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# Chain conundrum

## Question

More fun with infinity. Agree or disagree?

A chain with finite number of links needs support for its topmost link, or the chain will fall.

But the (much heavier) infinite chain will never fall because each link hangs securely on the one above it.

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unsure

the question is whether the amount of mass the chain has overcomes any gravitiational force required to move it. it will either coil up on itself into a black hole, or not move at all. i tend to think the first more likely, but both are possible.

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My vote is

The chain will not fall, or already has. Being of infinite length, it will either extend infinitely away from the planet making its gravitational pull negligible , or be coiled together infinetly and have no chance of falling as it is clustered together.

If however you have an infinitly long chain suspended in an infinitely large space which has a universal gravitational force exerted on all elements of the chain equally, and the chain doesn't snap under it's own weight or generate a gravitational pull of its own, I'd say the chain would fall infinitely.

There is no resistance to the gravitational forces on it (unless you define it starting from a discrete point where it is anchored and extend infinitely downwards) so the chain will accelerate at the rate the forces acting on it would dictate. However, since the chain is infinitely long it will fall indefinitely as all of it can never hit the ground (if there is indeed a ground in this environment).

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Disagree

I'm going to go more-mathematical:

I'll grant you the inductive step, but not the base case. Show me that any particular link is stationary, and I'll agree.

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I gather that the chain has mass, exists in some gravitational field ("much heavier"), and has at least one endpoint ("...each link hangs securely on the one above it...") I would also define infinity as {the set of all points on a number line} and my chain as {the set of all links along its own length}. My thought is that (1) all links (points) of my infinitely massive chain are able to serve simultaneously as endpoints and linking points, (2) all links of my chain serve both to support all other links and exist as any of the other links, and therefore, (3) the chain serves to support itself.

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there's no such thing as an infinite chain.

Also fun: determine the infinite chain's center of mass.

Edited by gavinksong
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unsure

the question is whether the amount of mass the chain has overcomes any gravitiational force required to move it. it will either coil up on itself into a black hole, or not move at all. i tend to think the first more likely, but both are possible.

It's a paradox, more mathematical than physical, of course, so it lacks a "right" answer.

I'm giving it to Phil for the "Black Hole" factor.

I'm into cosmology at the moment.

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