Chain Hangs Over a Pulley and Starts Moving

[haruspex]

I never "claimed" anything of the sort. What are you talking about??
And what are the "two offsetting mistakes" I made?

1. ?
2. ?

In post #26 you appear to have used, directly or indirectly, conservation of momentum and got the right result.
In post #19 you used conservation of energy and should have got the wrong result, but you erroneously took the overall descent of each link of the chain to be h/2 instead of h. That exactly canceled the error resulting from assuming energy to be conserved (because exactly half is lost).

 

[haruspex]

As @haruspex points out, it is not. Energy is lost to the inelastic collision of the chain with each new link that it picks up.

[In practice, there are complexities with the force analysis as well. Torque and angular momentum considerations can lead to a "push-off" from the table. That effect is Googleable]

Fwiw, I have never believed the kick-off-the-ground explanation for the chain fountain. It would explain a reduced loss of energy in the pick-up, but that should only make the whole movement faster. It reduces the tension needed to accelerate the added links.

To explain the mid-air arc we need an increased tension in the top of the ascent. This is where the moment of inertia of the links could do the trick. As each link rounds the bend it tends to lift the ascending side.

 

[hutchphd]

In post #26 you appear to have used, directly or indirectly, conservation of momentum and got the right result.
In post #19 you used conservation of energy and should have got the wrong result, but you erroneously took the overall descent of each link of the chain to be h/2 instead of h. That exactly canceled the error resulting from assuming energy to be conserved (because exactly half is lost).

I must plead guilty to vigorous (and unseemly) hand-waving. Thanks for calling me out.
This is a much more interesting and perplexing (to me) problem than I initially understood
Is the result for a continuous (presumably pliable but massive) rope salient? The requirement of "links" seems odd.

 

[haruspex]

I must plead guilty to vigorous (and unseemly) hand-waving. Thanks for calling me out.
This is a much more interesting and perplexing (to me) problem than I initially understood
Is the result for a continuous (presumably pliable but massive) rope salient? The requirement of "links" seems odd.

Yes, it should apply to a rope as well. I have seen it posed using string.

It's a tricky problem because the posing of it treats the chain/rope as infinitely thin and starting in a pile of zero size, whereas in each real scenario one has to think what is going on at the detailed level.
For a rope the length is constant, so, no matter how thin, it must start as a stack of horizontal sections. When lifted, these will acquire some horizontal velocity, and this will not contribute to overcoming gravity. If no KE were lost it would mean the rope acquires a transverse vibration. This may seem like a detail that could be somehow avoided by a different model, but every model I can think of - rope, chain, telescope - has a way of getting rid of KE.
The chain model, as in the chain fountain, salvages some of the KE by means of rotational inertia, as discussed in the chain fountain link.