Note: This post details my version of the paradigm experiment for what is Unit 5 (Constant Force Particle Model) of the Modeling Instruction Mechanics curriculum. My paradigm experiment is different from the official one (which uses a modified Atwood’s machine), and my students build this as their fourth model.
Hey guys, I want to show you something cool. Bring your new packet and a pencil and come next door.
Make sure the stools are out of the way or someone’s going to get themselves killed in this experiment. Stand on the other side of the table. Don’t stand at the end there; you’re going to get impaled. [Okay, with the talk of impaling people the expectations for "something cool" are much higher now than they maybe ought to be...]
What kind of motion is happening here?
I want to show you a few different things. [Start off with a simple and not quite so "cool" demo of pulling the cart (loaded with 4 bricks) with one spring. Yep, no one thought that was very cool. They'll still start talking about the motion. "It was speeding up!" NOTE: Make sure you do all of your cart pulling demonstrations with excellent form (move your feet in small, quick steps, over-exaggerate your close attention to the end of the ruler (holding the spring so the first coil is always lined up with the edge of the ruller and the last coil is at 30 cm), etc).]
Pretty great, right? How about this? [Same demo, except this time you only have the cart loaded with one brick. Much more dramatic! Now some kids start noticing that you're paying really close attention to the ruler. They start asking about why you're using a ruler. Throw it back to them and let them observe some more and then decide what you're doing with the ruler... they'll figure it out pretty quickly. Why would you care to keep the length of the spring constant? So that the spring force is constant? Yep.]
Okay, one more. [Pull out the "brass knuckles"... now they are starting to think this is cool... and pull 4 bricks with 3 springs. Again, more dramatic than that first measly run.]
What can we measure? How can we measure it?
We want to be able to describe this phenomenon more carefully. What can we easily measure, and what tools can we use to measure it? [The wildly varying answers that you get in the first experiment (first day of school and Constant Velocity Particle Model) are now toned down and focused in on what is important. They are starting to think about the motion and forces more specifically and don't see this question as so ambiguous and arbitrary anymore.] Common responses include:
Mass, measured in grams or kilograms using the electronic balance. [Well, they might say "mass" or they might say "weight" but either way they want to measure it on the balance, so they mean mass. And they will quickly agree that that's what they mean.]
Acceleration, measured in m/s/s using the slope of a velocity-vs-time graph made by motion sensor. [When they realize that the computer will also make them an acceleration-time graph, they might be tempted to use that. Until they see how messy it looks. A calculation off a calculation isn't as clean.]
Spring stretch, measured in centimeters, using a ruler. They typically suggest this idea as a way of getting at the spring force. Speaking of…
(Spring) Force, measured in Newtons, using… well, they actually often suggest measuring this by doing another experiment to find the spring constant of the springs and then using (we use for spring constant and for da stretch of da spring (they think of as being equal to , but I shift them to to make it easier later on in the energy unit when they are squaring that quantity (and squaring is another layer trickier in terms of notation))) to find the spring force. That’s a great idea, but would take a lot of time relative to the amount of time needed for this experiment overall. So instead, I suggest that if they used springs that all had about the same spring constant, and if they stretched the springs to the same stretch every time, they could just measure the force in “number of springs” (that is, F = 1 spring, F = 2 springs, etc). They agree pretty readily that this plan is a good idea. (Another common suggestion for measuring the force is to pull the springs with a spring scale. Those guys also usually agree that just counting springs would also be easier.)
Those are the usual suspects. Some others might end up on the list, but they are getting pretty good at focusing in on what is relevant to describing the phenomenon at hand. So the next question becomes… which of these can we easily change and choose values for vs. which of these depend on the values we choose for other quantities.
It becomes clear that we can easily change/choose the mass (add/subtract bricks on the cart) and the force (add more parallel springs), but that the acceleration value will depend on those quantities and not be easily chosen. We decide to leave the spring stretch constant so that we can measure spring force in “number of springs”. Sounds like we’re about to do two experiments.
