Tag Archives: Central Force

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.

Modeling Central Force: Day 3

Note: Since writing this post I have significantly changed how I start this unit; see new post here

I had three main objectives for my lesson today;

  1. Wrap up the modeling process, particularly figuring out that the proportionality constant for Tension vs. v2 should be m/r;
  2. Go through the awesome graphical derivation of centripetal acceleration
  3. Use the traditional spinning stopper lab as a goal-less problem.
1. We looked yesterday at the ‘A’ parameter of the quadratic, the one in front of v2. Generally students found A proportional to mass and inversely proportional to radius, which was good, but there was no clear pattern, so I started compiling data. This is one class’s data; looking at it this way made me excited for the possibility of having computers in my room so I can easily compile using Google Forms. This also allowed for sorting by radius or mass, which was useful. The students were quick to figure out that C should be zero, hence the yellow cell that indicates less-than-perfect data; we don’t trust that regression anymore. Despite that, their proportionality constant was better than most. Only half the class had reasonable constants; surely not enough for them to see the relationships perfectly. However, at least they were able to pick out the general trends. One thing I hope the Modeling Instruction workshop helps me with this summer is the perplexity of how some student groups get fantastic data and some awful with the same setup and instructions, when I often can’t seem to find the source of the problem (though time and class size contributes to the lack of problem solving as well). 
Velocity vectors on a circle

2. Next we went through the graphical derivation of

I had them working on their desks with whiteboards, and I guided them through the steps. I think that this derivation is fairly abstract for students’ to come up with on their own (though I dare you to get them to do so, please let me know if you do!). It went fairly well, and will be re-iterated when they do their pre-lab assignment which is essentially the same exercise. I like this one in particular because a)  it is super cool that you get a concise, elegant equation by doing a graphical proof, and b) I try to iterate that everything comes from somewhere (I use this for energy as well). Any equation we use can be re-derived using physics. These students are too used to 16 popping up in quadratic equations in Algebra II without any clue to the fact that it has significant physical meaning.

The last part was the best.

3. I ended today by demonstrating the spinning stopper lab and asking them to analyze it as a goal-less problem (which I haven’t done a ton, but I will be adding more; thanks to Kelly O’Shea for introducing me to this concept).

A student uses multiple viewpoints

The students ran with it, first figuring out that Ft=mg for the hanging mass, then using that to analyze the stopper. The cool thing was when one group (there’s always at least one in each class) figured out that the string must be at an angle as the y forces must balance. We then had a discussion about this, and I demonstrated that when the string goes quickly it is hard to see the angle.

Overall this last day went well, but would be better with solid data. Still, I definitely preferred using pendulums to the spinning stopper lab for the modeling aspect (to be honest, it’s not like I’m a veteran of modeling anyway…), though I would hope to find an even better way to model central force. Let me know if you have improvements!

Modeling Central Force: Day 2

Note: Since writing this post I have significantly changed how I start this unit; see new post here

Today we compared data.

I started class by asking students why the net force at the bottom of the arc was not zero. They realized quite quickly that forces should not be balanced since the direction of motion is changing (due significantly to previous emphasis on velocity as a vector, that is, when direction changes, velocity changes. Once again I credit Kelly O’Shea and her method for teaching BFPM for this). So then we drew a free body diagram and reasoned through which force should be bigger; tension up or mg down? They want to say down, but a quick discussion about what would happen if the net force was down dispels that idea. (Since the object is moving horizontally for the snapshot we have taken, a downward force would cause downward motion; clearly this is not the case.)

