This week we moved into chapter 4, which contains A LOT of interesting physics! Among other things, students see the importance of simple spring systems and how they can be used to model certain quantum mechanical phenomena like chemical bonds. The Einstein solid model is introduced, and will play a central role in chapter 12 (if we get that far). Here are some things I find especially noteworthy about chapter 4.
Tension is introduced in a pretty cool way, which was never the case in my undergraduate experience. In this framework, tension is the magnitude and direction of each part of the third law pair at any point where a rope, spring, cable, etc. is cut and separated into two pieces. The direction is always parallel to the axis of the thing that is cut. I can imagine an introductory framework in which the stress tensor is introduced, after an appropriate introduction to tensors earlier, for a particular material and geometry and students see how to let it operate on a direction (unit vector) to get the stress, and force if the appropriate area is known, for a specific geometry. The traditional introductory approach is really one for which certain directions are parallel (or antiparallel) so the full geometric beauty goes unnoticed. Other than general relativity, elasticity is the only area of introductory physics where fourth rank tensors are needed but I may be mistaken and ignorant of other situations. Anyway, this would be an excellent place to bring tensors and geometry into the course. In his blue book (Teaching Introductory Physics), Arons has some fantastic problems on tension and how it must be nonuniform when the rope (or rod, or cable, or whatever) accelerates parallel to its long axis.
The concept of mass density as a connection between the microscopic and macroscopic never ceases to amaze me! The fact that we can calculate, even if only roughly, the “size” of an atom by treating it as a classical particle and combine that result with an estimate of Young’s modulus found by “piling weights on the end of a wire” as Bruce Sherwood loves to call it to get an effective interatomic spring stiffness that can be used to predict and model some real physics is astounding! I think that’s probably the biggest “thing” in chapter 4.
Analytical solutions to simple harmonic motion are introduced. In this context, students see some of the basics of solving differential equations, which they will learn formally in their ODE course next semester. I think that sometimes we should introduce formal solutions in intro physics, but I waiver on that quite a bit. Numerical solutions always work, so maybe we should introduce those computationally and leave it at that. I don’t know.
We did a more or less traditional spring lab in which students measured stretches and calculated spring constants and experimented with springs in parallel and series. We also did a formal Young’s modulus lab in which students measured Y for two different wires. We only have one apparatus so I did a run-through as a demo and then the students (there are only eight in the class) did their two experimental trials the next day.
In all, this is a very meaty chapter with lots of good physics, computation, and stuff to talk and think about.
Tension is definitely an interesting topic! I remember being unsure (without being able to being able to put my finger on it) about tension when I took intro physics. It didn’t seem to act like a force—I think I wanted to add up all of the tensions at all points along a string, which would give an infinite answer. Only much later did I realize that you really do need to think of it as a tensor.
I fantasize about introducing the idea of the “river of momentum” (or even “river of 4-momentum”)—terminology courtesy of Misner, Thorne, and Wheeler—in this elementary context. It’s really interesting to look at an elementary mechanics problem, such as a weight dangling from a string, and visualize the constant flow of (4-)momentum that’s happening even in a static situation. I doubt that would go over well, since I would think it would be hard to convince a beginner of that idea.
I love this! I think Andy Rundquist has mentioned something about the “flow” of momentum in classical physics. I’ll check with him to see if I’m remembering correctly. In case you haven’t seen it, Thorne and Blandford’s forthcoming text (Modern Classical Physics, due out next year) is really like a “MTW lite” and introduces many of the same ideas from that book. I don’t recall seeing the “river” analogy. I’ll check both sources now though!
For me the notion of momentum flowing is all about how all interactions are really just momentum swaps. Our interaction with the earth is a constant momentum swap with mg as the flow rate.
Indeed, in the Newtonian setup. Note though that in GR there’s no force of gravity, so when you think about in that framework, gravity doesn’t do any momentum swapping. So there, if you have a mass hanging at “rest” from a spring in the lab, it’s really accelerating constantly upward away from the geodesic that it “wants” to take. So there the momentum flow is understood as the same one you would get if the mass and spring were out in deep space and you were pulling on it to give it constant acceleration, by the equivalence principle.
The idea of momentum flows and swaps is useful in both settings (though of course less intuitive—but more crucial!) in GR.
For example, take a column (something that takes compression), aligned along the x-direction. Attach two planks to the ends, sticking out perpendicularly in the +y-direction, and attach a spring under tension to the free ends of the planks. You get a rectangular shape, with one side under compression, the opposite side under tension, and the other two sides under a shear. Now consider the +x-component of momentum: this circulates around the rectangle, going in the +x direction in the compressed column (stress tensor element positive) and going in the -x direction in the spring under tension (stress tensor element negative).