Appending The Conservative Models

After investigating the causal relationship between torque and angular acceleration, I introduced the possibility to the class that perhaps we also needed to revisit the Energy Transfer Model and the Momentum Transfer Model. The students agreed that an object that is rotating must have energy. This was pretty easy to demonstrate.

I set up a situation in the class where two of the variable inertia disks that we created on the 3D printers were placed at the top of an inclined ramp. The internal marbles were placed at two different configurations inside the disks and then the students predicted which disk would reach the end of the ramp first. I was pleased to find out that the class appeared to agree that the disk with the marbles located closer to the radius would be the winner. I really think that our investigation with the variable inertia disks solidified the students’ conceptual understanding of rotational inertia and the importance of mass distribution.

I have not yet found a good experiment where students could discover the rotational kinetic energy relationship, so I decided to take them through a derivation based on linear kinetic energy. I then asked the students to do some whiteboard work. I asked them to demonstrate that the disks would indeed reach the end of the ramp at different times. Although this wasn’t strictly a constructivist approach, it was good practice in doing some fairly difficult algebra without numerical values – something the students traditionally are not very good at.

We then moved onto momentum. Again, I started by reviewing the Momentum Transfer Model for a particle. At this point the pattern had been fairly well established. The relationship between angular and linear quantities seemed to have taken hold because the students were quick to propose a mathematical definition for angular momentum. Our next goal was to figure out whether this was a conserved quantity.

Mr Holt and I had created a set of metal disks that could be attached to the rotary sensors. I decided to create our own, rather than (sorry Vernier) buy them as I thought that the commercial kit was over priced. It wasn’t too hard to create the disks, especially when you have access to a CNC plasma cutter!

The students attached one disk to the rotary motion sensor and then got that disk spinning. They then took a second disk that had small magnets attached to it, and dropped this disk onto the spinning disk. The students compared the angular velocity before and after the disks were combined and then calculated the angular momentum of the system before and after. The data we got was quite good with the class getting in the range of only about a 5% to 6% difference.

Wrapping it Up (or Un-Rolling It Down)

As a final deployment, I decided to try the deployment activity that Frank Nochese did with his students. It seemed like a good (and fun) way to wrap up our model (or as I have already argued – models).

Before doing the deployment activity, I reviewed all the model specifics with the students. My point here was to impress on them that what we had not really built a new model, but rather had extended many of the prior particle models to include rigid extended bodies. This generally only required that we consider the moment arm in all the particle models. I think this really helped a number of students see the connection between models that they felt they understood and all this rotational stuff that seemed a bit confusing.

I then set them up with the deployment activity, but I asked them to specifically solve the problem using both energy and net torque. There was some success, but I realized that the task was a bit much for the class. Once again, it is clear that I need to give them more practice with these problems that require multiple steps and that involve algebraic manipulation of symbols without numbers. Plenty of time to practice that!

Investigating An Unbalanced Net Torque

We started by looking at the fact that a disk experiencing a net unbalanced torque also experienced a change in rotational or angular velocity. The students used a rotary motion sensor to measure the angular position and the angular velocity of a disk experiencing a constant torque and the students immediately recognized the similarity between a particle experiencing constant linear acceleration and a rigid body experiencing constant angular acceleration.

So, the natural next step in our investigation was to determine the causal relationship between a net unbalanced torque and the angular motion state of a rigid body. The students discussed how we might set up an investigation that would help us understand this relationship, and I helped guide them towards a final investigation design where we used a rotary motion sensor attached to the wooden disk we had used in a previous investigation.

The rotary sensors from Vernier are rather expensive, but also quite nice. They come with a plastic spindle with three different pulley radii. The only issue that I have with these is that they do not include a useful screw for attaching objects to the pulley – they expect you to purchase their completely over priced accessory kit. I don’t suggest this. Instead, you can create your own and then use a 6/32 screw to attach them to the pulley – just be careful not strip the threads!

Working off a similar investigation when studying the causal relationship between a net unbalanced force and linear acceleration, the students recognized that we needed to include a force sensor for measuring the applied force. The students then had LoggerPro plot the points for the calculated torque (using a calculated column) and the measured angular acceleration.  The data isn’t super clean, but its good enough for the students to conclude that the relationship is most likely linear.

Dimensional Analysis – Inferring Rotational Inertia

I was amazed to discover that some students in the class set off to understand what the units of the slope could be reduced to. They immediately saw that the proportional constant (slope) included kg (mass) but that wasn’t all. After some work, the class had determined that the units for the slope were kg * m^2.

This led to a qualitative discussion about the inertia of rotating objects. We discussed hoops and disks primarily, and the class seemed to agree that the units made sense. Although I didn’t have any hoop-disk sets like the ones you can buy from various vendors, we did perform some thought experiments around mass distribution and rotation, but we needed to be sure that we were on the right track.

A Better Variable Inertia Disk

So last year I was looking for an investigation that would really help the students discover the importance of mass distribution for a rotating object. Reading the material on the AMTA website regarding the unit on rotational motion, the researches stressed the importance of connecting mass distribution to the rotational inertia. I found this variable inertia disk from Fischer Scientific and decided to purchase it. I commend these guys on making this, but frankly I decided that I could make a better one.

My colleague and I set about redesigning these disks. The improvements we made included a) better compartments for the metal marbles so that they didn’t move around, b) an index and lip so that the two sides fit more securely together, and c) more compartments so that we can test more mass configurations. After we made our designs, we used our Makerbot 3D printer to print out ten copies of the disks.

With these disks attached to the rotary motion sensors, the students were able to confirm that although the total mass of the disk did not change, the angular acceleration declined as the mass was moved outward from the radius of rotation. This was more of a qualitative investigation, but I think the students were able to clearly see how moving the mass farther from the axis affected the rotational inertia of the disk.

Deploying The Model (So far)

Now that the students had built a predictive model that described the quantitative relationship between rotational inertia, angular acceleration and net torque, they were ready to test it.

We returned to the investigation setup with the disk, but this time attached a hanging mass to the string and they attempted to predict the angular acceleration. The students immediately made the mistake (as I thought they might) in considering only the disk in the system. This gave me a chance to review with them the following process:

1. Identify the system schema (now it includes things that can rotate!)
2. Draw your force vector (free body/particle) diagrams and now also your force-moment arm (rigid body) diagrams.
3. Write the summation of forces AND the summation of torques for the system.
4. Do some algebra.

The trickiest part to figuring this out is getting the signs right due to interaction pairs and making sure that they agree. I ask the students to start with a diagram and then go through and label each force + or – by picking a force and then finding its partner and then making sure that the rest of the force directions agree.

Once they were able to get a prediction that we all agreed seemed correct, they ran the experiment. The class was able to predict the angular acceleration within about 10% error. We discussed the possible reason for discrepancy which led to some interesting discussions around modeling the rotational inertia of the attachment screw and washer, and also the frictional forces at work.

The students were pretty convinced that the model worked, and so next stop – energy and momentum in systems with rotational motion.