Forces that Change Direction in the Central Force Particle Model

My students historically struggled with the notion that some forces can change direction depending on the situation. I developed this worksheet to specifically help them recognize that forces can be up or down at the top of a vertical circle. I use this as my second discussion worksheet, after the standard first worksheet where they discuss the direction of the net force for circular motion. They have also already developed a formula for centripetal acceleration.

The day we discuss this worksheet is usually one of my favorites, as it’s designed to bring about tension in the classroom, only to be resolved at the end of the period. Like much of modeling, the magic happens in how the worksheet is used.

I start a 45 minute period by having students work for 10-15 minutes on the worksheet. This gives them time to familiarize with the situation, but, in my experience, not enough time for them to figure out the ‘punchline’. I then assign 2 groups to do part a, 2 groups to do part b, 2 groups for part c, and 1-2 groups each the rest of the parts, depending on how many groups there are. The two groups for parts a and b are particularly important; I try to either choose 2 groups that have drawn normal force opposite directions at the top of the loop, or guide one group to draw the opposite of the other.

I then have both groups for part a present simultaneously, and there is usually a raucous discussion about which board is correct. Just when the tension is highest, and unresolved, I say “ok, next board!”.

Students: “Wait, what? But….the answer….”

Me: “Trust me. We’ll get there. Next board!”

Again, with part b, I try to have groups who chose opposite directions for normal force. That way their equations are different in that one has a positive Fn and one a negative. (aside; I have them do Force Addition Diagrams, you can see examples of them for this worksheet on this post). Again, just when tension is highest as they argue which is correct, “Next board!”

They really don’t like this.

As a result, when the groups do part c, one gets a negative and one gets a positive normal force.



Me: “Which is correct?”


The resulting discussion is great. It is easiest to resolve at this point by having them make a force addition diagram that is quantitative. That way they can see that if Fg is 637 Newtons, and Fnet is 234 Newtons, both down, then Fn must be 403 Newtons up (note that the numbers now are slightly different than the boards in the link above; as I recall, the old numbers resulted in an odd coincidence that sidetracked conversations, something like centripetal acceleration being half of gravity). This becomes very clear when drawing the numbers on both FADs.

Once we have figured out that normal force must indeed be up for a-c, d and e follow fairly easily.

Usually when I do this worksheet I end up with kids fervently arguing, then feeling very satisfied at the resolution that finally comes toward the end of the period. That tension is what makes this discussion work so well.

One final note: in my AP Physics C course, I actually set this up by looking first at a qualitative situation with a banked curve, where friction could be up or down the incline. We have that discussion, then after we resolve part c of this worksheet kids recognize that they are the same type of situation, where forces can change direction depending on the speed of the object.


Elevator Situations: An Introduction to Unbalanced Forces

I’ve been meaning for some time to compile some items that I’ve developed over the years, since the courses I teach don’t lend themselves to using the straight up Modeling Instruction curriculum. I develop items mostly to fill gaps I see in understanding or content, depending on need. I wanted to be able to share this work with the larger physics community.

That said, I do need to issue a disclaimer; I think it is possible, likely even, that some of the materials I hope to post came out of conversations with others, or even after seeing someone else’s materials. If I’ve inadvertently stolen anyone’s work, please let me know and I will remove it immediately. To the best of my memory, however, the materials are original.

This first item arises out of a pretty standard physics problem; analyzing an elevator that is accelerating. I wanted a worksheet, however, that emphasized the similarity between speeding up while moving upward and slowing down while moving downward; likewise, slowing down while moving upward and speeding up while moving downward. I wanted to be able to point out that the common feature in these situations is the direction of the acceleration, and thus the direction of the net force. I also wanted to re-emphasize from our work with Constant Acceleration that a negative acceleration does not necessarily mean slowing down. Finally, I wanted to give students some easier situations before we moved on to ramps and angles.

I use this worksheet with my first year physics course that is similar to AP Physics 1 (it’s concurrent enrollment through the U of MN), and it’s the first thing we do after we build the model by pulling carts with spring scales (kinda like what Kelly does but with 1N spring scales). I also started last year doing it after model building with my ‘regular’ physics class, which is oddly between a standard HS physics course and a conceptual physics course.

