Silly but Interesting Question

[auforum]

If the Earth were hollowed enough so that a man could go to the center of it, assuming no magma, and be placed directly in the center of the Earth and assuming he was more or less symmetrical ( I know ... weird but go with me ) would he:

A: Float in the center because gravity from all sides is keeping him suspended
B: Be torn apart by the forces acting on him in all directions
C: A worm hole opens up and sends him to parallel universe
D: Something I have not thought of yet

I have to know!

Nate

 

[ray b]

A most likely case

B no 1G acc will not pull you apart

C NO

D maybe

 

[shrumeo]

it's 1 g here on the surface
not 1 g at the center

 

[kuenmao]

He would probably be torn apart. Assuming that the Earth is hollow, gravity would act pull him outward. However, the distance between the man and the center of masses of the Earth(cut into pieces) would be much smaller than the distance between a normal person on the surface and the center of the Earth. That means that the gravitational force would be much stronger than mg. Which explains why the person would be torn apart.
Just my guess, may be wrong.

 

[lvlastermind]

he would actually be crushed

 

[cookiemonster]

Gauss's Law... He'd feel nothing.

cookiemonster

 

[Dburghoff]

My guess is that he'd float. He'd be on the inside of a Faraday pail. Just as there's no electric field on the inside of a hollow sphere, there would be no gravitational field (at least due to Earth) on the inside of the gigantic massive sphere.

 

[kuenmao]

Actually...a person is not an infinitely small point, so he would not experience the same gravitational force at each point of his body. He'd be ripped apart.

 

[Janus]

A. He would float. He would also float if you put him anywhere within the hollow shell of the Earth. Gravitational forces cancel out at all points within a hollow sphere not just at the center.

 

[kuenmao]

Whoops, sorry about that, but the man should float. I was wrong in saying that he would be torn apart.

 

[shrumeo]

This is all assuming that there is some safe place for the man to be immune to the crushing pressures at the center of the earth?

Is so, he should float, or if he got pushed off to one side a bit, he might orbit the center of gravity, assuming his safe haven was large enough.

Ah, but how would the orbit of the moon affect him? That puts the center of gravity of the Earth moon system out from the center of the earth. Would he constantly be dragged around the walls of his safe haven?

 

[salamander]

Gravity does not orgin from some mythic central point in the middle of the Earth, that is a missinterpretation of the term "centre of gravity". Gravity depend on mass and distance alone. The formula g=GM/r^2 is a generalisation assuming that you are located outside the massive body, IE, the body is a point object.

My theory is this: gravity can not become stronger than 1 g along any line on any place around or inside a sphere with the same dimensions and properties as our own Earth. This can only occur at one Earth radius from the centre of our hypothetic, Earth-similar sphere, or if you wish, on Earths' surface. As we get deeper inside Earth, gravity from the mass "above" (what's up and down here anyways?) will start cancelling gravity from mass "below" us. I can't figure out the mathematics of this science I'm far too unskilled at this type of geometry (I just about know how to find the centre of mass in a pyramid). My guess though is that the sum of all acceleration any point body will feel allong a straight line through the centre of Earth, regardless of the bodys' position inside the Earth, is 1 g. I don't know how to describe this, but you're all clever people, I think you know what i mean.

This is basicly the same point as Janus proclaimed, except that in my model a person put anywhere BUT the centre of Earth would still be falling towards the centre of Earth. At the centre, it follows naturally from my discussion abowe, the acceleration along any line must be exactly 1/2 g one way, 1/2 g the other way. Thus acceleration cancel and one would simply float. I agree that the uneven distribution of mass of the human body may cause slight a problem to maintain position at centre of Earth, if there was actually a hollow room made at the centre. However that problem will probably have been solved a long time ago when we have developed the neccesary equipment to conduct this experiment...

 

[Chronos]

Easy. You would be weightless, feeling virtually no tugging or compression whatsoever... i.e., you would float. You would, however, drift around very slightly depending where the sun and moon were positioned. Just hope nobody up top kicks a pebble in the hole. That would hurt.

 

[krab]

... I can't figure out the mathematics of this science I'm far too unskilled at this type of geometry ...

This is basicly the same point as Janus proclaimed, except that in my model a person put anywhere BUT the centre of Earth would still be falling towards the centre of Earth.

The mathematics has been figured out, and it's not as you say, but rather as Janus says.

 

[AKG]

Well, Janus argues that all forces cancel out, but that doesn't mean he won't be torn apart. Imagine having your arms bound by two ropes. 1000N are pulling on your left arm, 1000N on the right arm. The net force on you cancel out, but this will probably tear you in two (and I'm not talking about it pulling your arms out of your sockets, consider yourself as a perfectly rigid body if necessary). Of course, maybe this is slightly different because the gravitational forces act on each point, it's hard to make an analogy with ropes that act on a single point.

 

[Goalie_Ca]

e) the "air" pressure would kill him! :rofl:

 

[salamander]

The mathematics has been figured out, and it's not as you say, but rather as Janus says.

Cool. Maths are amazing. Can somebody show me a proof? I'm really interested.

Cheers.

 

[Doc Al]

Well, Janus argues that all forces cancel out, but that doesn't mean he won't be torn apart.

Janus meant what he said: the gravitational force within a hollow uniform sphere is exactly zero at every point. That means that the Earth exerts no gravitational force whatsoever at any point within the cavity. You will not be pulled apart. You will not be pulled at all.

