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?
But then (and the point of posting this today) we can look at the graph and wonder what sort of function might fit these data. Looks kind of quadratic, right? Let’s put a parabola on there, with the coefficient we’re pretty sure we need. (I also put the data in the data zoo, at this page.)
The parameter P is about 0.02. And it looks (on the face of it) to be a pretty good model. Now we can go in a number of directions.
Here’s one: what does P mean? If you know some physics, you’ll be thinking about the Hooke’s spring constant, but it will not be obvious at all what the relationship is. But if you’re not a physics person, you can still try to make sense of it. For example, if P were bigger, what would the curve do? (Curve up more steeply) What does that mean about the spring? (The same number of slinks would hang farther.) What would cause that? (Maybe the spring is heavier, or the spring is less springy, a weaker spring.)
Another question is, why is this quadratic?
At this point, I want to go all meta and harp on this: it’s perfectly OK if you have no idea why it’s quadratic. If you could have predicted it, you can use the data to validate your idea. But if you didn’t, you can just use the model, or (my favorite) you can use the data and the fact that it appears quadratic to illuminate the situation and extend your understanding.
I bring this up in a blog likely to be read by math teachers because it is more obvious, with a physics context, that this could be a legitimate school task even if we don’t understand the underlying science. Put another way, there’s nothing wrong with answer analysis.
But I did get a request for explanations, and I had failed to post them where I said I would. So I will treat that question—why is it quadratic?—in the next post. Try to figure it out before you read it!
By the way, this is one of a whole bookful of physics labs designed to better incorporate mathematical modeling. A Den of Inquiry. Two volumes, no waiting.