I took the liberty of applying a creative commons license to the work, so it may be shared and/or adapted with attribution for noncommercial purposes under CC BY-NC 4.0. This is mostly so that it couldn’t be used for commercial purposes; feel free to use and change (and tell me about it)!

Without further ado, here’s the worksheet, enjoy!

Making Homework Work for My Students

Homework seems to be a perpetually debated topic in education circles. I’ve written about homework before, (and that post contains links at the end to some great reads on the subject), and this is really the “Where are they now” post a couple of years later. After thinking through how homework can work better for students and teachers, I got the chance to implement a new policy alongside a course I started last year, AP Physics C. I’ll now be expanding it to another course for next year, a college-at-my-school course similar to AP Physics 1.

So here’s the gist; @LCCTA tweeted last year about an interesting homework policy and I modified it for my purposes. I wrote this description that I passed around twitter for feedback, then I implemented it during the 16-17 school year. In summary, the policy goes something like this;

  • student does something to try to improve learning of physics outside of class (and there’s some choice built into those somethings)
  • student documents that something
  • student submits documentation and gives themselves a homework score

My primary reason for this policy is to extend an emphasis on learning (rather than answers) from my classroom to homework, and I think that was moderately successful.

Students responded well to the policy, based on the end of year survey;

Screen Shot 2017-08-09 at 7.08.36 PMScreen Shot 2017-08-09 at 7.08.49 PM

The comments indicated the students appreciated the flexibility and choice.

All that said, there were a couple of things I’d like to improve on. First, allowing students to submit a variety of media as evidence, including videos showing the work they had done, proved too cumbersome for me. A student flipping through pages in their notebook is difficult for me to actually look at, as it turns out (who’d of thought?). The new policy will ask them all to simply submit a Google Doc template including pictures of their work (all our students have Macbooks so this is not really an equity issue for our school).

Then there were a few students who just watched videos; I’ve made it more clear in version 2.0 that problem solving must be at least part of each check.

I’ve added a table for students to actually keep track of when they work and what they do; valuable information for all of us. Along with this I added a requirement to space out the work they do; cramming for 5 hours the night it’s do is anti-the point.

Finally, I’ve made a reflection portion that’s more specific, asking students what helped them, what they still have to work on, and what questions they have.

I’ve shared a draft of the Google doc template here, which includes all of the above changes alongside some things that have stayed the same from version 1.0.

I’m still conflicted to some extent about grading. I’m fairly certain at my school there will be too many students who prioritize other graded homework over studying physics if it’s not worth a grade at all, so last year I compromised by making homework a small, 10%, category. I’m certainly open to ideas on this.

I feel like I’m getting closer to the policy I’d like to have, but would love your feedback; please comment here or chat me up on twitter (@rutherfordcasey).



Choosing to Engage

I have Franco* in a course I’m teaching for the first time this year, Foundations of Physics. FoP was designed to help students who really struggle with math and reading, identified by test scores, to earn their required Minnesota physics or chemistry credit. It’s a below grade level class filled with students who largely have been unsuccessful in school in the past, and is certainly has an overrepresentation of students of color**. Franco has to some extent been a thorn in my side. He’s boisterous, distracts other students, and won’t stay in one place. He’s also demonstrated that he’s very bright, and he has a great sense of humor.

Earlier in the year I mentioned that we were going to perform a lab over again, and Tahvo* stood up and said “Another one!” Franco was among the students laughing; clearly this was a thing. I ignored it and moved on. A few days later it happened again, then again a few days after that. Finally I asked about it. “Just Google ‘DJ Kahled'” they told me, snickering. Turns out it’s a short clip that’s fairly mild (I expected way worse) that has become a meme. So here’s the thing; I could have assumed they were laughing at me. I could have reacted. By engaging in conversation instead, we’ve now gotten to the point where ‘another one’ is a class meme. It’s helped build culture in my classroom.

Another time Franco put an ear bud in my ear; it was NWA, which I recognized (I grew up listening to NWA, Easy-E, and Dre, coming back now with Straight Outta Compton). I’m not immersed in hip-hop culture now, but I can certainly have a conversation with kids and show a bit of interest. When they put in their earbuds, I can engage with them rather than react against them.

Today after class Franco was one of the last ones out the door so I stopped him.

“Hey, what’s your plans after school?”

