core and begins to slow down (as gravity is now pulling it "down" back towards the other end of the tube) until it reaches the other side, where it falls back towards the core again, repeating the process.

IIRC, the "pulling" in all directions at the core doesn't effect it since it's moving fast enough to not be all that effected by gravity. It's not moving at terminal velocity-by any means- but it moves past the core fast enough that it doesn't have much chance to be pulled "everywhere at once."

On that note, and this fully me here, I'm not sure gravity "works like that", it tends to pull towards the center of things. I mean, the core isn't being pulled towards the crust of the earth because of everything above is pulling at it.
 
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Actually if there were somehow to be a hollow chamber at the core of the Earth, wouldn't any object at the center of it be pulled AWAY equally in all directions (ignoring, more or less, mass differences in the material surrouding the hollow space) back TOWARD the surface?

Yes, pulled in all directions, but no, not toward the surface. This is what Shaw referred to above. Inside a uniform spherical shell, the gravitational pull in all directions cancels out to zero. Any pull toward the surface in one direction would be cancelled out by the mass in the opposite direction, so you'd effectively feel no pull at all from the shell around you. The only gravity you'd feel would be from any other mass that was inside the shell with you. (In this case, that would be the part of the Earth that's below your current depth. Everything above that depth would cancel out, so the inward pull you'd feel would diminish the closer you got to the center, until it was zero at the center.)

But as Trekker4747 says, if you're falling through the center of the planet, then your momentum takes you through it. And you'd rise up the same amount you fell down, like a pendulum, so if you started stationary at the surface, you'd slow to a stop at the opposite surface (assuming a perfect sphere).

Interestingly, the time the trip would take would be the same no matter what the path of the tunnel, even if it's a chord rather than a diameter (just as the period of a pendulum of a given mass is the same whether it's got a small arc or a larger arc -- because the longer it falls, the faster it gets, so it cancels out). Also interestingly, I believe that the full length of a round trip (the period of the oscillation) is equal to what the orbital velocity would be exactly at the Earth's surface (if the Earth were a uniform sphere with no obstructions, or if you were orbiting, say, an Earth-mass black hole at Earth's radius).
 
Actually if there were somehow to be a hollow chamber at the core of the Earth, wouldn't any object at the center of it be pulled AWAY equally in all directions (ignoring, more or less, mass differences in the material surrouding the hollow space) back TOWARD the surface?

In fact if you're inside a hollow sphere, the total pull of gravity you feel is 0! You wouldn't be pulled in any direction because for each point in the shell that's pulling you toward it, you have an equal amount of force pulling you away from it (from other points in the shell).

If you were outside such a shell and got pulled in (lets say through a hatch of some sort) you would continue moving at exactly the same speed you entered at until you smashed into the opposite side (unless you're lucky enough for someone to have built a hatch there too). If you managed to stop yourself somehow before going splat, you'd float there weightless forever.

Edit: Christopher beat me to it!
 
Actually if there were somehow to be a hollow chamber at the core of the Earth, wouldn't any object at the center of it be pulled AWAY equally in all directions (ignoring, more or less, mass differences in the material surrouding the hollow space) back TOWARD the surface?

In fact if you're inside a hollow sphere, the total pull of gravity you feel is 0! You wouldn't be pulled in any direction because for each point in the shell that's pulling you toward it, you have an equal amount of force pulling you away from it (from other points in the shell).

If you were outside such a shell and got pulled in (lets say through a hatch of some sort) you would continue moving at exactly the same speed you entered at until you smashed into the opposite side (unless you're lucky enough for someone to have built a hatch there too). If you managed to stop yourself somehow before going splat, you'd float there weightless forever.

Edit: Christopher beat me to it!

This would be true, of course, assuming the pull of gravity equally in all directions was LESS than the failure threshold of the material being pulled. Right? Otherwise, you're a mist spraying on the inside walls of the hollow . . .
 
This would be true, of course, assuming the pull of gravity equally in all directions was LESS than the failure threshold of the material being pulled. Right? Otherwise, you're a mist spraying on the inside walls of the hollow . . .

