Notation for Newton's Second Law: F, Fnet, Sigma-F. And Julius Sumner Miller.

Alex writes in:

I was listening to Julius Sumner Miller lecture on YouTube while doing some work last night and noticed that he said F=ma. 

So my question is using the term net force something that teachers just use? Was it something that wasn’t used then but is common vocabulary now?

Glad you reminded me about Julius Sumner Miller (which is still what my brain wants to call my colleague Julius Reynolds, because until him I'd never actually known a Julius other than Orange Julius).  I should watch some of those videos with my AP class when we're online in December!  Make sure you watch the one where he puts random stuff in liquid nitrogen, and forgets some things in the carafe.  JSM is a big, big influence on my teaching style, and I'll bet you can tell.  He didn't have a pet hippopotamus named Edna, though, more's the pity.

That's an interesting history question, Alex... there are people who research the history of physics education.  I wonder... I copied Gardner Friedlander, one of my go-to folks for AP Physics history, in case he has something to add.  (He did - see below!)

Before we start - I am not discussing whether your personal version of Newton's Second Law is the best one.  Of course it is, and all others are inferior.  I will not publish any comments that talk about why one notation is best.  I'm discussing the various notations, their history, their pros and cons.  If you use sigma-F but I don't, that doesn't mean you're better than me or vice-versa.  Really. We all have our reasons for our notation.

I do seem to remember that when I grew up - and I watched JSM in high school physics class all the time, on VHS - the phrase in common use was just "F=ma."  I looked back at the earliest AP solution set I can find, and it says sigma-F = ma.  (There was no equation sheet back then, before the days of graphing calculators.)  The 1983 New York Regents exam says "a=F/m."  I'd love some evidence that my recollection of "F=ma" being the common statement is correct... certainly I think that was the non-physics-class zeitgeist meme that people would say.

When I started teaching in 1996, I always used the notation "Fnet," emphasizing that only the NET force could be set equal to ma.  See, I discovered quickly the common issue that students would pluck a force value from the problem statement, pluck a mass, and smash them together into F=ma to get an acceleration.  I didn't like the sigma-F notation because students tended to add force numbers without reference to direction when they saw that.*

*or they would cluelessly wonder "what's that big ol' squiggly E, I dunno, it's got an F next to it, I guess."

In the AP physics 1 revolution, the curriculum design committee (led by some top rate physics educators) decided to rewrite as a=sigma-F/m, because this emphasizes that acceleration is usually the quantity that is the result of the various forces.  I had some discussions with a big group about whether and why that made sense or didn't make sense, and how that change should be explained.  On the equation sheet now, it reads a=sigma-F/m = Fnet/m.  I don't know, but I suspect, that this dual notation is to accommodate the two dueling camps of physics teachers: those who prefer students to write using vector notation "Fn + (-mg) = ma" because down is the negative direction, getting a negative acceleration if the acceleration is downward; and those who prefer students to not use negative signs but just magnitudes of forces, writing either "Fn-mg=ma" or "mg-Fn=ma" starting with the direction of acceleration.  The former is sigma-F=ma, the latter is Fnet=ma.

Gardner Friedlander says: He quoted several sources from the last 50 years that use all three versions of what should be set equal to ma: F, Fnet, sigma-F.  

His thought - which I agree with, now - is that the difference is probably not so much a historical trend, but rather the intended audience.  Julius Sumner Miller and Paul Hewitt were aiming at a general, non-mathematical audience trying to understand physical concepts, so used just F.  Mathematically based courses preparing students for continuation in the physical sciences used sigma-F.  And those aiming for an in-between audience, like AP Physics B, AP Physics 1, high school honors courses - they tended to go for the in-between notation of Fnet.

And in a final twist... I've started changing my notation to "netF = ma".  I've noticed for years that Fnet is confused with Fn for normal force; and that it's a misconception that "Fnet" is a separate force like the friction force or a tension that should be on a free body diagram.  By using the language of unbalanced and balanced forces (as the Physics Classroom does), the net force is just the unbalanced force; so netF emphasizes something different from Fnet.