## 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:

## Hanging Slinky Analysis 1: Sums to Integrals

Last time, we (re-)introduced the Hanging Slinky problem, designed a few years back as a physics lab but suitable for a math class, say Algebra II or beyond. We looked at the length of the hanging slinky as a function of the number of slinks that hang down, and it looked seriously quadratic.

I claim that knowing that the real-world data is quadratic will help you explain  why the data has that shape. That is, “answer analysis” will guide your calculations.

I beg you to work this out for yourself as much as you can before reading this. I made many many many wrong turns in what is supposed to be an easy analysis, and do not want to deprive you of that—and the learning that comes with it.

Slinkies are great. You can demonstrate waves. You can make them go down stairs. They are super-dynamic physics toys. They make a great sound.

But they are also pretty great when static. Consider, for example, a hanging slinky. How far down does it hang?

Well. It depends.

For this post, I’ll skip the question-posing part of this and go directly to what it mostly depends on: the number of coils (slinks) that are hanging down.

Let’s skip all the way to the data. Here is a graph of the length (in cm) of a hanging slinky as a function of the number of slinks. You should, of course, record your own data, if for no other reason than to experience the glorious difficulty of measuring the distance.

We can pause here and make sure the graph makes sense. What do you see in the slinky itself? How would you describe the spacing of the coils in the hanging slinky? How does that pattern get reflected in the data and in the graph?  Continue reading The Hanging Slinky

## Modeling Digression: Coin Weights

Too much philosophy, let’s get some data!

Part of the motivation for the recent posts on modeling is that I’m writing a paper for a friend. It takes off from the hexnut weight data and cubic model that appear in The (still unpublished, sigh) Model Shop. More on that soon, because I’ve found some interesting features in hexnut data. But first, another friend sent me a link to this post on doghousediaries I now share with you. Check it out.

Because I had just been doing hexnuts, I immediately thought about coins as an alternative: how do you suppose the weight of coins (Physics people: I will be reporting weight in grams. We both know that’s mass, not weight. We can handle that, right?) is related to their size?

I mean, hexnuts look more or less geometrically similar (they’re not, but that’s the other story), so you’d think their mass is cubic in the linear size: a nut that fits a half-inch bolt ought to be eight times the weight of one that fits a quarter-inch bolt. And that model fits pretty well.

So as teachers, we look for situations that stretch that understanding. EGADs has you cut out cardboard squares or circles and weigh them; since the difference in size is two-dimensional, the weight-size relationship is quadratic.

But what about coins? They look kind of two-dimensional, but the big ones tend to be thicker. So are they cubic or quadratic or something in between? Well. Whenever I come back from some other country, I usually have leftover coinage, which gets saved in that jar in the top drawer. And I have a good scale and (I know, I’m a measurement geek) a decent micrometer. So it’s an empirical question.