Modeling Central Force: Day 3

Note: Since writing this post I have significantly changed how I start this unit; see new post here

I had three main objectives for my lesson today;

  1. Wrap up the modeling process, particularly figuring out that the proportionality constant for Tension vs. v2 should be m/r;
  2. Go through the awesome graphical derivation of centripetal acceleration
  3. Use the traditional spinning stopper lab as a goal-less problem.
1. We looked yesterday at the ‘A’ parameter of the quadratic, the one in front of v2. Generally students found A proportional to mass and inversely proportional to radius, which was good, but there was no clear pattern, so I started compiling data. This is one class’s data; looking at it this way made me excited for the possibility of having computers in my room so I can easily compile using Google Forms. This also allowed for sorting by radius or mass, which was useful. The students were quick to figure out that C should be zero, hence the yellow cell that indicates less-than-perfect data; we don’t trust that regression anymore. Despite that, their proportionality constant was better than most. Only half the class had reasonable constants; surely not enough for them to see the relationships perfectly. However, at least they were able to pick out the general trends. One thing I hope the Modeling Instruction workshop helps me with this summer is the perplexity of how some student groups get fantastic data and some awful with the same setup and instructions, when I often can’t seem to find the source of the problem (though time and class size contributes to the lack of problem solving as well). 
Velocity vectors on a circle

2. Next we went through the graphical derivation of

I had them working on their desks with whiteboards, and I guided them through the steps. I think that this derivation is fairly abstract for students’ to come up with on their own (though I dare you to get them to do so, please let me know if you do!). It went fairly well, and will be re-iterated when they do their pre-lab assignment which is essentially the same exercise. I like this one in particular because a)  it is super cool that you get a concise, elegant equation by doing a graphical proof, and b) I try to iterate that everything comes from somewhere (I use this for energy as well). Any equation we use can be re-derived using physics. These students are too used to 16 popping up in quadratic equations in Algebra II without any clue to the fact that it has significant physical meaning.

The last part was the best.

3. I ended today by demonstrating the spinning stopper lab and asking them to analyze it as a goal-less problem (which I haven’t done a ton, but I will be adding more; thanks to Kelly O’Shea for introducing me to this concept).

A student uses multiple viewpoints

The students ran with it, first figuring out that Ft=mg for the hanging mass, then using that to analyze the stopper. The cool thing was when one group (there’s always at least one in each class) figured out that the string must be at an angle as the y forces must balance. We then had a discussion about this, and I demonstrated that when the string goes quickly it is hard to see the angle.

Overall this last day went well, but would be better with solid data. Still, I definitely preferred using pendulums to the spinning stopper lab for the modeling aspect (to be honest, it’s not like I’m a veteran of modeling anyway…), though I would hope to find an even better way to model central force. Let me know if you have improvements!

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