Write down your objectives
By now they also know that their objective should be in the form of “Find a relationship between…” so with little prompting, they list the objectives for these experiments.
- Find a relationship between acceleration and mass.
- Find a relationship between acceleration and force.
We agree to graph both experiments with acceleration on the vertical axis so that we can more easily compare how mass and force affect acceleration.
And we’re off: Taking data!
They now know just what to do for their experiments (they decided what to measure and how to measure it, after all!) and so they don’t really need any instructions, but in this experiment it is worth the investment of 3 or 4 minutes time to give them a few tips about how/how NOT to take this data.
The most important “tip” you give them is already over. If you didn’t model good form in pulling the cart during the demonstration at the very start of the pre-lab, there is, say, a 95 – 100% chance that some if not all of the groups will do wrong exactly what you did wrong when you were pulling the cart. So in your first demonstration, you should have been careful, attentive, and deliberate in how you pulled the cart.
Now is the time to give tips about common mistakes. They will laugh at seeing these since they will so obviously result in bad data, but if you don’t show them, at least one or two groups will do most of these. One or two groups still might do some of these, but they (and their group mates) will recognize what they are doing and probably correct it on their own.
Classic mistakes to show them:
- The “I’m too lazy to move my feet” mistake: Stand in one spot and try to pull the cart like that. Spring and ruler twist toward you as the cart moves.
- The “Come on, hurry up already!” mistake: Be impatient with the slow start and overstretch the spring trying to speed it up.
- The “Don’t mess this up!” over-careful mistake: Pull the cart with the ruller touching the cart the whole way.
There are others, but showing these few seems to be sufficient. (When you don’t show some mistakes ahead of time, you end up with a couple of really angry groups who, after graphing their data, find out that they need to start all over again because they didn’t exert the force consistently.)
Finally, give them a few more guidelines for successful data taking. Get one student calibrated at pulling the cart. That student should practice several times before you ever take any data. Don’t switch people pulling the cart. And make it easier on the puller by starting with the easiest data to take (one spring in the force experiment or 4 bricks in the mass experiment) and moving toward the hardest data to take (five springs, 1 or 0 bricks).
Use one spring when you do the mass experiment. Use four bricks when you do the force experiment.
Use the spring puller to pull with multiple springs (hooking them through different fingers works… but it’s not pretty).
Let’s agree to have the length of the spring always be 30 cm.
Creating and interpreting graphs
This is the first experiment in my class where they take non-linear data. (Of course, they don’t realize that right away.) In Honors Physics, they’ve been using a graphing program since the force experiments in BFPM. In regular Physics! they’ve been graphing by hand, so using LinReg is a new experience for them. There are some little foibles of using the program (maybe another post on that later), but overall… it rocks! Much easier than using Excel, still gets the idea of range in measurements, much quicker for them to figure out, and makes linearizing graphs intuitive and simple. We’re using LinReg for the first time this year, but I’m a fan already.
Graphing the acceleration vs force experiment
Okay, let’s plug our data into LinReg and check out the acceleration and force graph.
That looks good, but everyone got it to be… off by a bit. Shouldn’t the force be zero when the acceleration is zero? This graph makes it look like you can exert a force without the object accelerating, and we have a feeling that Mr. Newton would disagree. Oh, wait, you probably could pull on the spring a little before it started moving because there must be some friction. So the forces would still be balanced when you pull just a little. In fact, I bet you could tell me about how much friction there is on the cart…
So what could we graph in order to get a directly proportional relationship? [Something like spring force minus friction force.] So I could call that the net force, right? How much the forces are unbalanced? [Sure.] So if we graphed acceleration vs Fnet, what would we find? We suspect it would be directly proportional. [Sure, a little bit of a handwave in there, but they're right there with you on this one. And if it really bothers them, you can do the relevant extension at the bottom of the post.]
