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View Full Version : Hypothesise this science buffs.



MazDan
05-01-2009, 06:52
OK Im gunna throw something at you all that is, typically of me, a bit out of the ordinary.

A few hypothetical situations to consider..........

In all situations you need to imagine that a hole has been drilled right through the earth say about 2 metres diameter from one side going right through the centre and coming out the other. The hole is air conditioned and stays at a constant 24 deg C


1. What would happen to water if it was poured into the hole from one side?

2. What would happen to a solid, say an iron bar, if it was dropped into the hole? Would a stainless steel bar make a difference?

3. Could you move through the hole and come out the other side?

4. Would it make a difference where the hole started and finished to any of the above............ie starting at north pole and finishing at south pole.





Hopefully they will get some of your thought juices moving for a while. I have had much pleasure considering the possibilities over the past few weeks.





One last thing............can you try and keep the answers in terms that everyone can understand.

Cheers.

svacheme3
05-01-2009, 07:15
1) The water would get trapped at the point closest to the center of mass of the Earth, which due to variations in density is NOT the geometric center, so your hole would not intersect it.

2) Same thing.

3) Yes, provided whatever you were traveling in had enough power to move against the gravitational pull, the direction of which would be changing.

4) Yes, because as stated the center of mass and the geometric center are two different points. The equilibrium point would be different if the hole went through the center of mass instead.

Akoto
06-01-2009, 08:13
5. What would happen if you jumped into the hole on mushrooms and got stuck in the middle?

Pander Bear
06-01-2009, 08:53
im pretty sure if you drilled a hole, it would collapse due to gravity... it kind of makes the whole thing moot, rite?

Akoto
06-01-2009, 23:48
^Why wouldn't that be a problem for other holes? Assuming liquid magma was all taken care of, digging a hole of any incomparable size to the planet earth straight down isn't going to collapse because the forces act towards the center of the earth.

I guess towards the center there would be equal force on all sides of the hole, but that place is basically a moon sized ball of iron...I don't think a 2 meter hole is going to be a problem.

For the liquid portion I think we can use some imagination...one massive 50 mol frequency carbon tube perhaps?

INFaMaS
08-01-2009, 01:08
im pretty sure if you drilled a hole, it would collapse due to gravity... it kind of makes the whole thing moot, rite?

In all situations you need to imagine that a hole has been drilled right through the earth..

MazDan
08-01-2009, 01:32
Wow, i thought you guys could do much better than this.

what would the water look like in the middle if thats where you agree it would stay. Would it be in the shape of a ball? Would it cling to the sides?

What about the iron bar.........would it just float arround or be against something or what?

Would the stainless steel bar react differently?

Would something dropped in bounce around forever due to the opposing forces of gravity?

How big would you expect the zero gravity area to be and therefore would that affect something dropped into the hole as it comes out of the zero area and into a gravity affected area only to be pushed back through the zero grav area again.

qwe
08-01-2009, 06:28
^the questions were answered pretty well i think

1. the water would be sitting on the edge of the tunnel, because as someone pointed out, the hole would not go directly through the actual zero-grav center because the earth is not a perfect sphere and the mass is not distributed equally

if it DID go through the gravitational center, yes, the water would eventually be a ball in the center

iron bar, same case as water

the reason they float in the middle: gravity pulls them outward in all directions equally. however, if it gets pulled out just a little, the net gravitational force will then force it back to the center. so we have an equilibrium situation where the system returns to its initial state, "the bottom of an energy well"

stainless steel bar wouldnt react differently, i am pretty sure the earth's magnetic field would not be powerful enough to make a difference between magnetic metals and magnetic metals

yes it would bounce bounce bounce, but like anything, eventually settle

technically the "zero gravity area" is a point. but the earth is big. there'd be appreciable room for a nice chunk of water... think of many many swimming pools. and no matter how much you pour in, btw, they'd ALL stick to the center even if the volume of the water is greater than the volume of the "zero gravity zone" because the water would just fall onto the water in the zero grav zone

to find out the radius of the near zero gravity, i could write a program that treats a 3D sphere as a bunch of mass-points, and calculate the gravitational force at each point, inside and outside the earth. the limitation would be that the simulation would be easiest with a perfect sphere or ellipse, not the actual earth shape, and randomly distributing mass differently at different places would probably turn up an "inaccurate earth" as well

the size of the near zero grav zone would not affect what you drop into it. the earth is big enough. this is assuming of course that the hole actually intersects the gravitational force's center

if the mass is distributed unevenly in the right way, there may be no stable equilibrium point where gravity would pull liquid or objects into free fall within the earth. that is also a strong possibility to consider. then again, there could be multiple equilibrium points where free fall could occur as well

qwe
08-01-2009, 06:39
since you dont like the answers ill try to answer them myself (the original Qs)


