In the middle of our Balanced Forces unit, we do a couple of experiments to come up with equations for some of the types of forces we’ve been talking about while drawing qualitative free body diagrams. We tend to do them at the same time and to post-lab them together. I covered the gravitational force last time. Next up: spring force.

## Observe an interesting phenomenon

Bring your packet and pencil. Let’s go next door and look at something cool. [*Okay, actually, we are already next door looking at something cool because we just finished pre-lab-ing the Fg experiment. Anyway—*] Springs are awesome, right? [*Now I'm stretching the spring and they are itching to pick it up and start mashing it together.*]

I think you know what I’m going to ask next.

## What can we measure?

So, what can we measure?

**Spring force.** How? What tool can we use to measure it? (The scale thing from before.) The spring scale? (Yeah, that.) Alright. [*It doesn't bother them how circular this measurement idea is going to be. Oh, well.*] What else?

**How much the spring stretches.** Okay, so the change in length of the spring. (Yeah.) What can we use to measure it? (A ruler…) Great. [*At this point, there might be some confusion on their part between focusing in on what we can measure about the thing we literally just observed and asking different kinds of unguided questions (ex. "How much we can stretch the spring before we break it." — that might be interesting, but it is not the same as identifying what we can measure about the current phenomenon). In any case, they are usually pretty easily guided back to the process of defining the experiment.*] Okay, so we’re deciding to measure how much the length of the spring changes, right? Not the length of the spring itself.

What’s our objective? (Find the relationship between spring force and spring stretch.) So you know what to do now? [*Nodding.*] After you make all the measurements (of spring force and spring stretch), what will you do with the data? (Make a graph.) Perfect!

Hey, before you go—what symbol are you going to use for spring force? (Fs) How about for spring stretch? (Uh… s?) That’s sort of a terrible variable, right? It kind of looks like a 5, or like seconds, or something. Maybe ∆x or ∆L (change in length) would work. I usually use “d” because I know that eventually, one day, you’re going to end up squaring that thing, and it gets kind of annoying to keep writing the parentheses you need because of the ∆. (What does “d” stand for?) Um—”da spring stretch”? Not perfect, I know.

[*There are some quirks in taking data for this experiment. Some students will do something like taping down the spring scale so that it stays in place while stretching the spring with their hands. In that case, the "fixed" end of the spring is not actually fixed (as the force changes, the spring scale gets longer, so the "fixed" end of the spring moves, too, making the measurements that assume spring end location incorrect. The easiest way to take the data is to hold the spring in place (with a hand or something else) and to pull the other end of the spring with the scale itself. I don't tell them any of that before they start, but go around and guide them a bit as they take data. I want to let them have and create their own ideas about how to take the data. And they need to see that their idea really is a problem in order for anything I say to be meaningful for them.*]

## Graphing the data

[*When they start taking this data, they have already used LinReg once (in the Fg experiment just a few minutes ago), so they are already pros.*]

[*They quickly get their data input and start examining their graph.*]

What is the intercept on your graph? (Uh, 0.02.) 0.02 what? (Uh, cm. No wait, Newtons.) Is it 0 N? (No, it says 0.02 N.) Oh. I think it says 0 N, though. (???) Look at the range for the intercept. It includes 0 N, right? (Ok, yes.) So you can’t say it isn’t 0 N. (Well, ok.) Let’s change tactics—*should* it be 0 N? (Um.) What would it mean to have an intercept of 0 N on this graph? (Well, if the spring isn’t stretched, then there isn’t any force. Okay, yes. It should go through (0,0).) And your graph says it does, or at least says that you can’t say it doesn’t. So I think you don’t need to write down an intercept for your equation.

Now that you’ve got the graph all set, just make a quick sketch of it in your notes and write the equation of the line using meaningful symbols and units. When you’re ready, go ahead and make a quick whiteboard of your graphs and equations for both experiments.

## Board Meeting

When you’re finished with the experiment, go ahead and make a whiteboard of your graph and your equation. Make sure you use the symbols from your graph instead of “x” and “y”. Make sure you put units on your slope. [*Note: The board below has the graphs from both experiments at once (which I alluded to in the intro to the post). We talk about them one at a time during the meeting, so I won't discuss the first graph here.*]

Alright! Let’s have a board meeting! Do you remember what we do during these meetings after experiments? Remember that we’re looking to talk about what is the same about our graphs and what is different about our graphs so that we can flesh out the relationship we were hoping to find.

So. What’s the same? What’s different? (They’re all linear.) Everyone had a linear graph? Actually, wait. Did everyone have a linear graph that also went through (0,0)? (Yeah.) So in that case, it’s an even more specific thing. Everyone found the relationship to be directly proportional. What about the slopes? [*Are they all the same, like in our weight and mass experiment? They quickly point out that they were using different springs, of course, and so it makes sense that we all have different slopes.*]

So what does it mean to have a steeper slope on your graph, then? (Ummm… oh! It means it’s not as stretchy of a spring.) Cool. So that slope tells you something about how stretchy the spring is.

## Post-Mortem

You were looking for a relationship between spring force and spring stretch. Was there a relationship? (Yes.) What relationship did you find? (It was directly proportional.)

Cool. So—

And so—

What did that constant tell you? (That’s the slope. It’s telling you about the stretchiness.) Nice. So let’s call this constant the “spring constant”. How about just putting a quick subscript on that constant? So we can write the whole relationship as—

Remember that spot on the front of the packet where we made a table of the forces? There’s a column that we ignored before for equations, and you could store that one there.

—

**Other posts about the Balanced Force Particle Model:
**Building the Balanced Force Particle Model

Common Types of Forces

Force Vector Addition Diagrams

Balanced Forces before Constant Acceleration

Empirical Force Laws: Gravitational Force Experiment

Building Newton’s 3rd Law

*(Next in the series—coming relatively soon!)*

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