Allright...you Geology and Physics Guys

[john turner]

i would think that the mass all around you would cause sufficient force to tear you apart when you were in the center of the earth, even if you were just "passing through" as per the presented falling scenarios.

 

[Tituscrow]

Another interesting quirk of this problem...

Suppose your entry point was still at the North Pole, but your exit point was now New York, or Sydney, or Paris, or Manchester....the time taken for you to oscillate back and forth in your new tunnel would be exactly the same as the full North Pole/South Pole tunnel.

 

[mongo2]

Suppose you dug from the north pole

So this is an underwater drilling operation?

 

[BassyBill]

i would think that the mass all around you would cause sufficient force to tear you apart when you were in the center of the earth, even if you were just "passing through" as per the presented falling scenarios.

I was thinking you'd feel half your weight pulling on your head and half pulling on your feet in the opposite direction. Not enough to tear you apart, but still pretty uncomfortable. Not sure about this, though. I'm a chemist really.

 

[BassyBill]

Okay, here’s the model. The top row cells have comments to explain what’s going on. The variables you can play with are in red. My maths isn’t up to doing this the way a proper mathematician would do it, so I’ve used an iterative process in Excel to crunch all the numbers for me.

As linked here, I’ve used a mass of 50 kg, but of course this doesn’t affect the travel at all. The mass (our guy) is dropped from a height above the planet’s surface equal to the radius of the planet. This planet is similar to Earth in terms of radius and has the same gravitational field strength at the surface, but it’s different in a few other ways – no atmosphere, it’s a perfect sphere composed of a solid of uniform density, it’s not moving and it has a big hole through it!

As our little guy falls, he accelerates faster and faster as he approaches the surface of the planet and his weight increases during the first part of his 12 million metre journey towards the centre. You’ll see that sometime between 2010 and 2011 seconds, the ratio of his distance from the centre to the radius of the planet reaches 1 and he enters the hole experiencing normal Earth gravity (weighing his usual weight), having completed the first six million metres of his trip to the centre.

Now he’s inside the hole, his weight starts to decrease and the formula used to calculate g changes from this point on (see cells D2012 and D2013). He’s still accelerating, but his rate of acceleration gets less and less as he approaches the centre. Of course, he completes the second half of the journey in much shorter time than the first. But by the time he’s near the centre, his weight is so low that his velocity is very nearly constant and close to the maximum that he will reach. 2624 seconds into his trip, he is just under 10.5 km from the centre and is travelling at 11 km/s. The model has done its work at this point and stops here, but our guy will not. During the next second, he will pass through the centre (and be weightless at that exact point) and then fly on towards the other end of the hole and right out into space, converting kinetic energy back to gravitational potential energy and so slowing down all the way until he comes to an instantaneous stop exactly six million metres above the surface on the opposite side to where he started, just slightly under 2625 seconds after he passed through the centre. Then he’ll make the return trip following an exact reverse of his previous journey, and so on and so on...

Here's a link to the spreadsheet.

Link Removed

Here’s a few charts showing some details of the portion of his journey from the starting point to the centre of the planet. The top left chart is one quarter of a sine wave.

Link Removed

And here's links to the formulas used for g:

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/sphshell.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/earthole.html#c1

 

[BassyBill]

Another interesting quirk of this problem...

Suppose your entry point was still at the North Pole, but your exit point was now New York, or Sydney, or Paris, or Manchester....the time taken for you to oscillate back and forth in your new tunnel would be exactly the same as the full North Pole/South Pole tunnel.

Yes. That's an interesting point, I'd forgotten about that. It's been a while since I thought about this, and previously I'd only done so in a fairly simple, qualitative way, with some misconceptions, it has to be said. That's why I like threads like this one, you always learn something from the discussion.

 

[Dmwphoto]

It has become obvious to me that I have no business reading this thread...

 

[NWB]

Why would you admit you don't know, when it would be easy to google this information AND you get to say you do know the temperature of the Earth's core?

Because I'm too old to remember a lot of stuff and too lazy to google it.

At any rate, nobody actually knows the temperature anyway. There are no direct measurements. It is calculated based on a variety of factors such as seismic data, thermodynamics, etc. each with their own set of uncertainties. That said, I do accept those calculations as a reasonable estimate.