1. it'd sit on the edge of the tunnel, on the side of the tunnel closest to the most dense part of the earth. (since the mass is distributed unequally).

to get it to float, you'd have to know the density of all the regions of earth's interior, and via computational simulation find a point where gravity is negligible. there may not be such a point

2. Same thing as the water, just without the shape changing behavior

sidenote.. a sphere of water (or a sphere of any particles) falling towards a gravitational center will turn into an ellipse and (initially) preserve volume. this is because the gravitational force is medium on the middle particles, less on the higher particles, and stronger on the lower particles -- so an ellipse shape forms, and you'd see this as it falls down the tunnel (assuming you have camera(s) watching it fall)

3. with an elevator powerful enough, yes. i'd much rather bungee jump into the earth though. who has not thought of falling into a hole that goes through the center of the earth, bobbing back and forth :) maybe im just a geek, but thats always intrigued me since i was little

4. the earth isnt a perfect sphere, so yeah. you'd need less force to get through, if going between north/south poles, because that is a shorter distance than from one point of the equator to the opposite side

the main factor though, is, you have to make your hole through the earth at just the right place so that the mass distribution around the tunnel is about equal

so, a better thought experiment along these lines, is to imagine a solid sphere with such a tunnel. much less messy than the earth. physics is always easier when you get rid of complicating factors

in any case, we wouldnt be able to construct such a tunnel even with extremely high technology, much higher than we have: we are not going to be able to build an elevator wall that will withstand the earth's guts: i hear there is some strange matter down there, very hot very dense

the previous answers to these questions were accurate, hopefully i helped to understand further

MazDan
08-01-2009, 10:58
if the mass is distributed unevenly in the right way, there may be no stable equilibrium point where gravity would pull liquid or objects into free fall within the earth. that is also a strong possibility to consider. then again, there could be multiple equilibrium points where free fall could occur as well

This in particular is I guess what i was looking for.

Sort of trying to get your imaginations running so that you can surmise other weird and wonderful possibilities.


I was surprised to learn that there would be no magnetic force in play at the centre of the earth..........as I sort of figured it was the most direct route between the two poles I thought it might have been at its strongest and hence hold that bar really straight and hard to move.

Now you need to remember that i am not a scientist but in my head i was sort of hoping that any matter might actually have greater gravitational force on it and hence it would be appearing smaller than it would on the surface or maybe even squashed, lol. So the fact its zero gravity was dissapointing to learn.


Thanks for taking the time qwe, I guess my thought provoking ideas were not so thought provoking after all.






Further question............

Many many years ago I used to read a magazine called OMNI. It was all about stars and stuff.

There was a proposal put in the mag that we could make a magnetically fuelled rocket.

It would consist of a cylinder filled with magnets on the outside and a tube or rocket fitted inside the cylinder.

In a similar manner to that which makes an electric motor move, the poles would be reversed constantly so that they would be drawing towards when needed and forcing away as the rocket goes past.

I sort of thought by now that this technology would have moved forward.

Enki
08-01-2009, 11:25
Many many years ago I used to read a magazine called OMNI. It was all about stars and stuff.

There was a proposal put in the mag that we could make a magnetically fuelled rocket.
Omni was a pretty wonderful magazine. One can still find many issues archived online. I think the propulsion concept you mention is often called a magnetic sail (http://en.wikipedia.org/wiki/Magnetic_sail).

MazDan
08-01-2009, 12:45
Enki, thats interesting but no not what i meant.

Imagine an electric motor but instead of the magnets forcing the inside to turn it forces it to move upwards.

The launch cylinder was massive, like I think maybe 4 klms long.
Obviously being magnetic there was little to no friction other than wind friction.

