In a recent post, I strongly suggested that a physics class should be a place where students are actively involved in the exploration of the relationships that exist between different variables (force and mass, for example)- not a place where students are simply given a list of equations they are told explain how the world works. Let's continue down this line with an example.

#### Example: Simple Harmonic Motion and Hooke's Law

This is a lab from a college class I took last semester:

#### Analysis

This lab isn't terrible. I mean, who doesn't like bouncing springs?

In the first part, we were required to find the spring constant by examining the relationship between the force applied to the spring and the spring's elongation using a graph. That's not too shabby, right? Well...no...but...

What the lab *doesn't* require is any thinking about the relationship between force and elongation. You make a nice graph, but are told right in the instructions that the slope of the graph is this thing called the "spring constant." We aren't expected to know anything more about how the relationship between force and elongation and the spring constant works.

In part two, we varied the mass on the spring and measured the period of the spring's oscillation, which we then compared to the expected period based upon our calculations and a formula we were given ahead of time:

I didn't need to know much to write up the lab report:

- The period of a spring's oscillation depends on the mass attached to the spring.
- The formula we were given to find the period of a spring's oscillation works.

That's it. If I was an astute student I might've realized that the slope of a Force-Elongation graph will give you the spring constant- but we were walked through that step in such a way that it would have been easy to miss that tidbit. Never mind understanding what having a larger or smaller spring constant would mean in real life.

#### Rethinking the lab

So now you're thinking that I'm just a cranky-pants who likes pointing out the failings of other people's labs. Let me try to improve your perception of myself by explaining how I'd like to run a lab covering the same content.

**First**, I think it's important to identify what I want students to understand as a result of completing this activity. I'd like them to understand:

- The nature of the relationship between the force applied to a spring and the spring's elongation.
- The slope of a Force-Elongation plot is the "spring constant."
- The nature of the relationship between the mass hanging on a spring and the spring's oscillation period.

**Second**, I want the students to play be the primary investigators. I'm not going to give them a sheet explaining step by step exactly what they have to do. I want the students to handle that part. Maybe I give each group of students a few springs and a set of masses and simply set them free to play around and make observations for 10 minutes or so- after which we discuss as a class observations they have made and decide upon a path for further investigation. Maybe I give some guidance right away and tell them to investigate the relationship between the mass on the spring and the elongation of the spring.

**Third**, we draw some Force-Elongation graphs. We discuss the relationship between force and spring elongation (it should be pretty obvious it's a direct linear proportionality- i.e., if you double the force on the spring, you double its elongation). So now we know that . Next, we look at the difference in the graphs for each spring. Why are some lines steeper than others? What is the difference between a spring with a steep slope and a spring with a more gradual slope? Then explain the slope on a Force-Elongation graph is called the "spring constant." So now we've figured out that if we know the force acting on a spring and that spring's spring constant, we can figure out how much the spring will stretch: . Hey...that looks an awful lot like Hooke's Law...

**Fourth**, I'd play this video clip:

**Fifth**, I'd tell students to investigate the relationship between the amount of mass on a spring and the period of the spring's oscillation. We'd collect data, make some graphs, and hopefully come to the conclusion that .

If we stop here, we've already done a lot. We've discovered Hooke's Law. We understand a stiffer spring has a bigger spring constant. We know how doubling the mass on a spring will affect the spring's oscillation. At this point I could introduce the equation . Maybe we could then do the second part of the lab posted above and see how close the observed periods of the springs match the values calculated with that forumla. We'd probably notice all of our observed periods were off by a little bit. This opens up a discussion of why we all have this systematic error. Why are we all off? What could be off? Looking at the formula, there are really only two places we could have error: the spring constant or the mass. Maybe we draw a free-body diagram for the mass on the spring. At this point a student will probably suggest we need to draw a free-body diagram for the spring as well. Hmm...you know...this spring has mass too...could the mass of the spring itself be affecting the spring's period? Now we've independently figured out we need to consider the spring's mass as well. From there we could figure out a test to determine how much of the spring's mass we need to include.