 

[NWB]

i would think that the mass all around you would cause sufficient force to tear you apart when you were in the center of the earth, even if you were just "passing through" as per the presented falling scenarios.

I don't think so. The dense mass of Ni, Fe at the core around you would have a gravitational attraction in all directions so that each acting force would have an equal counteracting force resulting in weightlessness.

 

[Munjibunga]

... he is just under 10.5 km from the centre and is travelling at 11 km/s.

He's really hauling ass, in other words. That velocity is greater than escape velocity. It's a good thing he slows down by the time he reaches the other side of the earth, otherwise, he'd go shooting into space never to return.

 

[BassyBill]

He's really hauling ass, in other words. That velocity is greater than escape velocity. It's a good thing he slows down by the time he reaches the other side of the earth, otherwise, he'd go shooting into space never to return.

Yup. 11 km/s is about Mach 33.

 

[john turner]

I don't think so. The dense mass of Ni, Fe at the core around you would have a gravitational attraction in all directions so that each acting force would have an equal counteracting force resulting in weightlessness.

well, now, that's the thing. no gravity =/= equal but opposite gravity except if it was acting on a point mass. this would not be acting on a point mass, but on a mass with non-zero volume. maybe the forces would cancel each other out at the floating body's COM, but any displacement from that is going to result in shear, i would think, in all directions.

I was thinking you'd feel half your weight pulling on your head and half pulling on your feet in the opposite direction. Not enough to tear you apart, but still pretty uncomfortable. Not sure about this, though. I'm a chemist really.

i would think the forces from the earth surrounding felix's (our traveling guy) body would act to also introduce increasing shear as he fell - he's falling through successively larger donuts of earth-mass as he falls to the center, and these are going to pull at him from all directions with greater force as he approaches the center.

i need to model this. i think it would be interesting

edit : the issue here is (i think) that folks are assuming non-deformable point-mass bodies in these situations, but this is not appropriate here. our falling guy is a deformable, non-pointmass (i.e. volumetric) body that we're talking about. to appropriately model this situation would require some kind of def-bod simulation like a finite element sim.

 

[Tituscrow]

well, now, that's the thing. no gravity =/= equal but opposite gravity except if it was acting on a point mass. this would not be acting on a point mass, but on a mass with non-zero volume. maybe the forces would cancel each other out at the floating body's COM, but any displacement from that is going to result in shear, i would think, in all directions.



i would think the forces from the earth surrounding felix's (our traveling guy) body would act to also introduce increasing shear as he fell - he's falling through successively larger donuts of earth-mass as he falls to the center, and these are going to pull at him from all directions with greater force as he approaches the center.

i need to model this. i think it would be interesting

edit : the issue here is (i think) that folks are assuming non-deformable point-mass bodies in these situations, but this is not appropriate here. our falling guy is a deformable, non-pointmass (i.e. volumetric) body that we're talking about. to appropriately model this situation would require some kind of def-bod simulation like a finite element sim.

Assuming point masses is a huge assumption when determining gravitational field strengths of planets and stars and the gravitational forces between them.

Fortunately, the distances are so vast that the point-mass assumption works.

It's not unlike the derivations of the formulae to determine the behaviour of molecules within (ideal) gases. Build your model around a single molecule and scale it up...the number of molecules is so big that any anomalies are ironed out statistically.

As to your point about Felix being stretched/sheared...suppose he approached the event horizon of a black hole and the force was so strong it stretched him into a single line of his constituent atoms....

...anyone care to crunch the numbers and work out how long this line would be? Assume a 70kg human consists of 7^27 atoms, and each atom is a few femto-metres in diameter...

I make it to the sun and back over 70 times.

 

[BassyBill]

well, now, that's the thing. no gravity =/= equal but opposite gravity except if it was acting on a point mass. this would not be acting on a point mass, but on a mass with non-zero volume. maybe the forces would cancel each other out at the floating body's COM, but any displacement from that is going to result in shear, i would think, in all directions.



i would think the forces from the earth surrounding felix's (our traveling guy) body would act to also introduce increasing shear as he fell - he's falling through successively larger donuts of earth-mass as he falls to the center, and these are going to pull at him from all directions with greater force as he approaches the center.

i need to model this. i think it would be interesting

edit : the issue here is (i think) that folks are assuming non-deformable point-mass bodies in these situations, but this is not appropriate here. our falling guy is a deformable, non-pointmass (i.e. volumetric) body that we're talking about. to appropriately model this situation would require some kind of def-bod simulation like a finite element sim.

