Could humans at this time construct a “shell” of Earth material around the Earth, just outside the atmosphere? Thickness doesn’t matter, as long as it’s reasonably durable to debris.
Don’t ask why, just humor me. I know it would cut out the sun and doom us all. I just want to have an idea of how much material it would require, how much of the planet’s mass it would borrow, and what it would likely be made of (besides concrete).
If it were built, would it naturally rotate/revolve in synch with the Earth?
If you’re proposing a “roof”, I daresay it could be held in place by high-orbit satellites. Enough of 'em up there and we spread a fine-mesh net between them all. Shouldn’t be too hard. My only worry is that it’s affected by the satellites, and not the satellites by it - otherwise one wrong tug and the whole lot comes down.
Mind you, I am making this up as I go along. I daresay someone with an ounce of knowledge will be here shortly to provide a proper answer, but for now, if Sam Neill can present a space show just cos he was in a movie about a telescope, I used to write sci-fi short stories (ie: novels that never got finished) so I can waffle on self-importantly about satellites as long as I like.
What you appear to be proposing is a small-scale Dyson sphere. Everything you could possibly want to know about Dyson spheres and their cousins appears at this Web page. The short answer is, a “roof” of the type you are proposing (a solid shell) would have to be mightily thick not to collapse under gravity, and would take far more matter to build than we could conceivably get into a low-Earth orbit at the present time. And even if it were thick enough, every time it got hit by a meteorite, it would move out of place, and in the medium-run it would wind up hitting the Earth (and you thought a small asteroid would wreak havoc . . .).
I totally missed your last question the first time around. Also on the page I linked to is the answer to that question. Newton say, it’ll have exactly as much angular momentum as you give it. However, you wouldn’t want to give it all that much, because, combined with the already-strong gravitational forces, you’d wind up putting waaaaay too much stress on your poor shell. To clarify, your shell is shaped like a sphere. If it rotates, then it has an axis it rotates around, right? The parts at the ‘poles’ don’t move at all (relative to the Earth), while the parts at the ‘equator’ move quite a lot. So the poles are being pulled into the Earth by gravity, the equator is wanting to fly off to infinity (converting its angular momentum into straight-line momentum, the way Newton intended), and somewhere something’s going to give.
As to the material to make it out of, well, take a look at what they’re making skyscrapers and so forth out of these days. Then forget about it, because they’re making them out of reinforced concrete, which is prohibitively expensive to launch into orbit. That’s going to be your biggest expense, getting the stuff into orbit*, so go for something that’s very strong in comparison to its mass. Carbon fibre would probably be ideal, but I’m not an architect or an engineer.
Alternative: Build 200-mile-high struts and support the whole schlemiel from the ground. I have no idea which one would be more expensive; nobody’s ever tried to build a mile-high building**, let alone a 200-mile-high one.
** I know, Frank Lloyd Wright drew up a blueprint for one. Call me back when it’s been built, thanks.