## Hanging Slinky Analysis 2: The Pre-Tension Wrinkle

Last time, we saw how the length of a hanging slinky is quadratic in the the number of links, namely,

$\Delta x = \int \mathrm{d}x = \int_0^M \sigma mg \, \mathrm{d}m = \frac {\sigma M^2 g}{2}$,

where M is the mass of the hanging part of the slinky, g is the acceleration of gravity, and $\sigma$ is the “stretchiness” of the material (related to the spring constant k—but see the previous post for details).

And this almost perfectly fit the data, except when we looked closely and found that the fit was better if we slid the parabola to the right a little bit. Here are the two graphs, with residual plots:

Notice how the second one has smaller residuals, less of a conspicuous trend in the residuals, and a different function; using the names of the parameters in the illustration, its model is

$L(n)=Q(n-dC)^2$

That is, it’s just like the $y = kx^2$ model on the left, but translated to the right by an amount dC. Is the parameter dC real? If so, where does it come from?

Put another way, can we use our “answer analysis” perspective—which gave us confidence that a quadratic explanation was a good one—and find an actual physical phenomenon that would give rise to that rightward shift?

## Pre-tension

You bet we can. One name for the phenomenon is pre-tension, a word that in this context must always be hyphenated.

Here’s the deal: Springs have a relaxed, no-load equilibrium state. They try to return to that state. For many springs, if you stretch them out, they try to return to equilibrium by pulling back. And if you try to compress them (think of the springs in a suspension), they push back, trying to expand.

But slinkies are different. Their equilibrium state is fully compressed. Suppose you had a slinky up in the ISS, weightless (what a cool idea; I wonder if they’ve done this) and let it float. It would collapse all the way; it would look just like a slinky in its box, but without the box. If you pushed on it, you could not compress it further—not because the spring pushes back with its spring force, but because the coils are touching. It’s mechanically prevented from compressing further.

This also means that if you apply just a little teeny force to expand it, it won’t expand. You have to apply enough force to overcome the “excess” spring force that’s keeping the slinky collapsed. This minimum force is called the pre-tension, and let’s write it $F_p$.

At this point we can make a hand-waving argument about the data that’s actually pretty good: the parameter dC represents the number of slinks (about 4 with these data) whose weight is the pre-tension force. That is, it takes the weight of 4 slinks to start expanding a slinky. And sure enough, if you look at the picture, that’s about when it starts expanding.

Can we show that this is correct using math and graphs and Greek letters and everything? Sure. I certainly won’t deprive you. But for clarity, here are graphs of the Hooke’s-Law relationship between force and expansion for regular springs and for springs with pre-tension:

Graphs of spring expansion as a function of force in the regular (left) and pre-tension (right) cases. We’re using the form of Hooke’s Law from the previous post, where s = 1/k.