Yeah, I hear what you're saying. Although the guy's centre of mass would feel no gravitational pull, the other regions of his body would. Tidal stretching and squeezing forces may be the key here. I think the post of mine you quoted may be close - just thinking about things informally - but I don't have the math to prove it. Imagine making the mass of the planet into two equal touching spheres and then putting the guy at the boundary in a suitable hole, head one way, feet the other.

 

[Tituscrow]

I think you're also forgetting some important biology here too. We have evolved a skeleton that can withstand the earths maximum pull at the surface, in that we don't get 'pulled flat' by the huge ball of rock beneath us. It makes sense then to think that any subsequent combination of forces pulling from any number of directions would not give us even the slightest 'stretch.

Even a gentle breeze can blow a feather upwards, thus momentarily defeating gravity in a tug-of-war over a feather.

A small magnet will hold a paper clip indefinitely as the earth below struggles in vain to detach it and pull it earthwards.

There is a reason why gravity is the fourth and most mysterious force of all the fundamentals - it's just so weak, by many orders of magnitude, compared to the strong nuclear, weak nuclear and electromagnetic forces.

 

[BassyBill]

I think you're also forgetting some important biology here too. We have evolved a skeleton that can withstand the earths maximum pull at the surface, in that we don't get 'pulled flat' by the huge ball of rock beneath us. It makes sense then to think that any subsequent combination of forces pulling from any number of directions would not give us even the slightest 'stretch.

Even a gentle breeze can blow a feather upwards, thus momentarily defeating gravity in a tug-of-war over a feather.

A small magnet will hold a paper clip indefinitely as the earth below struggles in vain to detach it and pull it earthwards.

There is a reason why gravity is the fourth and most mysterious force of all the fundamentals - it's just so weak, by many orders of magnitude, compared to the strong nuclear, weak nuclear and electromagnetic forces.

Gravity's weakness doesn't really come into this, though. Yeah, it's very weak, but you'll still die if enough mass pulls on you hard enough in unsuitable directions.

Having said that and thinking further - you can solve this using the time honoured method of thinking of the sphere as a series of shells of decreasing diameter. As the guy falls down the hole and passes the surface of each imaginary shell, he'll only feel any force from the shells below him, not above. It's as if the larger shells disappear as he passes them. No crippling g forces at any point.

 

[neo 7]

I once had to bury a pet cat that died, and dug a hole about 3 feet deep. I remember being amazed at how cold the dirt was.

 

[Tituscrow]

Gravity's weakness doesn't really come into this, though. Yeah, it's very weak, but you'll still die if enough mass pulls on you hard enough in unsuitable directions.



Having said that and thinking further - you can solve this using the time honoured method of thinking of the sphere as a series of shells of decreasing diameter. As the guy falls down the hole and passes the surface of each imaginary shell, he'll only feel any force from the shells below him, not above. It's as if the larger shells disappear as he passes them. No crippling g forces at any point.

Concentric spheres, point mass and uniform density. That's the way to go.

 

[TOOL460002]

But doesn't the core rotate? That is what keeps our electro magnetic force field thing. So if you dug in to it somewhere, how do you know you wouldn't pop up at the bottom of the ocean?

 

[DerHoggz]

well, now, that's the thing. no gravity =/= equal but opposite gravity except if it was acting on a point mass. this would not be acting on a point mass, but on a mass with non-zero volume. maybe the forces would cancel each other out at the floating body's COM, but any displacement from that is going to result in shear, i would think, in all directions.



i would think the forces from the earth surrounding felix's (our traveling guy) body would act to also introduce increasing shear as he fell - he's falling through successively larger donuts of earth-mass as he falls to the center, and these are going to pull at him from all directions with greater force as he approaches the center.

i need to model this. i think it would be interesting

edit : the issue here is (i think) that folks are assuming non-deformable point-mass bodies in these situations, but this is not appropriate here. our falling guy is a deformable, non-pointmass (i.e. volumetric) body that we're talking about. to appropriately model this situation would require some kind of def-bod simulation like a finite element sim.

The gravitational attraction will never exceed 1g, nothing to worry about as far as shearing. There would only be slight variation in forces on extremities, fractional really, it would be essentially zero net. Felix is safe!