Side note #1: I want to mention why I didn’t do the typical rubber stopper lab (here’s a video if you are not familiar with that lab, ignore the flying pig part). I have done it for years, and I  found it at best to be an example of large systematic error. I actually had my honors classes write it up last year for that very purpose; how to write a lab with significant errors. I have never been able to get quality data, especially with students. I find that they even have trouble getting a good pattern for F proportional to square of v. Thus I wanted to try something new, and the pendulum method was something I had done previously as well and was suggested by another modeler on twitter as well (@BEPhysics). I found two main negatives for using this particular lab to build central force; 1) the data we take and the resulting analysis are only valid for the bottom of the circle, and 2) though it is true that F_net is greatest at the bottom of the circle, this lab falsely ‘proves’ that, in that force vs. time data shows a maximum at the bottom, but that is more due to the fact that the force detector is pointed vertically and thus only reads the vertical component of the tension. Even if this supports an overall correct vision, I don’t like the idea that kids would get the correct vision from an incorrect assumption. 

Then looking at the regressions as a whole, we re-confirmed that the data was quadratic, that is, that the net force on a pendulum at the bottom of it’s swing is proportional to the square of the speed at the bottom. That’s cool all by itself.

Next I had students looking for other patterns (what happens as r and m change?). I had them stay in their clusters first (see Modeling Central Force: Day 1 for more on this). This meant that they had different masses but approximately the same radius, and thus could look for mass dependence.

Side note #2: I have a lot of questions/problems with this part of the Modeling process that I hope will be addressed when I take the workshop this summer. I am still wrestling with the best way to have students share their data with each other to start finding the patterns. I also think that I need to bite the bullet and go get some large whiteboards; I have whiteboard desks and lots of space on my main whiteboard, but neither of those are portable, and using them has proven to be more of a workaround than something better than actual whiteboards.

I learned a lot about the guidance part of modeling in this particular instance. I really should have been more systematic in defining the radius and mass for each group. In one class, two clusters should have had different radii but were only different by 5 cm or so. More so, it would have been VERY helpful if each cluster had consistent masses, so that when they compared radius dependence they could do so with constant mass.

So here’s an example of what I was trying to do with the pairs/clusters concept, which was less successful than could have been due to my ambitiousness in directions.

This would have made the comparison process MUCH easier, I believe, as I could have them get into ‘clusters’ to check for mass dependence, and get into ‘groups’ for radius dependence. This is something I am going to consider trying later if I have other labs with two main parameters such as this.

Most groups had some sort of mass proportional to net force kind of data, but it was not obvious due flawed data. Data was VERY quadratic, but the actual parameters for the quadratic generally were not even close to what they should be. The A term (f(x)=Ax^2+Bx+C) should be m/r, but generally wasn’t. Some groups even had significant outliers; they had a larger mass than another group with the same radius, but the A terms were similar. Bad data was a big problem for this; in fact, NONE of the groups found the ‘correct’ terms for the quadratic (I found this out in analyzing the data later; we didn’t go so far in class to check). I didn’t have enough time to investigate reasons for this, though the fact that students and I can do the same lab with completely different results (meaning good vs. bad data) has been driving me nuts this year. Non-zero B values were a problem as well; though students were astute enough to realize the C had to be zero physically (Zero speed had to yield balanced forces, so if F_net was to be zero with v=0, C must equal zero) and thus it became a sort of litmus test for good data. C not zero? Now we don’t trust the rest of your regression. This worked reasonably well. B should be zero as well, but that was only true in the best of cases.

That said, in two out of three classes the data was convincing enough to show that F_net is proportional to mass and inversely proportional to radius. I used one of the other classes data sets to show the bad data class what was up (their data was truly horrible; no noticeable patterns at all! Very frustrating, both for them and for me). Thus

I asked what this seemed similar to; hey, it’s kind of like F_net=ma! Yes, yes it is. Meaning that

And this is where we ended for the day. (Note: It got much less frustrating for the wrap up tomorrow, which I will be posting asap)

Side note #3: I have been wondering to myself for quite some time about the rationale for linearizing. It seems as though it is something that made lots of sense 30 years ago, before we could graph any function in 2 seconds with a hand-held device. Now, however, why linearize? I don’t think I even learned how until teaching AP Physics, as in college we would simply deal with the actual functions, whatever they were. I would love some input on this. In this case, I linearized for one class and, as expected, the slope was essentially equal to the A in the quadratic while the intercept is equal to the B value. So why bother?