27 April 2013
Heresy: Ignoring the 2-pi term in the period of a spring for conceptual physics
Heresy, you shout. You just ignored the 2π term! You can't do that!
I remember being utterly shocked by my sophomore quantum physics professor exclaiming wildly that "factors of 2 don't matter!" We were a bit miffed that some homework problem predicted an answer that was off from a book value by about a factor of 2. What we, silly newbies that we were, didn't understand was that the homework problem was showing an alternate means of approximating a well-known value. We should have been amazed that we were that close; instead, we complained of inaccuracy. It took a couple more years before the "order of magnitude estimate" became a part of our collective soul. I want to begin the incorcism process a bit earlier.
In a laboratory in which we are trying to measure to a precision of 5 or 10 percent, a factor of 6 does indeed matter. So why do I write the period of a mass on a spring like this?
I'm teaching CONCEPTUAL physics to 9th graders. Most are finishing their first year of algebra. The idea of a square root is still new to them. And they react to π about as well as a bunch of trainee umpires.
Most of the questions I ask in conceptual physics are of the form, "The mass attached to the spring is doubled. Does the period increase or decrease? By a factor of 2? Greater than a factor of 2? Less?"
Well, the heretical equation above demonstrates the dependence of the period on mass and spring constant just fine. I've eliminated the 2π and I've broken up the square root signs above and below the fraction bar for clarity's sake. I'm not in the business of teaching mathematical notation. I'm pleased if my students can recognize that doubling the mass increases the period, but doesn't double the period. And if they can sketch a graph of period vs. mass for a constant k.
My hope is that when these students attempt AP Physics 1 in their senior year, they have a head start toward the necessary conceptual understanding. They can learn how to linearize graphs, to tease out the 2π factor, in future courses. For now, my class is understanding well the RELATIONSHIPS among period, mass, and spring constant. That's my goal, here in 9th grade, even if I burn at the stake the next time I see professional physicists.