"If you obey all the rules you miss all the fun." - Katharine Hepburn
Question #91780 posted on 06/03/2019 11:12 a.m.
Q:

Dear astrophysicists of the 100 Hour Board,

Imagine I'm standing on the moon and you're standing on Earth. We're both looking up into the sky. Wave - I can see you!

Okay, now imagine that above my head is a mass hanging from a chain. The chain, which is about 238,900 miles long, stretches from me (on the moon) to you (on the Earth); on the other end, there is a second mass dangling above you. In other words, the thing that's suspending the mass above my head is the force of the Earth's gravity acting on the mass at your end of the chain. And vice versa.

Is this possible? That is, could you, in theory, have a chain that spans the distance from the Earth to the moon and has a mass on each end, the two masses balancing each other out without touching the ground and the chain held in tension by the gravity of the earth pulling on one mass and the gravity of the moon pulling on the other?

How big would each mass have to be? How high off the ground could they get before gravity was insufficient to pull the chain taut? How easy would it be to disrupt such a system? (If it helps, you can assume a weightless chain, although I would definitely be interested in an answer that accounts for how mind-bogglingly heavy a 238,900-mile chain would be. And how much force it would have to be able to withstand.)

-Not a physics major

P.S. I actually drew you a lovely ASCII-art diagram of the scenario I'm describing, as I feel like it's hard to visualize. But it turns out the Board can't render it properly. Alas.

A:

Dear Naphm,

This does not directly answer your question, but you might enjoy this related question on whatif.xkcd.com if you haven't seen it already.

-El-ahrairah

A:

Hi!

Your question is awesome. I took the time to attempt a solution that includes a chain with mass. There's some math, so buckle up. First of all, here's the setup the way I think you have described it (there's a tl;dr at the bottom if you don't want to see this):

Original Setup.png

I've simplified your question and made the object between the Earth and the Moon a single "dumbbell" made of two masses (M1 and M2) and a solid bar of length L. Both masses are 10 km above the surface of the planet. I've arbitrarily chosen steel as the material that we're making this out of, because why not? Since it's a single piece, gravity will act on its center of mass as if it were a point mass (i.e. as if it had no shape and all its mass is in one place). The trick is to put the dumbbell between the Earth and the Moon so that its center of gravity is in a very special place: the Lagrangian point, which is the place where the combined gravitational effects of the Moon and the Earth combine together to exactly equal the centripetal force required to keep stuff in orbit. In other words, it's a place where you can set something and it'll stay there. It's right here:

Setup w CoM.png

First, let's calculate the Lagrangian, which I will call x. As I said, both terrestrial and lunar gravity have to combine to make the orbiting object go in a circle. In math-speak, that looks like this:

Calculating LaGrange (correctly).png

That's complex, so I threw it into WolframAlpha and solved it for x (inputing a term for velocity blah blah blah physics physics). The Lagrangian is about 90% of the way to the moon. So now that we know x, we need to pick masses that puts the center of mass of the dumbbell at the Lagrangian point so that it will stay there without falling toward either body. That equation looks like this:

Calculating CoM.png

Here we need to make some choices. I constrained the system by choosing M1 and the material for the cable (MC) so that those masses are known. Then WolframAlpha solved for M2 for me. I picked 1,000,000 kg for M1 and a steel cable. M2 came out as an absolutely massive number: 2.8E16 (28 quadrillion) kilograms. This huge mass is necessary to drag the center of mass almost all the way over to the moon against the placement of the first mass and the cable.

I was going to leave it at that, but this actually presents several problems. I'm going to leave out the math for the rest of this but trust me, it was impressive.

  • The fact that the dumbbell is also orbiting Earth so as to stay aligned between the Earth and the Moon makes it so that there's tons of tension on the Moon-side of the dumbbell. So much that it would tear the steel bar apart. I fiddled with the width of the cable until it was strong enough to withstand the forces and ended up with a 70- or 80-m thick cable. That's a lot, but that's what it takes.
  • That much steel weighs a bunch. The mass of the cable is now on the same order of magnitude as M2. I did a quick check and found that there is actually enough iron on the planet to make that much steel. We'd need something like one one-millionth of all the iron in the world, but it's there.
  • If the masses were spheres of steel, M1 would have a 3-m diameter. Ok, that's fine. But M2 gets up to almost 10,000 m in diameter. Not sure how we'd get it up there or get it moving fast enough to stay in orbit.
  • If you read El-ahrairah's response and click the link, you'll find a more thorough explanation of the fact that you wouldn't be able to just stand under the thing and look up at it. Because the Earth's surface is moving relative to the dumbbell, you'd fly past the dumbbell at a pretty good rate of speed (~1000 mph).
In other words, this is totally impossible and could never be done. But if we could just will a dumbbell in place, it would probably look like what I described above (more or less, the details here are likely wrong since I'm sure I overlooked something and didn't get a second opinion, but I'm pretty sure the general idea is going in the right direction. Anyone who sees mistakes or things I overlooked should definitely submit a comment.)
 
tl;dr: With a 1,000,000-kg mass suspended 10 km above the Earth, you'd need a 28-quadrillion-kg mass on the Moon side connected by an 80-m thick cable to keep it from ripping itself apart. And then you'd still have tons of problems.
 
Cool question.
 
-The Man with a Mustache