Enki
08-01-2009, 13:07
http://en.wikipedia.org/wiki/Mass_driver MazDan, is this it?

MazDan
09-01-2009, 06:08
Thats the one Enki.

I see its considered to expensive to run, but with fossil fuels running out, I wonder if it still may be the future.

socko
09-01-2009, 06:48
It's been a while since I took physics, but due to angular momentum, wouldn't whatever object you dropped into the hole tend to bump against the eastern side, assuming it doesn't go through the poles?

qwe
09-01-2009, 09:30
I was surprised to learn that there would be no magnetic force in play at the centre of the earthno no no: you did not learn that, you read it in one of my posts :) i do not know for certain whether the magnetic force could overpower the gravitational force in this circumstance, just to clarify


I was surprised to learn that there would be no magnetic force in play at the centre of the earth..........as I sort of figured it was the most direct route between the two poles I thought it might have been at its strongest and hence hold that bar really straight and hard to move.
well the magnetic force is due to a layer outside of the innermost core, rotating, causing a magnetic field. surely someone in physics knows... would there be magnetic field lines INSIDE the (sphere representing the) rotating metallic layer, or are there only magnetic field lines on the outside? i do not know the mechanism by which the rotating layer produces the magnetic field. still in electromagnetism

btw, i don't think you should say that your Qs were not thought provoking. humanity did not know the answer to these questions at one point in time, or even that such questions could be posed: isnt it thought provoking just to learn, especially to learn that the universe is actually different than what you inferred from your logic? sometimes that gives me a headache, but it's worth it! (the "truth" that is)


due to angular momentum, wouldn't whatever object you dropped into the hole tend to bump against the eastern side
well, 1) what makes the eastern side more special than the western side?
and more importantly, while north and side have meaning because of the magnetic poles, what does east- or west- even mean in this case? any point on the equator has an opposite point--the complete symmetry cancels any possibility of one side being favored, unless we consider uneven mass density

(edit) oh, i believe you are referring to the fact that an object in motion will remain in motion, *conservation of momentum*

so then, you are asking, if we dropped the ball on one side, would it get all the way to the other before "bouncing" back?

the answer is no, for one simple reason: the air in the tube would slow it down. if there was no air, and the mass was distributed totally evenly, yes, the ball would reach the other side--in fact, if you dropped it 2 meters above the sphere's surface, if totally unimpeded, it would fall all the way to 2 meters above the other side's surface! however, in the real world things are messy, the (gravitational) forces do not work out so symmetrically when there are complicating variables like wind resistance

MazDan
10-01-2009, 00:43
well the magnetic force is due to a layer outside of the innermost core, rotating, causing a magnetic field. surely someone in physics knows... would there be magnetic field lines INSIDE the (sphere representing the) rotating metallic layer, or are there only magnetic field lines on the outside? i do not know the mechanism by which the rotating layer produces the magnetic field. still in electromagnetism



Im not sure about the relationship or if indeed there is any however as your probably aware, in order to make an electro magnet, you simply put some windings around a metal pole and connect it up.

Its definitely moe magnetic in the middle than the outside.

Could the same apply to the centre of the earth?

Is it the travelling of electrons over the earths crust that causes magnetism? Is this not similar to the movement of electrons in the electro magnet?

qwe
11-01-2009, 00:02
http://earthsci.org/education/teacher/basicgeol/platec/magfield.gif

and, wiki's answer: http://en.wikipedia.org/wiki/Dynamo_theory


the process through which a rotating, convecting, and electrically conducting fluid acts to maintain a magnetic field. This theory is used to explain the presence of anomalously long-lived magnetic fields in astrophysical bodies. The conductive fluid in the geodynamo is liquid iron in the outer core, and in the solar dynamo is ionized gas at the tacocline
...
In the case of the Earth, the magnetic field is induced by the convection of liquid iron in the outer core, and maintained by the constant induction of magnetic field. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the Coriolis effect caused by the rotation of the Earth. The coriolis force tends to organize fluid motions and electric currents into columns (also see Taylor columns) aligned with the rotation axis
... ...
The field is similar to that of a bar magnet, but this similarity is superficial. The magnetic field of a bar magnet, or any other type of permanent magnet, is created by the coordinated spins of electrons and nuclei within the atoms. The Earth's core, however, is hotter than 1043 K, the Curie point temperature at which the orientations of spins within iron become randomized. Such randomization causes the substance to lose its magnetic field. Therefore the Earth's magnetic field is caused not by magnetized iron deposits, but mostly by electric currents in the liquid outer core.

