# Rotational Dynamics Are Difficult to Learn… and to Teach

One of the most difficult units for AP Physics students is rotational mechanics. It can also be one of the hardest to teach. With that said, I really like rotational mechanics, but I want my students to enjoy it more. I want them not be be bogged down in some of the more difficult theoretical aspects of the rotational motion. There are several reasons for this topic being challenging I think – the use of the polar coordinate system, the introduction to the concept of a distribution of mass as opposed to just thinking about objects as point particles, the cross product of vectors producing a result pointing in the Z axis…yikes, it can get pretty heavy.

As a teacher, it is also tricky to come up with good investigations that allow students the opportunity to explore rotational dynamics. They do exist, and some are really good, but I also think that many don’t exactly build a strong link between the conceptual and mathematical models. Rolling a ring and a disk and sphere down a ramp can be a great demonstration that things that roll certainly do not behave like blocks or carts moving down a ramp, but its hard for the students to explore further – to build a predictive model that is mathematically precise. The normal way of things is to jump into the math and reveal the underlying mechanics through fairly complex derivations.

# Simulating The Angular Momentum Principle

This year I decided to turn to a computational approach. I wanted the students to computationally experiment and discover the fundamentals of rotational mechanics. In order to do this, we started from the Angular Momentum Principle. This is the approach used in the wonderful textbook Matter and Interactions:

Στ ·Δt = I·Δω

The idea here is simply that a torque acting on an object (that is free to rotate around a pivot point) for a specified amount of time is going to cause an object to experience a change in its rate of spinning – called its rotational velocity.

# Simulating A Physical Pendulum

Having the students start with this principle, they could learn some interesting mechanics of rotating bodies by building a simulation of a physical pendulum. A simple pendulum is “simple” in that it represents a single point mass hanging on a massless string. A physical pendulum has physical dimension, and so the body has to be modeled as having a distribution of mass. This gets really tricky to analytically model – especially the time dependent functions describing its motion.

But with a computational approach students could easily apply the momentum principle frame by frame. The students in my class were able to create a simulation of a physical pendulum in just over one class period. The pendulum that they modeled was a bar with weights at either end. The weights had different masses so that the bar would swing.

https://tychos.org/scenarios/250

The students set about building their simulations in Tychos – a web based simulation builder. As the students coded the simulation, they were continuously engaged in discussions about the mechanics of the code and the mechanics of nature. I asked the students to change the masses, or change the length of the bar, or the mass of the bar. They would run their simulations and discover that something didn’t quite match what they thought should happen. Students would then look for errors in the code, or they would question their own assumptions about the physics involved.

When they finished their simulations, and they were confident that their simulations were working, I had the students then use the simulation to investigate the energy of the rotating system. After some discussions about the transference of energy from the gravitational field to the kinetic energy (rotational) of the bar and the weights, the students discovered that their simulations also seemed to model the conservation of energy.

We didn’t stop there though. We had to see if the simulation did indeed match reality. This is called deployment in the modeling pedagogy. Using a meter stick, some disks that I had made for an earlier investigation, and a Vernier rotational sensor, I constructed a real physical pendulum.

To test the simulation, students were asked to identify all the inputs that would need to be measured and then entered them into their simulations. The students then used the conservation of energy to calculate rotational speed of the bar when the bar was released from an angle of 60 degrees – with the heavier weight above the lighter weight.

The students calculated a final rotational velocity of 2.8 radians/second. We then ran our test and graphed the angle of the bar as well as the rotational velocity of the bar (see graphs below). They measured a rotational max velocity of about 2.6 radians/second. The students were pretty satisfied with this result.

angular velocity of the bar

angular position of the bar

I then had the students graph the results of their simulations, and it was quite satisfying to see that the simulations and the sensor readings were matching really well.

graphs produced in Tychos

This led to some more interesting discussions around simulating axel friction and air resistance. We also discussed the shapes of the graphs and what might be going on with their mathematical structure. This was far more interesting than just having them read out of a textbook and then solve some problems on paper, and my guess is that the students gained an understanding of the physical mechanics by having to debug their own simulated mechanics. They were able to investigate errors in their own code, but by doing so, see how the position of the weights, and the masses and the angle changed the behavior of the simulation and the motion graphs. I think the students also really enjoyed this approach, and based on how they were talking about their simulations and the experiment, they demonstrated to me that they had grasped a deeper understanding of rotational mechanics.

# Simulating Circular Motion: An Inquiry Approach

This semester I have been very busy working on a new approach to teaching Physics. This has actually been part of an effort that has spanned more than three years, but this year I have really embraced this change and I have much to share.

This post is a preview of many to come. I am going to write several posts documenting my efforts and experiences throughout the school year. Hopefully these posts will help me capture what I have learned in the process, and perhaps will be a guide for anyone else who might be interested.

## Analytical Models Emerge From Computational Models

With the help and insights of others, I have been mapping out a new scope and sequence for teaching Physics that incorporates computational modeling as the primary method of modeling Physics. Rather than looking at computational modeling as an “add on”, I have been exploring the idea that analytical models are emergent. Computational models are more fundamental and analytical models emerge from those computational models.

The basic approach here has been to start with computational modeling, and then to allow the students to discover the analytical models that are revealed. This has been an exciting “unveiling” of physical patterns for the students. The other thing that I have witnessed is that students seem to intuit the principles of Calculus, even though my students are not at that level in mathematics training. More on this in a future post.

In future posts, I will report out on how this has been going and what I have learned in the process. For now, I simply want to share an example.

## The Inquiry Lessons: Discovering Centripetal Acceleration

One of the aspects of this project has been to re-invent many of my lessons. I have been creating inquiry based lessons based on an approach known as POGIL. I am not POGIL trained, so I wouldn’t say my activities are actual POGIL activities – they are POGIL “inspired”.

In the first lesson, students learned how to simulate an object moving in a circular path. I have attempted to incorporate an inquiry approach where students are guided using questions, as opposed to holding their hands. This approach can be messy, but I have found it has always lead to great conversations, unanticipated insights from the students and it gives the students a sense of discovery.

I have included a link to the lesson which you are free to copy, modify, etc. without any restrictions.

Lesson 1: Simulating Spinning Motion

Keep in mind that my students had already learned how to code movement, so if you are new to computational modeling, I will soon be writing some posts that introduce students to this approach and to computational modeling in general.

In the second lesson, students simulated circular motion using angular quantities. They then explored how they could represent the tangential velocity and then they explored what the acceleration was. This led them to discover that the simulation revealed that the acceleration vector pointed to the center of the circle. Through some guided inquiry, they discovered a number of interesting details, such as the acceleration increased when the radius declined, and that the tangential speed had a significant affect on the acceleration.

Here is a link to the lesson:

Lesson 2: Circular Motion

I do use a tool, that I am actually partly developing with some friends as the simulation software, called Tychos, but you can modify the lesson to use any coding platform you like (of course I like Tychos, but I am a bit biased!)

Please feel free to comment here to give me feedback on the lessons if you feel inclined. I am certainly on a learning path myself, and I am sure there are many improvements that could be made!