Imagine having your arms bound by two ropes. 1000N are pulling on your left arm, 1000N on the right arm. The net force on you cancel out, but this will probably tear you in two (and I'm not talking about it pulling your arms out of your sockets, consider yourself as a perfectly rigid body if necessary). Of course, maybe this is slightly different because the gravitational forces act on each point, it's hard to make an analogy with ropes that act on a single point.

A better analogy would be a person in outer space exactly between two identical planets. The net force on every bit of that person is zero. He experiences no gravitational force. He is not torn apart.

 

[shrumeo]

Oh! was the hypothetical Earth a hollow sphere? I didn't realize. I thought we were talking about the real earth, and maybe somehow you could get to the center and be protected. But if it's hollow, and there isn't an Earth's worth of mass in all directions, then that would be different.

Hmm, reminds me of ANOTHER sci-fi book, maybe it was a Douglas Adams book (or maybe Asimov. But there WAS an artificial "planet" with an atmosphere but inside, it was a hollow sphere. Tha author described how you couldn't see the other side through the haze of the atm. and how you took little ferries and such through the center, where there were stations and floating islands and such.

 

[Doc Al]

Oh! was the hypothetical Earth a hollow sphere? I didn't realize. I thought we were talking about the real earth, and maybe somehow you could get to the center and be protected. But if it's hollow, and there isn't an Earth's worth of mass in all directions, then that would be different.

Perhaps you should pay closer attention?

And perhaps I should have said that the gravitational field anywhere inside a uniform spherical shell of any thickness will be zero. Assuming the Earth to be uniform, just hollow out a spherical cavity in the center (any size you wish). The gravitational field due to the mass of the Earth is exactly zero anywhere inside that cavity.

 

[krab]

First a note to those who imagine one would be pulled apart. This only happens if there is a force gradient, so that the force is one way on one side of the body and the other way on the other. That's NOT the case in a hollow sphere. In that case, every tiny sub volume is pulled in different directions simultaneously, so the net force cancels at every point of your body. You float.

To Salamander: The proof is fairly simple. Divide the mass of the hollow sphere into thin concentric spherical shells, and consider one shell at a time. You have a point mass at some point P inside the sphere, not at the centre. Draw a cone with apex at the point P, base where it intersects the shell; there will be two cones and two intersections of the cones with the sphere. Now find the net force from the gravitational effect of these two cone bases on point P. The nearer one would have a stronger effect, but it has a smaller area than the farther one, and since the area is proportional to distance squared while the force is proportional to one over distance squared, so the forces from the two intersections are the same but opposite in sign and so cancel. (That's not quite true for finite-sized cones, but you can make the cones as small as you want; the rigorous proof is an integral of course.) Add up all the cones possible that result in covering the whole sphere and you will see that also the whole sphere has no effect. So this applies to all your shells. So the net force is zero.

A much shorter proof is by using Gauss' law, but I'm not sure you have learned this law.

 

[ArmoSkater87]

To put it really short...FLOAT

 

[shrumeo]

I thought Gauss's Law was about magnetic flux and electric fields(?) Does this mean it also applies to gravity?

As in draw a cylinder around a wire, take an area integral, and that will tell you the flux. Can you do that for gravity?

Dern, it's been way too long since I've sat in a physics class.

 

[salamander]

krab:

Yeah that stuf seems logical. I can probably find the integral proof if I do some thinking, hell I think I will, haven't much else things to do these days. My problem was that I was assumiung that the hole would be made just at the place where the person was, and just big enough to fit him. My fault, missunderstood the discusion. Sorry.

And no guys. Nobody is going to be ripped apart anywhere.

Cheers.

 

[quarkman]

I thought Gauss's Law was about magnetic flux and electric fields(?) Does this mean it also applies to gravity?

As in draw a cylinder around a wire, take an area integral, and that will tell you the flux. Can you do that for gravity?

Dern, it's been way too long since I've sat in a physics class.

Someone correct me if I am wrong, but electric and gravitational fields are similar, they are conservative (irrotational). In other words the (gradient operator) cross (field vector) equals zero. Another one of Maxwell's great equations.

As for the Earth as a shell being discussed, I prefer to look at it this way:
The sphere is symmetrical, therefore all the forces are canceled out (am I explaining this correctly?)

 

[GOD__AM]

A. He would float. He would also float if you put him anywhere within the hollow shell of the Earth. Gravitational forces cancel out at all points within a hollow sphere not just at the center.

Possibly in the hollow core you describe something different might happen. It seems to me (and I may be wrong) that the person would experience a slight centrifugal force that would hold him to the shell. I would guess that the larger the hollowed out area the more g-force pushing him to the wall as the relative velocity increases. He would feel weightless until he came into contact with the wall of the shell. This is assuming of course that the hollow still rotates with the rest of the planet.

 

[HallsofIvy]

He would probably be torn apart. Assuming that the Earth is hollow, gravity would act pull him outward. However, the distance between the man and the center of masses of the Earth(cut into pieces) would be much smaller than the distance between a normal person on the surface and the center of the Earth. That means that the gravitational force would be much stronger than mg. Which explains why the person would be torn apart.
Just my guess, may be wrong.

No, the mass "close" to the center of Earth decreases as the cube of the distance from the center so the gravitational force decreases as r(r³/r²= r).
Assuming uniform density, the force would be equal in all directions and the "tidal force" on a "non-point" person would not be large.