Convection of molten iron within the outer liquid core, along with a Coriolis effect caused by the overall planetary rotation, tends to organize these "electric currents" in rolls aligned along the north-south polar axis. When conducting fluid flows across an existing magnetic field, electric currents are induced, which in turn creates another magnetic field. When this magnetic field reinforces the original magnetic field, a dynamo is created which sustains itself. This is called the "Dynamo Theory" and it explains how the earth's magnetic field is sustained.


another pic
http://www.nature.com/nature/journal/v421/n6918/images/421027a-f1.2.jpg

Pander Bear
11-01-2009, 02:13
I wish zorn would come here and slap sense into us.

MazDan
11-01-2009, 02:18
I only sort of understood all that qwe.
Does it mean it could be more magnetic in the middle or not?

Akoto
12-01-2009, 17:51
...the Earth's magnetic field is caused not by magnetized iron deposits, but mostly by
electric currents in the liquid outer core.

Well I learned something new


Does it mean it could be more magnetic in the middle or not?

Since magnets get hugely more powerful when you get closer I would have thought yes, but I guess it turns out from what qwe has discovered that the magnet that is the planet earth is really just a hollow shell of a magnet, meaning that the strongest magnetic pull is closer to us on the outside, and since I don't have a regular problem with iron bars being stuck to the floor I think the whole thing is moot.

The total magnetic force of the planet must be something pretty high, but the force acting upon something as small as an iron bar is entirely negligible.

qwe
13-01-2009, 12:21
every statement in this thread is still indeterminate, until zorn collapses the statements into one truth or the other... ah the puns

Akoto
13-01-2009, 20:23
One sec, let me try my zorn call:

"Gravity and electromagnetism are the same thing. Supersymmetry is the study of inc blotter pictures. L. Ron Hubbard is the greatest scientist ever to live"

qwe
14-01-2009, 16:06
i wonder how it feels to be a science geek celeb. roll out the red carpet LOL we are probly embarrassing him. his fault though, going to school for 15 years so that he can answer all of bluelight's idle astrophys Q's :)

zorn
17-01-2009, 05:09
lol.... :D Alright, alright, here's my answer to most of the unanswered questions I saw in the thread. If anyone's wondering anything else, just ask it -- the thread is long & messy and I probably missed some questions in there.


well the magnetic force is due to a layer outside of the innermost core, rotating, causing a magnetic field. surely someone in physics knows... would there be magnetic field lines INSIDE the (sphere representing the) rotating metallic layer, or are there only magnetic field lines on the outside? i do not know the mechanism by which the rotating layer produces the magnetic field. still in electromagnetism

There are magnetic fields inside the rotating layers. Look at the picture of the magnetic field lines around the Earth. Magnetic field lines never terminate; they always make circles. Since they're looping around the outside of the Earth, they have to link up into loops in the inside. And the density of field lines tells you how strong the magnetic field is in a given place. The actual deal is that the field lines loop around places where there's current. (This is one of Maxwell's equaitons: the curl of B is proportional to the current density.) So what you will see is the field lines will "bunch up" together down through the Earth's core. They will get squeezed to roughly the width of the rotating core creating the field. So you'll have a stronger magnetic field inside the Earth than outside. It'll keep getting stronger until you get inside the core, at which point it'll tend to level off in strength. However -- I doubt the field would ever get so strong to be really all that noticeable. The Earth's field is extremely weak relative to the kinds of magnetic fields you're used to seeing in permanent magnets or electromagnets.

The details of how the rotating metallic layers create a magnetic field from near-scratch (there just needs to be a small "seed" field) are very confusing and difficult to understand qualitatively. There are computer models that replicate it quite well... though models of the Earth's actual field (as opposed to simple dynamos) were only achieved quite recently.


It's been a while since I took physics, but due to angular momentum, wouldn't whatever object you dropped into the hole tend to bump against the eastern side, assuming it doesn't go through the poles?

