Day 21: System Schema and Differential Equations

Physics: Newton’s Third Law! There was a quiz, we went over it. It was a good jumping off point for discussing N3L on my terms instead of the book’s terms (Forces come in pairs, interactions of exactly two objects, simultaneous, same magnitude and opposite direction), and then we went to system schema.

A system schema is a diagram that identifies all the objects with unique names, identifies all of the interactions that are happening, and labels each force force pair with a unique pair of names.

Example: Box on a table
Step 1: Name each object something unique. In this case I named the box B and the table T. The “entire Earth” also needs to be identified as an object (named EE), because gravity is a thing that exists, but I don’t usually have students include it in the picture.

Step 2: Each object, including the entire Earth, gets its own bubble. The objects whose motion you care about modeling get an extra dotted line inside the bubble to indicate that. I hadn’t drawn that yet in this photo, but it is there in the next one.

Step 3: Draw double-headed arrows between the objects for each interaction that is happening and give them unique names.

The notation I use for naming forces (for everything that is not friction and normal) is a capital F with the subscript that first says what is applying the force, has a dot, and then the name of the object that is feeling the force. I tell my students to use F_(of.on). So the forces between the box and the entire Earth are F_(EE.B) and F_(B.EE). Since many surfaces end up with two defined interactions (normal and friction), those get special notation. Normal gets an N_(of.on) and friction gets f_(of.on). I make the ‘f’ for friction exaggeratedly curvy to make it very clearly different from the capital F in handwriting.

Yes, technically there is obviously a force of gravity on the table and a normal force between the surface of the Earth and the table, etc. You could go down a rabbit hole of all the forces on other things going down through the building and the building’s foundation, but since none of those objects have the dotted line inside, I don’t want to model their motion, so I don’t care about what’s going on with them. At all.

Here are a few more examples of System Schema for different systems:

I could have added the harness in there between the horse and the cart if I was feeling particularly pedantic, but I lumped the horse and harness into one object. Anyway, from here, it is easier for students to make the free-body diagrams of the objects of interest, because the forces on the objects of interest are already identified and named. All that’s left is checking out the picture and figuring out the direction of the forces for a FBD. Knowing which are interaction pairs help with direction, particularly for things like friction of blocks sliding past one another or when you have pulleys doing weird things. Just remember to give the strings on each side or between each pulley its own name.

The students had a hard time sorting out the difference between a surface and the entire Earth, which is fair, and if the surface is the ground there is no reason why you can’t have the normal force and friction between the EE and the objects. That’s really not a problem. But there are situations where the surface is something else, like on a ramp or in an elevator or on an airplane or a hot air balloon or at the top of a loop in a roller coaster where you have to think of the surface interaction(s) and the interactions with the entire Earth in different ways and in different directions. That is why I like to separate EE from Surface (or table or floor or ramp or whatever you name it).

Honors Physics: Today we had to do a lot of monkeying with the schedule and when things are due. I hate to have too many things due at once, and I’m perfectly fine shifting things around as long as it gets done eventually. I’d rather just push out the due date a few days than have them all stressed and miserable and asking for extensions, anyway. Then we reviewed analyzing systems and it took all period, so I pushed the quiz back another day to give us more time to review more tomorrow and try more goal-less problems. I think I’m going to cool off on the momentum ideas for this first exam and focus on the CVPM/BFPM and CAPM/UBFPM. Maybe have one question with an IF diagram, but really double down on momentum after the exam when we also discuss energy for elastic collisions and having them choose between the two models.

AP Physics: Air resistance is tough. The textbook uses a v-squared relationship, but all the AP questions I found used a linear relationship for modeling purposes that required solving a differential equation. It’s just separating and integrating, and the linear relationship leads to an exponential result, so it’s not tough, but modeling it is a challenge mathematically. I screwed some things up when going over it, so back-tracked and did some more simple problems to work back up to the more difficult ones, but I really need to spend some time with air resistance and solving DEs. Last night I felt like I had a handle on it, and I do for the simpler situations, but I got myself into some trouble trying to do a more complex example problem today in the first class.

The second class I approached things much more cautiously. I started by asking if they had questions on the homework and letting them know I needed time to go over the orange questions (i.e. the most challenging questions) in the textbook but would have them tomorrow. Then we talked about the basic idea of changing acceleration with velocity, did easy examples of population growth, of radioactive decay, and slowly increased the complication of the DEs rather than jumping right to a hard one. That was a silly thing.

Non-Academic Situation: We had a whole bunch of student disciplinary events happen in the past week, and one of the students from my dorm was dismissed today. I’m going to miss him, but I hope this makes him consider his choices more carefully in the future. I’m working through processing it. I really hoped he would retain his position and work to improve, but apparently this was not his first major disciplinary event in his time here or even this year. Dang it, kids. Make better choices!