Force Vector Addition Diagrams (or, Components No More!)

The graphical solution bug has really gotten me this year (and in the best possible way). I’ve apparently done such a good job of pushing the graphical solutions that one of my classes stopped me in my tracks while I was showing them how to solve force problems by breaking the forces into components and doing 2 perpendicular N2L analyses.

“Why can’t we just use the vector addition diagram that we already drew?”

“Yeah! We can figure out how long that side is, and then we can figure out how big the gap must be. It’s not that hard.”

Not that hard? This skill (solving 2-D force problems, especially when the forces are unbalanced) has typically been one of the most difficult in the entire year of regular Physics! classes (high school juniors, typically in Precalculus) even though it wasn’t (and isn’t) any kind of big deal in the Honors Physics classes (high school sophomores and juniors, typically in Algebra 2 Honors or beyond).

So when we first start looking at unbalanced force problems and my regular Physics! students tell me to chill out because they already see an obvious way to solve them, who am I to intervene?

What are Force Vector Addition Diagrams?

Force vector addition diagrams (a lot of students end up writing “VAD”, though I never do that) take the free body diagram (FBD) and add up all of the forces as vectors (so, head-to-tail with the size and direction of each arrow being very important). If the forces are balanced, then you should end up with nothing left over when you add them all together (\overrightarrow{F_{net}} = 0). That is, when you add all of the forces together, you should end up back at the same spot where you started. So with balanced forces, the vector addition diagram is a closed shape.

If the forces are unbalanced, there will be a gap in the diagram since the vectors will not add up to zero. The size of the gap represents how unbalanced the forces are. The direction of the gap represents the direction of the acceleration (same as the direction of the net force). We generally draw the net force vector as a bigger, outlined arrow to distinguish it from the forces acting on the object.

The order of addition doesn’t matter (just as in “normal” addition). So there can be multiple correct answers to the addition problem that (to the kids) don’t look the same as each other. The result will look the same, but the shape created by the vectors might look different. Some students have understood that for a while and some are just starting to realize the subtleties there.

Both of those diagrams show correct vector additions from the same correct FBDs (the second one had some trouble with the numbers, but the shape was correct).

Break out the Protractors: Using Vector Addition Diagrams to Solve Problems

At first, we always draw the diagrams to scale. As long as you know enough about the directions and/or lengths of the vectors, you can figure out the direction(s) and/or length(s) that you don’t know.

Here’s a great student solution from a quiz. It was early on in our experience with vector addition diagram drawing, so she wrote a lot of her thoughts out as she talked herself through how to set up her work. In light pencil marks near the top (right-ish), you can see the sketch of the general shape she did before trying to draw it to scale. She writes out her conversions between her measurements in centimeters and the values in newtons.

Drawing the diagram (especially to scale) can help struggling students check to see whether their answers make sense.

You can see the erased lines from previous attempts at solving the problem. Once she had drawn the angles, she realized that there must have been something wrong with what she had done. She persisted in figuring out what was wrong until her entire solution was correct.

As they became more comfortable with the diagrams, many students started drawing just the sketch, labeling the angles, breaking the shape into smaller shapes (triangles, rectangles) and using right angle trigonometry to find the unknown lengths and directions. They saw the opportunity to use trig without my suggesting it to them and while they already had another, perfectly valid, way of solving the problem. I like that so much better than feeding them components while I teach them an algorithm.

Some students (especially the juniors in Algebra 2) haven’t learned about sine/cosine/etc yet. It would be easy enough to just show them how to use the button on the calculator, but they are (so far) more than happy to keep using the protractor and a scale drawing to solve problems. They already have a method that makes total sense to them.

Now that we are in the new semester and two units (soon to be three) removed from when we learned about unbalanced forces, I’m starting to see a few students “invent” the idea of looking at the x- and y-components of the forces and doing a Newton’s 2nd Law analysis in component form. After spending so much time with the vector addition diagrams, they started to see the pattern about finding the size of the horizontal sides and the size of the vertical sides in pretty much any diagram they drew.

Using these diagrams as an “exclusive” (until they figure out otherwise) way of solving force problems has been great for my students this year. Compared to past years, many many more of them find success in situations with unbalanced forces and the level of understanding (for all students, including the strongest ones) is deeper. Using one of these diagrams to solve a problem shows much more understanding about how the quantities are related than does churning through an algorithm and writing down a lot of equations. So, win-win.

14 thoughts on “Force Vector Addition Diagrams (or, Components No More!)

  1. Vectors has been one of my fears for next year (switching to modeling). Our workshop leader mentioned that he adds in part of the modeling content from trig for his classes. I think I like your idea of just using the protractor. I’m becoming heavily swayed by Dan Meyer’s mindset of don’t give it to them until they ask for it. Great post again!

    1. I have at least one junior in Algebra 2 who both (a) still has no idea what sine and cosine are and (b) is one of my strongest students (earning one of the highest grades in the first semester and working doggedly to do the same for the second semester, too!). I actually think she has a better understanding of what’s happening with forces because she is always thinking about size and orientation of the vectors and never relying on her calculator to do the lifting for her. So there’s definitely hope for vectors sans trig (and for the ones that know it or as they learn it in math, they eagerly bring it in on their own).