Graphing the acceleration vs mass experiment
Alright, one graph down and one relationship found! Now let’s see how that mass data worked out…
Whoa, that’s definitely not linear! [It's... a parabola (almost unanimous first instinct from students)!] Ok, great. So this is the left side of the parabola, right? And if you kept adding mass, eventually the graph would go back up. [No.] Oh, so it’s not a parabola, then, huh. Okay, so then what is the function family (or parent function or whatever the heck they are calling it across the hall nowadays)? [A lot of hemming and hawing (and some exclamations of Exponential! Oh wait...) later, and we eventually settle in on "1/x" (which is not a function as it is not even an equation, but we're getting closer to saying what we mean bit by bit).]
So you think the acceleration is directly proportional to one over mass. [You can see the gears turning in their heads as they consider whether that's the same as the "one over x" that they said before.... and eventually, "Yes."] Okay, so if you graphed [pointing] acceleration vs [pointing to the horizontal axis now] one-over-mass, you would expect it to be directly proportional. [More gears turning and thinking... and then, "Yes," though you might have to go through that again a couple of times while they process it.] So we would really like to graph acceleration vs one-over-mass and check that it gives us a linear, directly proportional graph. Here’s where LinReg gets really awesome.
Let’s go back to the data screen.
We can get it to graph things that it calculates from our data. So instead of mass, we want to try 1/mass.
The data might disappear because of some quirk in the programming. Just click back and forth between the radio buttons once and it will come back. No worries. But then… Oh no, it is all zeroes now! Oh wait, it is just really small numbers. We can change the number of decimal places. And… there. All better.
Now we go to graph it again and… it asks for updated units.
Right, right. So if the units for mass were kilograms, then the units for 1/mass must be 1/kilograms. Enter that in… And then the graph appears.
Wow! That looks linear, now! Great work, guys. Save that graph.
The Biggest Idea we’ve had so far this year
So now that we’ve got the relationship between acceleration and net force and between acceleration and mass, we need a way to combine them. We kept mass constant for the force experiment and we kept force constant for the mass experiment. We need to put these relationships together, but not using substitution. We really need to superimpose them. We need to overlay them with each other since each experiment kept the other variable constant. We’re looking for what the relationship would be if we changed both mass and force at the same time. They suggest something like or .
We massage that into . We remember that we can replace with “equals, times a constant” and come up with .
I tell them that the Newton (which they have been using since BFPM—the Balanced Force Particle Model) is defined so that . After looking at it for a minute, they decide that the constant in must be equal to 1 (with units that turn into Newtons). So we can simply write it as .
Whoa. Think about that for a minute. If you know all of the forces on your object (and you know what object it is, so you know the mass), then you can predict where that object will be in the future. That’s some powerful stuff, man. That seems like a pretty big idea. Actually, that seems like the biggest idea we’ve had (so far, at least). In fact, that’s one of the three biggest ideas we’ll have all year.
Now, would it surprise you to hear that someone else has already found this relationship before we got to it? Darn you, once again, Isaac Newton, for having been born hundreds of years before us! So, okay, okay, other people call this thing ”Newton’s 2nd Law”.
Optional extensions and add-ons
If you have a lot of class time, a lot of student interest, or just a group or two that finishes more quickly than others, there are of course more little investigations and paths to go down. Students could try to find the acceleration of the cart when the net force is just friction (just push the cart and make the velocity-vs-time graph) and see how it relates to your acceleration-vs-spring force graph. Does knowing that value help you make a shift of the graph so that you can find a directly proportional relationship?
Students could find the value of the spring force in Newtons (do that experiment that they wanted to do above) and see whether the slope on the acceleration-vs-force graph has anything to do with the mass in kilograms of their cart+bricks. Etc, etc.
Oh, and a final mug shot from some students who were taking data and wanted to make it onto the blog:
Nice work, guys.