“I don’t know…I gotta figure that out I guess.”

“Yeah, well, you got a lot of potential. You could do something really great. You just gotta figure out what that thing is.”

“Dang, you don’t hear that from many people.”

You don’t hear that from many people. Meaning, most people have written Franco off and he knows it. He’s been unsuccessful for undoubtably a variety of reasons. But he’s still a kid, and he still deserves to be believed in. I can engage in his positives, rather than react to the negatives. I can choose to engage.

*None of the student names in this story are their real names. 

**I’m happy that our school district is working hard to correct this, both by ending below-grade level courses except for those required for students to graduate off their IEP (including this being the last year of FoP), and by implementing an Excellence in Equity team to find solutions for our students we have traditionally failed to help. 

The Seeds of Climate Change Denial (or Newton’s 3rd Law)

Today in class we were working on practice 3 from my Momentum Transfer Model (MTM) packet (which is a combination of standard modeling questions, Kelly’s questions, and a couple I wrote maybe). In first period we did problems 2 and 3, below, in a row. After problem 3 I gave students one of my standard speeches about how some students decide physics is about choosing an answer and then switching it because they know they’ll be wrong. I try to emphasize a number of times throughout the course that a good scientist isn’t one who’s initial intuition is correct; she is one who is willing to take a step back, consider the science of the situation, and then make sense of it. Anyway, in 1st period, almost every student choose that the VW experienced a greater force in 3a, despite the fact that we did quite a bit of work with Newton’s 3rd law, work I still think is pretty effective, back in November. The problem is, students haven’t made it second nature yet to start with science and end with conclusions. They start with their gut and go from there.

Screen Shot 2015-04-23 at 11.22.09 AM

During 2nd hour I inadvertently gave my speech between questions 2 and 3. Every kid said that the forces were equal in 3a. Every kid.

This helped me to realize that even when students understand a concept at some point, they still tend to revert back to gut-first thinking over time. In this case, the students were attempting to make sense of the situation, but without considering the physics involved they tend to confuse acceleration (the affect of the collision) with the force of the collision itself. Instead, if they start by applying Newton’s 3rd law, such that the forces are equal, then apply Newton’s 2nd to show that the effect of that same force depends on the mass of the object, it makes sense that the forces are equal but that the effect, the acceleration, is different.

So it’s not really that big a deal in the scheme of things when students’ don’t correctly analyze a collision, but it’s a much bigger deal when we have policymakers denying climate change. The problem is summed up in a picture from a tweet that came across my feed the other day;


Teaching science should mean giving students the tools to analyze situations and make logical conclusions while at the same time emphasizing how to do so in a variety of situations. If we fail to give students the tools of science, or show them how they apply, or challenge them with different situations, or reiterate their prior understandings, we leave room for them to revert to gut-oriented thinking. I hope that my preaching about starting with science rather than intuition can help in some small way to move our society from one where people try to ‘prove’ their own biases to one where we use data on a real quest for truth.

Starting Circular Motion

I’ve been looking for a better way to start circular motion for quite some time. Though many people use the spinning stopper lab, I found it difficult to get quality data or even a decent trend, even when I did the experiment. I tried using a pendulum setup, but I wasn’t happy with the hand-waving about non-uniform circular motion. I really like this particular treatment because it first focuses on the conceptual aspect of centripetal acceleration as toward the center, and then follows that up with lab quantifying the acceleration. It also gives some really nice opportunities to review and refine some lab techniques like uncertainty propagation and linearizing (both could be dropped or modified for your purposes) that my students will need as we progress through the year.

Day 1

We spend the first day investigating the accelerometers in Labquest 2s so that we understand the direction of the acceleration based on the graphs of x, y, and z acceleration versus time. (Note: certain this could fairly easily be adopted to use Arduinos or student phones; but note that asking students to risk their phones on a spinning apparatus in day 2 is tough). First the students turn on the built-in 3 axis accelerometer and observe that it reads approximately 9.8 m/s/s in the vertical direction, no matter the orientation. Then we watch the video below up to the two minute mark to explain why that is, and to help understand how the accelerometer is collecting data.

Next we do a series of short trials to confirm directions of positive and negative for each axis. We then turn off the z axis accelerometer as we will be working only with x and y.

