We went over our last few questions on angular momentum, including a really good conversation about whether bouncing off a see saw or sticking to a see saw will cause a larger final angular velocity.
We then did some practice AP Physics 1 questions. It was great to hear all the different ways students came to their answers.
OK, so we talked a little bit about angular momentum yesterday, and I didn't write about it. I walked past the rotating platform and dropped a book on it. We noticed that mechanical energy wasn't conserved, and we didn't have a good way to calculate the dissipated energy. We noticed that momentum wasn't being conserved, because the rotating platform didn't slide forward. But it did rotate, so something was being conserved. We decided it was angular momentum, and, by analogy, came up with a formula for angular momentum. We played with the bicycle wheel on the rotating platform, which we could explain with angular momentum, along with the classic ice skater spin.
Today, we whiteboarded some problems and then watched one of my favorite videos: Veritasium's bullet-block experiment. It is so motivating and a great way to make students apply energy, linear momentum, and angular momentum to a problem.
We started today with rotational energy. We talked yesterday about how a Tic Tac can bounce higher on the second bounce than the first, but today we wanted to see the conservation of energy in another situation. We did lots of trials of things going down ramps--cars, spheres, disks, rings, and hover pucks. Some of us didn't believe that one of the PASCO cars and a hover puck would go down the ramp with the same motion, but a quick test (see above) showed us that we can ignore the rotational energy in the wheels of one of those cars. All spheres went down the ramp with the same motion, but how did spheres and cars compare? That was a HUGE debate. We drew an LOL diagram for objects going down ramps and tried to, with our definitions of translational and rotational kinetic energies, to explain what happened in lab.
We started today by investigating moment of inertia. I gave each group two metersticks of about the same mass (the range of masses of the metersticks in my room was large: 95 grams!) and 2000 grams of masses. I then challenged them to make two sets of objects with the same mass and same center of mass but wildly different moments of inertia. Oh, and the mass can't be anywhere in the 45 cm-55 cm range on the meterstick because that's where we'd put our hands to rotate the sticks horizontally back and forth to get a sense of its rotational inertia around the 50-cm mark. They came up with a few different solutions. We tried them all and found the biggest difference in rotational inertia came from the biggest difference in the distance from the pivot point.
Then we took off all the masses from the meterstick with the large moment of inertia. They now had their meter stick with two 500-gram masses pretty close to the center of the meterstick. Using the units of the moment of inertia as a hint, I asked them to guess where they could put much smaller masses on the other meterstick so the two metersticks would have the same center of mass and same rotational inertia around the 50-cm mark but different masses. A picture of one of their solutions is above.
We then worked on a few problems with unbalanced torque. Then I took a page out of Kelly O'Shea's book (again) and used Tic-Tacs to show that sometimes they bounce higher after hitting the table when they're spinning. We talked about how there must be rotational energy.
We had an idea from yesterday that an unbalanced torque could cause an angular acceleration. But how are they related? Today was all about collecting data for an angular acceleration vs. net torque graph. We talked about the various ways to find the angular acceleration, and we seem to have a lot of different methods. We saw a clear linear trend, but the slope didn't make much sense. What would be measured in kilogram meter-squared? We'll try an experiment on Monday to see if we can understand that slope better.
We finished up our balanced torque problem today, but today was mostly spent how to describe rotational motion. An object that isn't spinning is easy to describe, but how do we describe a bicycle wheel that's spinning at a constant rate? Students in groups tried to find a number that would describe the spinning rate. We came up with two different numbers: the velocity of the edge of the bicycle wheel and the revolutions per minute (rpm) of the bicycle wheel. We talked about the strengths and weaknesses of each approach and came up with an even better measure of the velocity of the wheel.
We then looked at a setup in the picture above; we came up with predictions of what the angular position-time and angular velocity-time graphs would look like. We weren't sure if they were right, so we went into lab to collect data. Our angular position-time graphs looked just like parabolas, so I guess the angular acceleration is constant. I wonder how angular acceleration is related to torque?
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.