      And thanks! 🙂

  2. I love the vector addition diagrams!

    I’m doing my own version of the modeling materials and in a different order than the original MI but that order is similar to yours. I have a few questions pertaining to sequence.

    When do you introduce vector addition diagrams (BFPM or UBFPM)?
    If you introduce them in BFPM, do you also use them (in a general sense) for unbalanced problems there?

    When/how do you introduce the term Fnet?

    How do you deal with forces at an angle and when are they introduced?

    Thanks so much for all you do!

    1. Great questions! 🙂

      Vector addition diagrams come up in the BFPM unit. The timing varies, just depending on when the class finds a need for them, but it often happens when they are whiteboarding the FBD practice set (the one where they draw a lot of FBDs for a variety of situations). We’ll start picking on whether the diagrams look balanced or unbalanced (and whether they should look balanced or unbalanced), and that’s easy enough for when there aren’t angles happening, but it becomes really problematic when there are forces at angles. So we start talking about ways to decide whether the diagram looks balanced or unbalanced, and it eventually leads to the idea of adding the forces together (as vectors) and seeing whether we end up with zero or with something left over.

      So Fnet comes up right away, there, though I usually don’t bring up that term then (unless it makes sense in the given conversation). They have a strong idea about unbalanced forces already (and how unbalanced forces will affect the velocity of an object, qualitatively at least). One class this past year called it something like “the ultimate grand supreme arrow of destiny” (or something) for a while. The big outline-y thing that points in the direction that the forces are unbalanced (and that is the vector sum of the forces).

      Does that help? Please ask again if I haven’t quite answered it yet! 🙂

      1. Your way of explaining and discussing how the process works is very helpful (invaluable!).
        The humor is also appreciated (supreme arrow of destiny?).

        I do have some follow up questions.
        If VAD does comes up during the 5 common problem set, do you explain it briefly or do you have to go into more detail? How do you have students practice using VAD? I imagine it can be done by going back and using the 5 problem set itself but, it has been my experience that learning VAD takes a lot of time for students to become proficient and often students need help using (or being reminded how to use) a protractor. I’m afraid to get too far away (time and concept wise) from the model packet.

        Also –
        Do you wait until PMPM to address other angled vectors using VAD such as displacement and velocity?
        Where does work at an angle come into the models?

        On a related/different topic, I have to teach the details of friction interactions (which I don’t see in the MI materials) and thought an exploration to graph the ratio of normal and frictional forces would be a good introduction. It seems to fit in just before your 5 problem set in BFPM (at least the idea fits due to the details of the questions). I’m going around in circles a bit because I need the students to have an understanding of balanced (and therefore equivalent) forces to make the friction lab work but I don’t think that will happen until after the 5 problem set. I also don’t want to get too far away from solidifying the balanced/unbalanced force ideas and VAD to put it in after the 5 problem set (and before going on to the remainder of the packet). Any ideas on where to put the friction exploration in the grand scheme of the models?

        On a final note, I’ve been toying with the idea of waiting to introduce N3L until MTM (which I would do following UBFPM). I like the idea that both (N3L and MTM) point out/reinforce the force interaction between objects. And I like that it’s as if we’ve ‘zoomed out’ on the BFPM and UBFPM single object focus. It also ‘feels’ right (to me) to close a loop I began with EMT. (Zoomed out with EMT, zoomed in with CVPM, BFPM, CAPM, UBFPM, and then zoomed out for MTM.) It follows with my idea of beginning with the big picture and always keeping it in view while examining some of the details. I would do CFPM and PMPM following this as special cases of the whole ‘circle’ of models. What difficulties/conflicts do you see as possible with these ideas?

        Sorry I’ve thrown so much at you at once (and sorry to say I actually have more questions in different topics).

        1. I’m not sure what you mean by the 5 common problem set. I was talking about this page in my BFPM packet:
          where they draw a bunch of FBDs. During whiteboarding, we almost always get into a discussion about how to tell if an FBD with forces that are at angles looks balanced or unbalanced, and we end up with a qualitatively drawn vector addition diagram (no protractors yet). Later in the same packet, we start using them to solve problems quantitatively (both with and without forces at angles—I don’t separate that into a different topic since it is all part of the same model). If individual kids need help using a protractor, I help one-on-one as needed. I don’t really do significant direct instruction about pretty much anything (all year).

          Once they become pros at using force vector addition diagrams, the other uses (for 2-D momentum problems, in projectile motion, etc etc) become obvious when they arise.

          Re: friction—It is easy to fit another empirical force laws experiment (probably investigating many things since they won’t immediately focus on what things that they can measure will actually affect the friction force—so velocity, area, normal force, etc) in with the Fg and Fs ones. I just don’t do it because I don’t see the value in the amount of time it takes.

          Re: N3L—it is one of the most difficult-on-a-gut-level ideas in the course, so I think getting it out there as early as possible (so you have time to play with it as much as possible) is key. I wouldn’t take it out of the force models, but I know in my classes we talk a lot about it in the MTM unit as well (starting right away—the relationship they figure out from our paradigm experiment sounds suspiciously like N3L… something about what the 2 objects do to each other is equal but opposite…). I think bringing it in as much as possible is a good idea. They need a lot of time to really train their gut on that one (or to convince the tiny Aristotle in their guts to cede to the tiny Newton in their brains on it, at least).

          And… No worries on the lots of questions. 🙂

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