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 (). 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.