Socko's right. This would be the most noticeable thing. Unless you drilled the hole directly through the N-S rotational pole, if you jumped into the hole you'd be smashed against the eastern side of the shaft.

The reason is the Coriolis force, incidentally the same thing that's responsible for hurricanes (and other very large-scale phenomena) rotating one direction in the northern hemisphere and another in the southern. It's easy to understand qualitatively where it comes from if you think about it: the Earth is actually spinning west-to-east, or counter-clockwise if you look down on it from above the north pole. Let's say you drill your hole through the equator, in the middle of the Earth, and jump in. If it were standing still, no problem. But the Earth keeps spinning, so as you fall, the hole rotates around, and you hit the side of it. You might think you'd hit the western (rear) side of the hole, as it "catches up" to you. But in fact you'd hit the eastern, since when an object spins, stuff on the outside moves faster than stuff closer to the center. So you're moving in the eastward direction faster than the deeper parts of the hole are, and you end up hitting the eastern wall.


I was surprised to learn that there would be no magnetic force in play at the centre of the earth..........as I sort of figured it was the most direct route between the two poles I thought it might have been at its strongest and hence hold that bar really straight and hard to move. The magnetic field will be the strongest there, yes. But it wouldn't "hold the iron bar in place." You're used to very small magnets which act in this way. But the Earth is huge, and as a result the magnetic field near the center will be basically unchanged for miles and miles. An iron bar in the center will have a tendency (though not strong) to align itself with the Earth's field. But it won't feel any significant pull towards the center or away. The magnetic "pulling" is caused by the gradient (change with distance) of the magnetic field. The Earth is big, and so its field changes only over hundreds of miles, and so doesn't have much of this force.

to find out the radius of the near zero gravity, i could write a program that treats a 3D sphere as a bunch of mass-points, and calculate the gravitational force at each point, inside and outside the earth. the limitation would be that the simulation would be easiest with a perfect sphere or ellipse, not the actual earth shape, and randomly distributing mass differently at different places would probably turn up an "inaccurate earth" as well As long as you ignore density variations in the Earth, it's a lot simpler than that, actually, thanks to a cool theorem. For any spherically symmetric distribution of matter, it turns out that two useful things are true: First, if you're outside the spherical shell of matter, its gravitational effect on you is just the same as if all the matter were located at its exact center. And if you're inside the spherical shell of matter, its gravitation effect on you is zero -- no matter where you are "inside."

So if you hollowed out a hole in the center of the Earth, then everywhere in the hole would have zero gravity, not just the exact center. And it's also easy to calculate the strength of gravity anywhere along the chute through the Earth. You just calculate how much mass M is as close/closer to the center of the Earth than you are -- ie, how much mass M is inside the sphere around the center of the Earth whose surface passes through you. Then you calculate out what the gravitational force from that mass M would be if it were all gathered at the center of the Earth -- easy, you just use G*M*m/r^2. And that's it -- all the mass "outside" you can just be ignored; its gravitational effects on you completely cancel out.

qwe
18-01-2009, 03:44
i see what socko was saying now. the earth's rotation. and i see, they'd hit the eastern side if they fell past the center of the earth before colliding with the wall yeah. thanks zorn

about the mass at a point in the tunnel... if you are 50% down the tunnel (halfway to the other side) the gravity is zero. if you are at 25% or 75%, wouldnt the acceleration due to gravity be 50% of the surface's, or aka 4.9m/s^2? so g can be found simply by the percentage of how far down you are between the surface and the center of the earth... multiply this percentage by 9.8m/s/s and that is the acceleration downward?

zorn
18-01-2009, 07:00
You'll hit the eastern side if you hit the wall before the center of the Earth. Since you start on the outside of the Earth, you're moving east faster than the inside of the Earth is! (Tangential velocity = Earth's angular velocity * radius.) So as you close in on the center you move towards the east relative to a straight tunnel/line.


about the mass at a point in the tunnel... if you are 50% down the tunnel (halfway to the other side) the gravity is zero. if you are at 25% or 75%, wouldnt the acceleration due to gravity be 50% of the surface's, or aka 4.9m/s^2? so g can be found simply by the percentage of how far down you are between the surface and the center of the earth... multiply this percentage by 9.8m/s/s and that is the acceleration downward? Well, you need to know the density profile of the Earth. Suppose most of the Earth's mass was in the core, then being 25% through the tunnel (halfways to the center) would make gravity stronger (since you hadn't gone "past" much mass, and were closer to most of the mass.)

