## The Hanging Chain

One of the many simple, charming exhibits in San Francisco’s Exploratorium is a tall metal chain that hangs over a spinning pulley. That’s the whole exhibit. It looks like this:

That video conveys the size and scope of the exhibit, but makes it hard to see what’s going on in detail. Fortunately, YouTube user grahamj21 created this simulation:

The chain, though spinning past, appears to writhe, stationary, in mid-air. This only happens at the bottom of the Exploratorium’s chain. If you tap it on the side, the side itself quickly goes back to being straight, while a moment later the bottom of the chain begins its strange contortions.

When we hit the chain, we send two traveling waves out in either direction from our tap. (The first video incorrectly identifies them as standing waves.) One wave travels in the same direction as the chain. It moves away quickly and we don’t see it. The second wave moves against the chain. This second wave would appear to move fairly slowly to us; the chain might be pulling it up while it is trying to propagate down.

Let’s further suppose that the wave wins this race and swims all the way down the up-flowing chain. When it reaches the bottom, something interesting happens. The further down the chain we go, the less tension. As the tension goes down, the wave speed goes down, too. Finally, at the bottom of the chain, the wave speed and the chain speed match. The wave gets stuck there, and we enjoy watching its pained death throes.

This is the story I’ve told about the chain since the first time I saw it, years ago. I’ve never stopped to check if it makes sense, though. Would the wave speed at the bottom of the chain be a meter or two per second, as the observed speed of the chain is? For simplicity, let’s imagine the chain is stationary and estimate the wave speed at the bottom.

The chain has a mass per unit length $\lambda$, is subject to a gravitational acceleration $g$, is a height $h$, and has a wheel with radius $r$. The wave speed $v$ must be some function of these variables.

Dimensionally, speed is just a length divided by a time, and does not include mass at all. $\lambda$ cannot come into our formula for $v$ – there is nothing else able to cancel its mass dimension. Of the remaining quantities, only $g$ has units of time, so the velocity must depend on $\sqrt{g}$. We can conclude

$v = \sqrt{gl}f(r/h)$,

with $f$ some unknown function of the dimensionless quantity $r/h$. When we look at the chain, we see that $h >> r$, so $r/h$ is a small number, and it’s reasonable to assume $f$ is well-approximated by a first-order Taylor series, $f \approx \alpha r/h$. This gives

$v = \alpha r\sqrt{g/h}$.

I have no particular way to estimate $\alpha$, but we can hope that it’s not too far from $1$. This would mean

$v \approx r\sqrt{g/h}$.

Setting $g = 10 m/s^2$, $h = 3m$, $r = 0.2m$, we get that, roughly speaking

$v \approx 1m/s$.

This is a pretty reasonable speed for the chain to be moving, based on the video. The estimate shows that our hypothesis about why the chain writhes is a reasonable one, without having to go through a complicated calculation. The problem is far from solved. The moving chain accelerates, especially at the bottom, so the tension and wave speeds are higher than for a stationary chain. We know very little about why the chain writhes in quite the way it does, and can’t predict its motion accurately. But we know that our rough understanding makes sense.

The first time I did this problem, I tried calculating the shape of the hanging chain, getting the tension from that, and evaluating the phase velocity $v = \sqrt{T/\lambda}$. This was kind of ugly because it involved solving a transcendental equation on the computer, and it gave me almost the same answer. Such a calculation is hardly superior, since I’m guessing at the dimensions of the chain anyway.