#### Overcoming the traditional lab format

If you randomly visited physics courses in high schools and colleges across the nation, you'd most likely see a *lot* of labs similar to the first lab. Traditionally physics labs have been designed so you're given a formula and are asked to make observations that fit with the formula. This is despite the fact that the student-led investigation requires deeper thinking, encourages greater engagement and thinking about the concepts, a better understanding of how the world works, and an understanding of what an equation actually *means*.

Why should this be so? I believe it's because traditional labs are easy. Print out a sheet with a step by step procedure. Hand out the supplies. Make some measurements. Maybe make a graph. Answer a couple quick questions. Done. The student-led investigation is tricker to share and explain. The entire process I described in the student-led investigation could be preformed without any worksheets whatsoever. It's harder for teachers looking for a new lab to stumble on a description of this type of lab. It's really easy to hit up The Google and find a lab handout, save it, print it, and pass it out. Student-led investigations also lead to potential student errors. Students may struggle. It may take more class time. Sometimes you'll get data that doesn't turn out as well as you'd like. This can be scary and frustrating for teachers. And yet...

Struggling with what this or that graph is telling us, or being forced to think about where errors came from, or having to defend your results and process requires a lot of **thinking**. Critical thinking. And helping students learn to think critically is worth the extra time and effort. As a bonus, they'll also actually understand the physics better, which is also a good thing in a physics class.

Great post!

While I was reading over your post, I thought of this one:

* Given a spring, ruler and three 20g masses, predict the length x that the spring will stretch when a mass of 15g is hung from it. Justify your answer using a careful analysis of data that you collect, including graphs and equations.

This covers 1 and 2 of what you'd like them to understand, because they're going to have to verify that the relationship between force/mass and the displacement is linear, model that line as a best fit line/equation with a specific slope for that spring, and then interpolate to find the desired length.

If you have time, you could give them little guidance on how to do the analysis and then follow this exploration with a short lecture on Hooke's law then have them repeat the lab but with a slightly different question:

* Given a spring, ruler and three 20g masses, predict the length x that the spring will stretch when a mass of 120g is hung from it. Justify your answer using a careful analysis of data that you collect, including graphs and equations.

Same procedure, so students who didn't do a deep enough analysis can improve their write-up, but advanced students can also explore the difference between interpolation and extrapolation, and start to intuit the idea of an elastic limit.

The oscillation frequency/period of the spring seems like such a separate concept, I'd leave it out of that first activity. I like the way you have it in "Fifth", but I think the leap to drawing a force diagram for the spring might be too big for most students; I'd want them to notice it's not just a systematic error in the period vs mass relationship, it's an error of *missing mass* (observed periods are longer than calculated/predicted periods, therefore either k is wrong -- unlikely -- or the oscillating mass is bigger than what we're using to calculate period). What mass could be missing from our calculation? What else in the system that is oscillating has mass? And then go on from there as you describe.

What do you think?

Doh, for the first exploration activity it should be "when a mass of *30g* is hung". Otherwise it would be extrapolation too!

Or better yet, make it something strange like 32g the first time, and 124g the second time. Don't let students get away with simple averages!

@

Micah: I think your suggestion for how to introduce the investigation is pretty good. It's a different entry point into the content- I was starting with trying to establish the linear relationship between force & displacement and then focus on how the spring constant manifests itself in real life. I think if you started with your suggestion (which I like), I'd throw in some investigation where they had maybe three different springs and three 20g masses and had to work out what the spring constant tells you about a spring. Either way I think students would get the relationships down, which to me is the most important part. I'm not sure that even a brief lecture about Hooke's Law would be necessary- other than a quick class discussion on the fact that this F=kx thing they just figured outisHooke's Law. But maybe that's what you meant anyway.As for the oscillation frequency section- I agree it's a separate concept, and I thought about either leaving it out entirely or doing a separate post about it. In the end I included it because it was included in the "traditional" lab I used as a jumping off point for this post. In the classroom I think I'd give it some separation from the first part of the activity- though I do think you could flow fairly smoothly from the first investigation (Hooke's Law) to the second (simple harmonic motion).

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