But let's assume the Earth's density is constant with radius (it's not, but I don't think this is an incredibly horrible assumption for a zeroth-order guess.) Say you're halfways to the center. How much mass is closer to the center than you? Well, the mass inside a sphere of radius R is rho*4/3*pi*R^3 where rho is the density of the Earth. Here R=R_earth/2, so


mass inside = rho*4/3*pi*(R_earth/2)^3 = 1/8*rho*4/3*pi*R_earth^3 = 1/8*Earth mass

We can ignore the mass not inside, and treat that inside as if it were at the center of the Earth. So the gravitation acceleration is


G*M_inside / R^2 = G*1/8*Earth mass / (R_earth/2)^2 = G*1/2*Earth mass / R_earth^2 = 1/2*g

We did the last step since g is by definition the gravitational accel at the Earth's surface, given of course by G*M_earth/R_earth^2.

So yeah, IF the Earth's density were constant, then when you're halfways to the center, gravity is half as strong. To take into account non-constant density, you just integrate the density to get the mass inside and then do the same calculation. Depending on the distribution of mass, you'll get a different answer. Since the Earth is IIRC denser in the center, gravity will fall off slower than linearly. If the Earth were less dense near its center, gravity would fall off faster than linearly.

Akoto
18-01-2009, 07:44
Sheesh, there's a lot I didn't even think about.

How far down the tube might someone make it before crashing into the wall?

The Coriolis effect would be greatest at the outside where you jump in, and if the earth spins at 1670km/h it would only take a second for you to smash into the side.

Because you have the same speed as the earth, and for the same reason you don't go flying backwards if you jumped on a bus, you would fall all the way towards the center before meeting with the wall opposite the axis of rotation, right?

Air resistance slowing your movement westward would be negligible because it would be pushed by the earth too (no prevailing winds in a 2m hole), but I'm having a hard time picturing what might happen mid way to the center as the earths rotation slows to 1/2.

Would you crash into the western wall too?

Edit: Forget all that. Zorn beat me to asking my question.

zorn
18-01-2009, 09:24
You have the same speed as the Earth at the outside. But the Earth isn't all moving at the same rate. The surface spins at 1670 km/h, but think about halfways down to the center. There, it's moving at only 835 km/h. (Think about the length of the distance a point travels around the center of the Earth -- on the outside it's double that of halfways to the center... but both travel those distances in the same time, 24 hours.) But, you're moving at 1670 km/h eastward when you start and you keep moving at 1670 km/h eastward the whole time -- once you've jumped your velocity E/W doesn't change. So inside the Earth you're moving faster than it and you crash into the eastern wall.

To find the time to hit the side of the wall is simple. Let's say the tunnel is 2*R meters wide. The Coriolis acceleration is given by

a_coriolis=2* (v cross omega)

omega is 2pi/24 hours, v is the speed you fall. Assuming v=terminal velocity for a person ~ 50 m/s, then

a_coriolis=0.007 m/s^2

Then to hit it would take you

time = sqrt(R / (1/2*a_coriolis) = sqrt(R in meters) * 17 seconds

You'd hit it at a speed of 0.12 m/s*sqrt(R in meters).

So for a 30-meter radius tunnel, that's 90 seconds to hit the wall, and hitting it at ~2.4 km/hr.


well, 1) what makes the eastern side more special than the western side?
and more importantly, while north and side have meaning because of the magnetic poles, what does east- or west- even mean in this case? any point on the equator has an opposite point--the complete symmetry cancels any possibility of one side being favored, unless we consider uneven mass density PS -- The Earth spinning west-to-east breaks the symmetry. :) But this is good physics thinking! A lot of physics is seeing & taking advantage of symmetries. Everything you can take care of with a symmetry is one less thing you have to calculate...

rational_animal
19-01-2009, 08:44
Nice answer zorn, however...

