The test was long today, but after the test, we had a little time to talk about cantilevering. I took a meterstick and let the first 30 centimeters of it hang off the edge. I then took a mass with a hook on it and hung it at the 25-centimeter mark. The meterstick didn't move. It was still in both translational and rotational equilibrium. We moved the mass farther and farther from the edge of the table, until, finally, the meterstick started to tip. I told them the mass of the the mass with a hook on it and then asked "What is the mass of the meterstick?" We'll play with this setup more tomorrow.
One of the three problems for homework was a doozy, so we talked about it for a little while. If you aren't given an obvious pivot point, what do you do? Well, it seems that in rotational equilibrium, any point on the object could be our pivot point. Since it's not rotating around any particular point, it must be rotational equilibrium around every point in the object. We then whiteboarded the problems using our model of torque.
We saw, further on, a question was going to ask about a force pointing at an angle. How do we deal with that? We weren't sure, so it was off to lab. The lab didn't give great results, but we figured out how to incorporate the angle between the force and the distance to the pivot point into our model.
We started balancing objects on metersticks on Friday. We graphed the relationship they found, and since most groups change the mass and then measured the distance the mass had to be placed at, we had a lot of graphs that looked like this:
Most of the boards had good equations, but had a hard time figuring out the units for the constant of proportionality. I picked a group that had gotten what we all agreed should be the constant of proportionality. I asked this group for the mass of their fixed mass and its distance from the pivot point. They had a 50 gram mass that was placed 45 centimeters from the pivot point, which is what I wrote out below their equations.
At this point in this class, there was a loud communal groan. "Why do you keep doing this to us, Mr. Lerner?" one student asked.
"I didn't do this; the physical world did."
It does seem like the mass times the distance equals the mass times the distance. But is it really mass? Couldn't it be force?
We got out the force meters, keeping our fixed masses at the same position. (Next year, I'm going to use spring scales.) Everyone just pushed down on one side of the meterstick until I came around and asked what they could do on the other side of the meterstick.
We realized that clockwise and counterclockwise seem to be the way to talk about these influences, not left or right or up or down. These clockwise and counterclockwise influences need to balance to keep an object in rotational equilibrium, just like forces must balance to keep an object in translational equilibrium.
We ended the period working on a few questions, one which seemed really easy and two which seemed really hard. Let's whiteboard those tomorrow.
We finished talking about the central force model by summarizing how we could apply kinematics, forces, energy and momentum to circular motion. Talking about momentum with the central force model was new to me this year, but, if you think about the planet-star system as having no non-negligible external forces, then the center of mass stays at rest or at constant velocity. When the planet goes one way, the star goes the other way. Such an obvious way to see them both rotating around the center of mass of the system.
Then we drew force diagrams for this situation:
We weren't sure how to draw the tension forces; don't they cross each other? And how do we show that the little mass is hanging down on the meterstick in the middle of the meterstick? When we moved the little mass, the whole setup was ruined.
We broke the particle model. We need to model this meterstick as something other than a dot. Off to lab to figure out how to keep something balanced when we care about where the force is being applied.