Students are told they will be spinning in one place with arms outstretched, with very smooth, fast steps, as quickly as possible. They will be starting the data collection while spinning to eliminate the spin up process from data collection. I ask them to graph what they will see; most think it will be sinusoidal. Then students actually do the trial, and sketch what they see. It takes a lot of individual conversations here for them to see that the primary acceleration is in the negative y direction (based on how we hold the Labquests). Spinning faster and smoother helps this, and I have to point it out for a lot of groups, who then confirm with more trials (and more falling over from dizziness). It’s a good time.

Next we have a quick conversation about how the acceleration is negative y, so that means it’s….wait, what? Toward the center? “Hey everyone, go grab a bowling ball and a hammer.” I instruct them to make the ball go in a circle using small taps with the hammer. No, not spin…actually travel in a circle. Then I ask what direction they have to tap it in order for it to go in a circle. “Toward the center.” I go through a quick note about how applying linear taps speeds the ball up or slows it down, and that the net force from the taps is in the same direction as the acceleration. Thus for our circle, we apply a net force inward, and as a result the acceleration is inward. They don’t like this, not one bit.

Now is the right time to talk about turning in cars. I ask them to get in a car with me, and I slam on the gas. What way did the car accelerate? “Forward.” What way did you *feel* like you were moving? “Backward.” We do the same treatment of slamming on breaks, and talk about how really our bodies are just trying to keep on doing whatever they were doing, so we feel like we move the opposite direction as the actual acceleration. Ok, so now we are going to turn left. What direction does it *feel* like you are moving? “Right.” So what direction are you accelerating? “Huh. Left. Towards the center of the circle.”

And that’s enough for today.

Day 2

Now that we have the basic idea about centripetal acceleration it’s time to quantify it. We brainstorm; what factors affect how strongly you feel pushed to the outside of the car? (but are you really being pushed to the outside? No? Good) They come up with speed and radius pretty quickly. This part does have to go pretty fast, as data collection is tough to get done in our 48 minute periods. If there’s time we have a conversation about how investigating the radius and it’s affect on acceleration is tough because speed also depends on the radius. So we settle on changing the speed of rotation and measuring the resulting acceleration. On what you ask? Only the best equipment for my physics students.

To measure v; How far does an object go around this circle? “The circumference, 2 \pi r .” Ok, so we’ll call the time it takes to go around once the period T, so the speed is v=\frac{2 \pi r}{T}.

Thus we measure the period to calculate the velocity (which we’ll do later). We use the statistics function on the Labquest to measure the mean of the acceleration, only while the acceleration is moderately constant, and use the standard deviation for the uncertainty. We collect data for 20 seconds since we now can’t avoid the spin up. Some very important student instructions;

  • Only use the linear section of the y acceleration graph
  • Each trial involves hitting play, starting to spin, maintaining that constant rate of rotation, then starting to count revolutions to measure the period. T=\frac{total time}{number of revolutions}
  • It’s hard to actually spin the chair at a constant rate. I’ve seen a variety of techniques, but most groups either reach from above on the back, keeping their hand on the back the whole time (as discrete pushes show up as very obvious waves on the acceleration graph) or spin from below with quick, regular pushes.

This year I had some timing issues so we really only had time to collect data this day; in future years I think we’ll have time to go over the calculations of speed and uncertainty using Google Sheets here as well. I walk them through how to use the period data to automatically calculate speed using Sheets. We also have a conversation about uncertainty in the speed; it’s a propagation of the uncertainty in the radius and the period. So we estimate those uncertainties, then use sheets to calculate the maximum possible speed using the maximum radius and minimum period for any particular trial. It’s really nice to do this now, as we do an experiment later using photogates where we have to similarly propagate the uncertainty.

Day 3

Start today by graphing acceleration vs speed in linreg. In most cases their are two pieces of evidence that the trend isn’t linear; it looks a bit curved (though this depends a lot on the group), and the intercept is usually significantly negative.

cent acc vs speed


As much as possible I have the conversation about these factors with each group, but as there gets to be more of them I toss it on the overhead and we hash it out there. We talk through why the intercept should be zero, and use the combined evidence to try linearizing. Below is a student spreadsheet with a wonderfully linearized graph.