Well no... assuming an ideal case (perfectly round and uniform sphere), the net pull on each individual atom of an object in the middle of a hollow shell is 0. Since the pull on everything is zero then a material can't be pulled apart.
 

Not long ago the History Channel's show The Universe had an ep focusing on gravity and they talked about this gravity train concept. Including how it would always take 42 minutes to go between any two points on the Earth's surface. Fascinating.

I can't help thinking of the line from the Pink Floyd song "Have A Cigar".

We're going riding the gravity traaaaiiiiiin.

Robert

Yes indeed I had this problem as an exercise in optimal control. We had to find using variational principle the optimal path between 2 arbitrary points. Except the optimal path would always result in shorter traversal time than 42 minutes with exception of two points directly opposite of each other.
 
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What I don't get in the Wikipedia article is the wording "all straight-line gravity trains". How can you build a gravity train that runs in a straight line unless that line is from pole to pole? 42 minutes should be a significant timespan for Earth's rotation to affect the trajectory, requiring a non-straight tunnel.

I could see all sorts of shapes for "underground orbits", just as for overground ones, but how can there be more than one straight-line orbit inside a rotating sphere? Or am I taking "straight" overtly literally?

Timo Saloniemi

Edit: I think I get it now: the article refers to all types of gravity train, not just freefall ones. A straight hole with a rail-riding train would suffer slightly increased friction from the coriolis forces pushing it against the rails, as opposed to an ideally shaped hole. But the effect would probably be insignificant compared with the overall friction increase caused by the fact that a train in a straight, non-pole-pole hole isn't in freefall but is forcing itself against the rails at varying strengths of force at varying points of the journey.
 
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Wow, I've learned a lot. Thanks guys. Keep the comments and ideas coming.
 
What I don't get in the Wikipedia article is the wording "all straight-line gravity trains". How can you build a gravity train that runs in a straight line unless that line is from pole to pole? 42 minutes should be a significant timespan for Earth's rotation to affect the trajectory, requiring a non-straight tunnel.
...
Edit: I think I get it now: the article refers to all types of gravity train, not just freefall ones. A straight hole with a rail-riding train would suffer slightly increased friction from the coriolis forces pushing it against the rails, as opposed to an ideally shaped hole. But the effect would probably be insignificant compared with the overall friction increase caused by the fact that a train in a straight, non-pole-pole hole isn't in freefall but is forcing itself against the rails at varying strengths of force at varying points of the journey.

No. We are talking about freefall tunnels. That's the only way to get the 84-minute round-trip period. But for the purposes of the thought experiment, we're assuming a frictionless tunnel -- no air, no contact with any kind of track. Let's say there are maglev tracks keeping the car from touching any of the walls. Any Coriolis effect would be too minuscule to overcome the magnetic repulsion.
 
Edit: I think I get it now: the article refers to all types of gravity train, not just freefall ones. A straight hole with a rail-riding train would suffer slightly increased friction from the coriolis forces pushing it against the rails, as opposed to an ideally shaped hole. But the effect would probably be insignificant compared with the overall friction increase caused by the fact that a train in a straight, non-pole-pole hole isn't in freefall but is forcing itself against the rails at varying strengths of force at varying points of the journey.

The article only refers to really one type of train i.e. the scenario just described by Christopher above. The key assumption is that any point of its travel the train suffers no friction loss.

Of course the 42 minutes is for straight line path between any 2 points but that is not the best solution. The optimal path for any two points can be described by a hypocycloid.

zzoptimalbs5.jpg


If the two points happen to be directly opposite of each other (like pole to pole) the optimal path is the straight line.

The derivation isn't trivial but the solution for the minimum time i.e. the time to traverse the optimal path is actually very simple.

tminah8.png


R: radius of the planet.
g: gravitational acceleration at the surface.
S: Great Circle distance from one point to another.

For example the S from Atlanta to Seattle is 3521000 m. Plug that into the formula above you'll get the time of 1434 seconds or 23.9 minutes.

If S is half of the circumference of the earth the formula produces the expected 42.21 minutes.
 
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