a physics question

[caraher]

21. This would be a nice homework problem for an upper-level college course

I don't want to calculate and answer right now but I think the answer depends crucially on assumptions.

One interesting one is the thermal properties of the "unobtanium" - can we assume the temperature of the gas matches the temperature of Earth's interior at any given depth? Or do we assume it is also a perfect insulator, so that the properties of the air in the tube are driven by the atmosphere alone?

I think it's safe to ignore the rotation of the Earth.

JT45242's calculation is a useful starting point. One issue they acknowledged is that "g" diminishes as you go deeper into the Earth. Exactly how it depends on depth depends, in turn, on how you model the interior of the planet. If you treat Earth as having a uniform mass distribution then local "g" is going to be proportional to the distance from the center of Earth, until you hit the surface. But it's understood that the density decreases with the distance from the center, which complicates the calculation of local "g."

Poking around a bit online I found some lecture notes including a calculation of "g" vs. distance from the center of Earth. (The vertical axis is labeled in meters but is obviously supposed to be km.) "g" actually is fairly constant from the surface to around 4000 km, rises to around 10.7 m/s^2 at 3500 km from the center, and smoothly goes toward zero from there (not perfectly linear, but close enough). A serviceable model would have g constant at about 10 m/s^2 to a radius of about 3200 km, then drop linearly to zero as you go to zero radius. So the formula in JT45242's post is pretty accurate until you get to 3200 km radius - assuming a constant air density.

But... that's not a great assumption, either. Air density is not constant in the thin shell of air above our heads, so it wouldn't be constant to a great depth, either. Other folks have commented on air being compressible, and the high pressures JT45242 calculated would certainly suggest that you can't take properties such as density to be constant.

As an aside, some folks note that "g"=0 at the center and suggest that implies a low pressure. But that's not how pressure works. Consider a sliver of air in that 1 square inch channel maybe an inch from the center of Earth. To be in mechanical equilibrium, that little slab experiences a (tiny) downward force of its own weight plus the pressure exerted by all the air above it, which needs to be balanced by the pressure beneath it. What is true is that when you get near the center, the additional contribution to pressure from the weight of a given "slab" of air gets smaller. But there is still the pressure from all the air above that needs to be considered.

When I'm less tired I'll dust off my thermal physics texts and see what I can figure out. It seems likely that you'd wind up leaving the gas phase (model the air as all nitrogen) at some depth. But I don't think you get a solid. If we take JT45242's answer for pressure you get something on the order of 80 MPa, and the phase diagrams I see for hydrogen require a pressure of at least 1 GPa to get solid nitrogen for any plausible temperature for this problem. Though if you wind up having a phase change, that will dramatically change the density... which in turn might really affect the pressure.

I think this problem is more interesting for the discussion of assumptions than for the answer!