Once the graphs all have linearized graphs, they whiteboard them. There will be a number of groups with data that makes no sense; I think they generally missed one or more of the “Important student instructions” bullets above. We talk about it, and I have them take a look at other groups’ data. The following discussion centers first on the quadratic nature of the data. Either someone does a unit analysis of the slope or I point out how nasty it is (\frac{m/s/s}{m^2/s^2}), so we simplify it to 1/m. Eventually someone notices that the smallest radius has the largest slope and vice versa. I ask them to combine the evidence of the units of the slope with the radius–>slope information into a claim about the slope, and we end up with a_c=\frac{v^2}{r} (note that facilitating this discussion is significant, but material for a different post).

I then emphasize the evidence that we’ve used to get to that point; the curve in the acceleration vs speed data and the negative intercept leading us to a quadratic relationship, and the units and radius comparisons leading us to an inverse relationship between acceleration and radius. We finally test it against our original musings; as we go faster around a curve, does it feel stronger? As we decrease the radius, does it feel stronger? It’s good that our equation matches our experiences.

In addition to the reasons stated at the beginning of this post, I love that the kids have a blast doing the lab. Playing with spinning chairs is fun for people of all ages.

On Personalized Learning

The term “Personalized Learning” has rubbed me the wrong way for quite some time. Admittedly this likely stemmed from stories like how Carpe Diem school ‘personalized’ learning by putting students in cubicles;

Carpe Diem cubicles

Raise your hand if you have a job where you work in a cubicle all day and feel energized, appreciated, and passionate about your work.

I can’t imagine much worse for my children. I want them exploring, interacting, discovering, and, most importantly, interested in learning. I don’t want them moving onto the next algorithm after earning a badge.

Recently it appears the Edtech community has strayed from the cubical, Khan Academy model of Personalized Learning in favor of something more nebulous; the basic idea that students can work at their own pace with the teacher guiding and tutoring on the side. This often comes with mantras such as ‘student choice’ and ‘individualized learning plan.’ These aren’t bad things, but I submit that students working primarily on their own, at their own pace, is.

Which brings me to my recent revelation about why personalized learning as a primary structure for learning bugs me; it’s still passive. It appears that most of the ‘content delivery’ is still about students absorbing information from a source passively, then working exercises or doing practice of some sort to work towards mastery. When I think of an ideal math lesson, on the other hand, I think of rich tasks that take collaboration and significant critical thinking, such as Fawn’s Barbie Bungie Jump (listen to the kids cheer in the video!), Dan’s 3 act lessons, or Desmos’s Central Park. Shooting for productive struggle, I want to walk into a math class and see kids pointing at each others work, arguing, and even cheering.  Summarized, I want math class to be engaging in the sense that students actually want to be there. 

If I want students to be learning through collaboration and dialogue, then, generally speaking, I want them moving along at about the same pace. I do have times, particularly towards the ends of units, where students are working on problems independently for solidifying problem solving or receiving remediation as needed. This, however, is usually a few days to a week, as compared to the other three to five weeks in a unit where students learn primarily through collaboration. To be sure, my beliefs here are rooted in the decades of research on STEM education which has demonstrated consistently that a variety of methods centered around active learning are the best ways for students to learn. Additionally, in talking to my wife about this, she gave an incredible insight; “What skills can you gain from class time that you can’t gain from studying?” Precisely. Studying on one’s own helps to learn content, but collaboration, argumentation, sense-making through inquiry, and many other skills are emphasized when rich activities are the focus during class time.

There are some other things that bother me about personalized learning. It appears to be rooted in the theory of Learning Styles, which isn’t really a thing, as it turns out. (see also here and here). Students, generally speaking, learn from some types of teaching and don’t from others. Identified preferences in how that learning takes place hasn’t been shown to make any real difference in the actual learning that happens.

Then there is this post which makes the claim that the ‘factory model’ of education that many personalized learning proponents want to upheave is really the first experiment in personalized learning.

Finally, I agree with Dan Meyer who states that personalized learning is fun like choosing your own ad experience is fun. (Spoiler alert: It’s not).

I do believe that proponents of personalized learning mean well, and I believe that aspects of the model woven into a class at the right time can be useful. In the end, however, I choose rich, engaging, interactive tasks over learning at one’s own pace.