When not to cancel common factors

Today I was helping some students of mine get ready for my final. These are science teachers who are in an alternative licensure program for physics. We were talking about the equations for the resonant frequencies in various systems and we developed some great insights about how to approach the equations involved.

One of the things we talked about were the similarities and differences between these two equations:

f_\text{same}=\frac{nv}{2L},\,n=1,2,3,\ldots,\; \; f_\text{different}=\frac{n'v}{4L},\, n'=1,3,5,\ldots

Here the left equation is for resonant systems where the two ends are the same (a string tied at both ends, or a tube open at both ends) and the right equation is for systems where the ends are different (a tube closed at only one end). We had derived both of these equations earlier in class and the students were asking why one has a 2 and one has a 4 in the denominator. As I was reminding them of the derivation, it hit me that they really both have a 4 in the denominator but we divide out a common factor of 2 in that case. In this post I’m arguing that it’s a mistake to do that, because, if you don’t, you get only one equation where the math leads to the physical principles.

Here’s what I mean. To derive either equation, I encourage my students to find patterns of nodes and antinodes that obey the appropriate rules. Really there’s only two rules: make sure the ends are correct (both nodes? both antinodes? one of each?), and nodes and antinodes should alternate.

Here’s what their whiteboards start to look like while studying a system with nodes on both ends:


N A N
N A N A N
N A N A N A N

After three they tend to get the pattern. Now we do some analysis. Prior to this point we’ve worked out that the distance between any two nodes and antinodes is a quarter of a wavelength. For the pattern above we find that there’s always an even number of those quarter wavelengths:

L=2n \frac{\lambda}{4}\to L=\frac{2n v}{4 f}\to f=\frac{n v}{2L}

When they analyze the other type of system they find that you always have an odd number of quarter wavelengths, leading to the right equation up above.

Here’s my point, if you don’t cancel the common factor of 2 in the second step above, you end up with

f=\frac{nv}{4L}\,n=2,4,6,\ldots

which looks the same as the equation for the systems with different ends, only you use even numbers instead of odds. What my students and I discussed this morning was that if you leave it that way, you have one equation for both applications. The “4” that they asked about is immediately traced to the fact that there’s is a quarter wavelength between nodes and antinodes. You use evens for one and odds for the other because of the nature of the ends.

I like this way of thinking about it better than what I usually do (the first equation of this post). Later today we were discussing another situation and I found another time not to cancel something. This time we were studying the diffraction pattern of a single slit. We thought about ways that the infinite sources inside the slit could conspire to give a dark spot. We reasoned that if the top half cancelled the bottom half, we’d be in business. Of course, this is how a lot of texts explain the first dark spot in the single slit pattern.

When you get to the math of it, you find that the very top portion of the slit cancels with the portion just below the mid line. The analysis at that point is very similar to finding a dark spot in a double slit pattern (d\sin\theta=\lambda/2) only you replace d with half of the spacing:

\frac{a}{2} \sin\theta=\frac{\lambda}{2}

There’s nothing wrong with this result, but I commented to my students that many texts simplify it to a\sin\theta=\lambda. My point is that leaving in the factors of 2 lets you see the physics: it’s a/2 because the top half and the bottom half cancel each other. It’s \lambda/2 because you’re trying to get destructive interference.

What are some other examples of times with math simplification leads to physics obfuscation?

 

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About Andy "SuperFly" Rundquist

Associate professor of physics at Hamline.
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6 Responses to When not to cancel common factors

  1. bwfrank says:

    I really like this post. This really focuses attention on how mathematics in physics is meant to represent the relevance of physical quantities and their relationships, not just mathematics as an equation for calculating. I think your experience is also a good example of we continue to learn physics (even intro physics) more deeply through our interactions with students. I’m going to think about this for a bit, and see if I can find any other examples.

  2. Bret Benesh says:

    Hi Andy,

    I love this post, too. I had similar insights when my future elementary education students were trying to figure out why “Invert and multiply” gives the correct answer to fraction division questions. We solved some problems only using a definition of division first, and then tried to backtrack to figure out how multiplication by the reciprocal popped up. It was very difficult to backtrack when the students simplified as much as possible.

    Students need to be taught that algebra is a tool to be used when convenient, not an iron-clad law.
    Bret

  3. Tamara Knudtson says:

    I am a high school physics teacher working on my master’s degree in education. One of the classes I took this summer was called mathematics for struggling learners, something I took to be able to better help my physics and physical science students with their math issues. We discussed many of these types of math issues you illustrated here. We talked a lot about things like Bret mentions in class and we even tied many issues that students have back to their lack of truly understanding math. We discussed how we need to ask students about math and truly listen to them.

    Thanks for your illustration.
    Tamara

  4. Joss Ives says:

    Andy,

    The inelegance of the two equations as you originally wrote them is something that has always bothered me. They are the only two equations which show up on an intro equation sheet that need a bunch of extra notation to specify the cases to which they apply. UGLY

    After thinking about leaving the factor of 1/2 in for your interference example, it occurred to me that this idea is not all that different than using a=F/m instead of F=ma. What we’re trying to do is write the equation, not in its most concise form, but in the form which best represents the physics the equation is trying to represent (great use of represent twice in a sentence Joss!).

  5. Brian Lamore says:

    Nice post. It’s not an equation but I think one common example of oversimplification is Newton’s 3rd Law. Everyone can state the rule, but I find many don’t really understand what it means. But stating the rule might earn some students a passing grade. (Oops, did I really go there?)

    I was setting up a N3L experiment in my lab during an “open house” sort of deal (I don’t remember what but visitors were milling around). I was going to crash together 2 carts and measure the force on each with force probes. One car had an additional 500g mass on it. Everyone who stopped by to inquire got the answer wrong or didn’t know. But everyone knew the “action-reaction” rule by heart. Like the Pledge of Allegiance or the lyrics to a Xmas song or their phone number…

    Perhaps the fact that this “action-reaction” rule is used in common metaphors makes this an unfair example, but there it is.

    • Andy "SuperFly" Rundquist says:

      I love your line about the Pledge of Allegiance, that’s so true! N3L is interesting to me because I think it’s the most important of the 3 laws. I like to say that 2 things are true about the universe: 1) things swap momentum. 2) interactions determine the swap rate.

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