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’ll split them over two posts here. First up: weight and mass.
Observe an interesting phenomenon
Bring your packet and pencil. Let’s go next door and look at something cool.
Are you ready? Here we go. [I dramatically put the mass hanger onto the spring scale. Then a mass onto the hanger. Then another. Even though it's only the third week of school, the students are ready to ham it up and react as though this phenomenon is the most utterly cool thing they've seen in at least several weeks.] I know, right? So—
What can we measure?
We’re going to do this just like the first experiment. We’ll follow that same sort of procedure of setting things up. Actually, we’ll always use that same sort of procedure, and soon enough you won’t even need me to walk you through it.
Anyway, what could we measure about this really cool thing? [The first suggestion is pretty much always to measure the weight.] What do you mean by weight? (Gravity.) Gravitational force? (Yeah.) Okay, great. What tool could we use to measure it? (The scale thing.) Oh. Huh.
What does this scale measure? [A lot of answers, but they usually settle in on it measuring the spring force. They want to say right away, too, that it's the same as the weight.] Okay, could everyone sketch a quick FBD for the hanger, then? [They do.]
Is it balanced or unbalanced? (It’s balanced.) How do you know? [A lot of answers. "It's not moving." But I'm holding out, if possible, for "it's at a constant velocity."] Great, so how does the spring force compare to the gravitational force? (They’re the same.) So we can use the spring scale to measure the weight (Fg) as long as we keep the hanger at a constant velocity.
What else can we measure? [Sometimes mass is suggested quickly, sometimes not. Sometimes the students haven't caught onto the whole scheme here, and they start saying things like, "how far you can stretch it until..." which is asking a different question of the situation, not saying what you could measure about what you've already seen. They can be redirected pretty easily. If they don't get to mass quickly, sort of idly looking at the masses and putting them on the table in front of me gets them to see the writing on the masses and be prompted to it, usually.] What’s the difference between mass and weight? [Interesting ideas from various places, here.] Which changes when you go to the moon? Your mass or your weight? (Weight.) Okay, so they’re definitely different. How can we measure the mass? (Just use the numbers on the things?) That works! [Using a balance to get the values also works. I just go with the path they elect each time.]
Okay, so what relationship can we find? The relationship between….? (Weight and mass.) Great. You can write that down as the objective.
So you know what to do now? [Nodding.] After you make all the measurements (of weight and mass), what will you do with the data? (Make a graph.) Perfect!
Graphing—now with more computers!
Instead of graphing this by hand, we’re going to start using a computer program called LinReg to do that work for us. Using LinReg makes graphing easier (no more hand-drawn graphs), and it also forces us to think about the range of every measurement. For example, when you’re reading this spring scale and want to say what the force is, you can say it’s definitely between what and what? [The student nearest me will look at the spring scale and say something like, "Uh, 0.5 N and 0.7 N."] Great. So when you’re entering data into LinReg, it will want you to write that as 0.6 N for the measurement and 0.1 N for the uncertainty (half the range). Then when you graph it, it will put range bars on every data point to help you see how big the data point is. (Better (more precise) equipment would result in smaller data points). It’s also going to make up 99 extra experiments by using random points within the range of each of your measurements, and it will use that to figure out a range for the slope and intercept of your graph. Pretty cool, right?
You should be able to find it on all of the computers, and it’s pretty self-explanatory, but I will also walk around and make sure you’re figuring out how to use it. You can open it right away and take data right into LinReg. Just make sure you save your file every once in a while.
So everyone knows what they’re going to do now? (Yep.) Right. Ready, set, go!
[Eventually the groups start telling me they are finished, and I walk around to each one and check in on their graphs. I ask them to change the window so we can see (0 g, 0 N). I ask whether the data looks linear. I ask them to do the linear fit, then drag through to see the fictitious teams.]
What is the intercept? Does the graph go through (0 g, 0 N)? [A variety of answers. Many say, "no" because the intercept value is not given as 0 N.] What about the range? See the range it gives for the intercept? Is 0 N inside that range? [Oh! Yeah, it is.] So you can’t say that the intercept is not 0 N. But more importantly, should the intercept be 0 N? [No. Or, wait, yes.] Why? [Because if you don't put any masses on the scale, it shouldn't have any weight.] Okay, so we feel pretty good about that intercept, then. So you don’t need 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 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 second 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? (They’re all 0.01 N/g.) That’s so weird! You all had about the same slope? (Yep.) Huh. But you all used different masses and different spring scales, right? (Yeah.) There must have been something in common, still, about what you used. [At this point, I usually shift over to the spring force discussion so they can see that they all had different slopes for their graphs in that experiment. They quickly point out that they were using different springs, of course, and we shift back into thinking about what must have been the same about our materials in the first experiment. Eventually, they start talking about how gravity was the same and we get to the idea that we all used the same planet for the interaction.]
That’s so neat. So, hang on. What if we all went to the moon and did the experiment again? Would you all get the same slopes as each other again? (Yes!) And would it be the same slope as you got this time? (No!) Would it be steeper or less steep on the moon? [Some thinking takes place. Then they come to the right conclusion pretty quickly, having talked their way through what the slope must mean.] Alright. Can I try to sum up what you’ve all just said? (Uh-huh.) Great; you can put the boards back, then.
You were looking for a relationship between weight and mass. What relationship did you find? (It was directly proportional.) Do you know that symbol, for directly proportional, from math? (No.) It’s like this—
No, it’s not a fish. It’s the directly proportional symbol. Anyway, I think you do have this next idea from math, though. We can replace that sign with “equals, times a constant”. I mean, the directly proportional things are basically equal to each other, but with some scaling factor. (Okay, right.)
And since we all found the same slope, it seems like this proportionality constant is some particular number (at least while we’re on Earth). Are you ready to have a name for it? (Sure.) So we give it a special symbol (g) and we call it gravitational field. We can’t call it gravity, because Fg is gravity, so we have to be careful about how we refer to it. We could call it “g” or “little g” or “gravitational constant” or “gravitational field”—just not “gravity”. (Okay.) If we had more precise equipment, we could have gotten smaller data points (that is, the boxes that represent our points would have been smaller because the ranges would have been smaller), so we would have had a smaller range for that slope and a more precise value for it. Do you want the accepted value for g near the surface of the Earth? (Yes.)
I think that’s in the range for our slopes, right? (Yes!) Nice. So using our new symbol, here’s the way that people usually write that weight and mass relationship—
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. You can write down the value for g somewhere around there, too. You’ll probably need to refer to it for the next week or so, but you’ll just know those facts pretty soon because you’ll use them a lot.
Other posts about the Balanced Force Particle Model:
Empirical Force Laws: Spring Force Experiment (Part 2 of this post.)