Yesterday we wrapped up balanced forces (BFPM) and started unbalanced forces (UFPM). The paradigm lab of UFPM is a modified Atwood machine lab. We did the pre-lab discussion of measurable/controllable factors that might influence the system's acceleration and then took data at the end of yesterday's class. This morning, we had our post-lab discussion. Each group whiteboarded their results and also added the results from each of the two sub-labs to a chart on the chalkboard. This gave us a good look at how similar the results were (quite) over a range of system masses and hanging masses. We clearly saw that the gravitational force on the hanging mass was very consistent with the slope of the acceleration vs. system mass line, and deduced the equivalence of Newtons and kg-m/s

^{2}from that. Checking this result against the data from the other sub-lab, we found that the slopes there compared favorably with 1/(system mass). We, especially my group, struggled with trying to understand the meaning of the compound unit m/s

^{2}/(1/kg)because we've gotten very used to being able to explain proportionality coefficients using the framework: "additional{something} for every {other thing that we change}" and that doesn't work really well for these units. We tabled the discussion on the grounds that this equation is a fundamental part of how the universe works, rather than the definition of a variable.

^{1}

Craig mentioned that he asks students to consider the situation in which the cart is given an initial velocity away from the pulley instead of being let go, and this causes more thought than you might expect, but has the advantage of bringing up velocity/acceleration confusions and asking students to wrestle (again) with the fact that acceleration and velocity can be in different directions and acceleration can be non-zero even when velocity is zero.

Craig also mentioned that this is when he builds a Model Summary with his students. We haven't been building Model Summaries as a group, sadly, because I'd like to discuss how best to guide that discussion.

Mark briefly discussed the elevator lab and then we whiteboarded worksheet 1 and did our student-mode board meeting. There were some good hypotheticals and extensions posed to the various groups. My group drew problem 6, and two of us got entirely tangled in the question of how there can be a traction force and a friction force (in opposite directions) between the same two objects, which left Lucas on his own finishing the problem and drawing the whiteboard. (Sorry, Lucas!)

After lunch, we had a teacher-mode discussion of the worksheet

^{2}, including an awesome discussion of ambiguity, pro- and con-. For some of the problems, I'm content with the ambiguity because of the rich discussion it sparks. For problem 6 and its ilk, though... I think I'll take the word friction out and just go with "drag forces" opposing the motion instead. I want students to understand that friction and traction are related, but I don't think the circular discussions of rolling friction vs. traction add much for high school students. Plus, as Michael pointed out several times, our models deal with everything as point particles, so they don't deal well with situations (like torque and rolling resistance) that require extended objects. We hashed out a definition of inertia that was mostly satisfactory along the way, having skipped that part of the post-lab discussion for the bowling ball activity in BFPM, and then moved on to the friction lab.

I had always set up this lab by reasoning through the qualitative experiment of trying to slide your hand along the table while pressing down vs while lightly resting your hand on the surface, but with some trig one can separate the effects of the gravitational forces from the effects of the normal force (by varying the normal force without varying the mass of the object sliding across the surface). Three groups decided to test this, two by varying the slope of the surface and our group by varying the angle of the tension with respect to the surface. If your students want to investigate this, have them tilt the surface: tilting the tension force was tough to manage with any reliability whatsoever. Everyone finished collecting data and writing whiteboards with about an hour to go and we discussed the effects of speed (no relationship) and area (also no relationship) on frictional force. The first was a little surprising for some and the second ran into seemingly contradictory prior experiences with wide and narrow tires. In case anyone reading this had any doubts: telling people that it's not the width of the tire (or the contact area) that matters but the roughness of the surface and the deformation of the tire didn't have any effect on their conviction that the width/contact area was a useful proxy for the factor(s) that increase(s) your traction when you switch to wide, off-road tires. Tomorrow we're going to discuss separating the effects of weight and normal force on the frictional force and come up with a name for the slope of the F

_{friction}vs F

_{normal}line.

^{1}I'm not sure I'm content with that explanation, because it sort of feels like this is (part of?) the definition of force, especially in light of Arons' operational definition of force.

^{2}I think I want to ask students to include interaction diagrams/system schemas and kinematics graphs with these problems. Then again, I think I want to add those to just about every worksheet in these two units, so...

## 1 comment:

I think one reason that (m/s/s)/(1/kg) is hard because (1/mass) doesn't have a name or a unit. We have been trying to give that a name over at my blog:

http://wp.me/p1yR3O-hA

Check out the last comment. It is so relevant to this.

I think another reason by be related to what Hestenes says about its definition. He suggests it is implicitly defined, not explicitly That is it is an abstraction across many ideas and observations.

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