>>If it were standing still, no problem. But the Earth keeps spinning, so as you fall, the hole rotates around, and you hit the side of it. You might think you'd hit the western (rear) side of the hole, as it "catches up" to you. But in fact you'd hit the eastern, since when an object spins, stuff on the outside moves faster than stuff closer to the center. So you're moving in the eastward direction faster than the deeper parts of the hole are, and you end up hitting the eastern wall.>>

What about the conservation of angular momentum of the falling object? At all radial distances the angular velocity is the same. You seem to be using the conservation of linear momentum only in your explanation. I would think that if angular momentum is conserved then your inertia will have you falling down the shaft without hitting the sides.

I wonder if there are any miners here with experience of drilling deep holes who may be able shed a small amount of empirical light on the mater? Do your drill shafts experience shear forces? My guess is they don't, but I'd like evidence one way or the other.

qwe
19-01-2009, 18:30
>>I would think that if angular momentum is conserved then your inertia will have you falling down the shaft without hitting the sides.>>

like zorn said, you fall down the hole while earth's surface is moving at X speed, halfway down the earth will be moving at X/2 speed, but you will still be moving at X speed

(when i am talking about these speeds, i mean velocity vectors in the direction in which the surface of the earth is moving towards)

maybe explaining it in different words will help

Papa
19-01-2009, 23:54
I don't think the falling object would have its angular momentum conserved. At the center of the earth its angular momentum L=rxp would be zero as r is zero, while it was definitely nonzero on the surface. Not conserved. Without friction though, the falling object would regain its original angular momentum when getting to the other side of the tunnel. I think zorn's correct... right?

rational_animal
20-01-2009, 14:33
In the absence of any external forces momentum (both kinds) is conserved - its a law of physics.

If I throw a ball straight up it comes straight back down (viewed in my frame of reference on the surface) thanks to inertia. The variation in height makes no difference. This is the same for dropping the ball down a shaft, the direction makes no difference.

In this problem you get a harmonic oscillator - its a well known example.

zorn
26-01-2009, 13:51
What about the conservation of angular momentum of the falling object? At all radial distances the angular velocity is the same. You seem to be using the conservation of linear momentum only in your explanation. I would think that if angular momentum is conserved then your inertia will have you falling down the shaft without hitting the sides. No, you're confusing conservation of angular momentum (true) with conservation of angular velocity (not true). Yes, the Earth spins with constant angular velocity. But this is because the Earth is a solid object; there are forces holding it together.

The angular momentum of something at radius R, rotating with the Earth, is of course
R*P = R*mv = R*m*w*R = m*w*R^2 where w is the angular velocity of the Earth. This is conserved, so if you decrease R (fall towards the center of the Earth), then your w must increase (you seem to be rotating faster than the Earth).

As it must, this gives the exact same answer as using conservation of linear momentum: you crash into the eastern side of the tunnel.



(Incidentally, this is where things like planet rings (eg Saturn's rings) come from. You have to overcome these shear forces to make something stick together, so its parts at different radii will all orbit at a constant angular velocity. There's a range of distances where gravity isn't strong enough to overcome those shear forces. When you have stuff orbiting a planet out at those distances, you get rings, instead of moons.)

zorn
26-01-2009, 13:56
If I throw a ball straight up it comes straight back down (viewed in my frame of reference on the surface) thanks to inertia. The variation in height makes no difference. This is the same for dropping the ball down a shaft, the direction makes no difference. It doesn't, however, not exactly, because the surface of the Earth is not an intertial frame of reference. It's being accelerated, the Earth is spinning it round and round.

You can do physics in a non-inertial frame of reference, you just need to include various "pseudoforces" that show up because of its non-inertial nature. The Coriolis force (http://en.wikipedia.org/wiki/Coriolis_effect) is the simplest & most important of those.

qwe
27-01-2009, 05:07
>>There's a range of distances where gravity isn't strong enough to overcome those shear forces. When you have stuff orbiting a planet out at those distances, you get rings, instead of moons.)>>

you wouldnt happen to know the function, given mass m of the planet, that would return x (outer radius) for the area where rings form? or know how to construct it? assuming all ideal conditions and spherical symmetry etc

zorn
27-01-2009, 14:37
^^^ IIRC it's called the Roche limit. Wikipedia or Google should have the formula for you.