Unifying types of interactions when teaching forces

Update: The code below is now in my GitHub repository.

This post is about a quick writeup on interactions that I made this weekend for students. Since modern physics so often loses to more concrete (and older) material when teaching physics, I wanted to give my students a flavor of the kind of particle physics done in the last 200 years. I thought that I’d let them read this and decide which interactions we should take into account when drawing system schema and force diagrams. I’m not completely satisfied with my treatment of the normal force and the Pauli Exclusion Principle within the article. My language is also a little overblown, but believe it or not, I did tone it down and cut out adjective chunks like “low-energy effective” and replace with “everyday” when describing interactions. Anyway, PDF and XeLaTeX source are below.

While I’m at it, I should also describe the way I teach system schemata. Because I taught energy first, I didn’t want to distract with forces. Also, interactions are not exactly forces; forces describe the effect of an interaction on a particular object. Hence, I felt content to label interactions with simple codes for the type of interaction. Last year I used \vec{\boldsymbol F} with a superscript for the type of interaction (with the “dealer→feeler” as a subscript). I like it for uniformity and the fact that it emphasizes that each type of force is indeed a force, but unfortunately, few college-level textbooks do this. When the emphasis is solving problems, a more succinct notation wins, so I thought that this year I’d give students a choice of force symbols. I hope to learn more about how students approach physics with their choices and to notice which approach is more effective at making connections between interactions, interaction energies, and forces. Examples (In the “Force Symbol” column, the first symbol is my version from last year and the second is what I typically see in problem-solving-focused mechanics textbooks):

Interactions, Interaction energies, and Forces
Interaction Interaction symbol Interaction energy symbol Force symbol
gravitational g U^{\textrm g} \vec{\boldsymbol F}^{\textrm g} or \vec{\boldsymbol W}
normal n N/A \vec{\boldsymbol F}^{\textrm n} or \vec{\boldsymbol N}
frictional f N/A \vec{\boldsymbol F}^{\textrm f} or \vec{\boldsymbol f}
tensional t N/A \vec{\boldsymbol F}^{\textrm t} or \vec{\boldsymbol T}
elastic el U^{\textrm el} \vec{\boldsymbol F}^{\textrm el} or ?
electric e U^{\textrm e} \vec{\boldsymbol F}^{\textrm e} or ?
magnetic m U^{\textrm m} \vec{\boldsymbol F}^{\textrm m} or ?

From where do our everyday interactions come? (interactions.pdf)

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\title{From where do our everyday interactions come?}
\author{Brian Vancil}
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\caption{Known and guessed-at interactions in nature}
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We know of only a few fundamental interactions in nature, and one human project in physics has been to explain everything that we observe in terms of simpler interactions.  If you look at Figure~\ref{intdiag}, as you move down the diagram, you will find interactions that describe more and more specific circumstances.  Each is useful, but on their own they explain very little.  In the history of science, humans have worked upward in Figure~\ref{intdiag}, taking scientific models that appeared to be different but were really different aspects of the same thing and unifying them into a single simpler theory that explained more.
For instance, the electric interaction is responsible for the following everyday interactions:
\begin{description}
\item[normal interaction]The repelling squishiness of matter as two objects push against each other is due to two sources: (1) electric interaction involving negatively charged electron clouds around positively charged nuclei and (2) the Pauli exclusion principle---not really an interaction!---between the electrons.  Something as simple as sitting in a chair involves a normal interaction.  So does air pressure and air resistance.
\item[tensional interaction]The electric bonding of negatively charged electron clouds to positively charged nuclei creates an attractive intermolecular interaction so that when a substance is stretched by a bit, it tends to pull back together.  Parts of a rope pull on other nearby parts of a rope through a tensional interaction.
\item[elastic interaction]An extreme form of the tensional interaction, in which matter changes its shape by a lot.  Springs and elastic are good examples.
\item[frictional interaction]Friction is not completely understood, but it involves the grinding together and shearing of irregular surfaces along each other.  Hydrocarbon molecules also play a role between the surfaces.  Something as simple as walking across a floor requires friction.
\item[electric interaction]Electrically charged objects attract or repel each other depending on whether their charges are opposite or the same, respectively.  If you have ever experienced static electricity, you know this well.
\end{description}
As an example of how the human project of searching for simpler explanations progressed, the \textbf{magnetic interaction} is familiar from magnets, but it was discovered in the 1800s that electric and magnetic interactions are really both part of a single \textbf{electromagnetic interaction}, which is responsible for the static electric interactions already mentioned, electricity, magnetism, and even light (really the entire spectrum of electromagnetic radiation).  In the second half of the 1900s, physicists learned to describe both electromagnetic interaction and the \textbf{weak interaction} (responsible for many forms of radioactive decay) by a single theory of the \textbf{electroweak interaction}.

Attempts have been made to unify the \textbf{strong interaction} (responsible for both the nuclear interaction that holds protons and neutrons together in the nucleus of an atom and for the interaction that holds quarks together within protons and neutrons) with the electroweak interaction into a single interaction.  These theories go by the name of Grand Unified Theories, but all of them predict types of matter that we haven't seen yet.

Also in the 1900s, physicists worked to unify the gravitational interaction with the other types of interactions to create a so-called Theory of Everything.  Most of those attempts failed, but we humans have learned a lot from the failures, and we are still at it.  In addition to the human project of unifying interactions, there is also an opposite human project of using these interactions to describe more and more complex systems of particles, everything from neutron stars to superconductors to everyday materials.  Will there ever be an end to the human drive to organize and explain the universe?
\end{document}  

Teaching about temperature

It’s been a while since I took thermodynamics and statistical mechanics, so I’m using this post partly as a way to refresh my memory—please tell me I’m wrong! It’s also a way for me to start a conversation on definitions of temperature, some of which are abysmally bad. Because science has a misconception problem, we ought to think of a good way to progress toward more sophisticated notions of temperature throughout one’s educational timeline and to make clear when we break from the naïve notions and why.

One has only to google “definition of temperature” to realize the problem: Once one skips the dictionary definitions about temperature being a scale of hotness (which, as I will argue, is arguably better than what follows), one gets to definitions that say something about how temperature is a measure of the average translational kinetic energy of the atoms in a substance.  (I won’t link to these because I don’t want to increase their Google PageRank.)

Bad definition 1: Temperature is a measure of the average translational kinetic energy of the atoms in a system.

Compare to NGSS draft 2: DCI PS3.A: “Temperature is a measure of the average kinetic energy of particles of matter. The relationship between temperature and the total energy of a system depends on the types, states, and amounts of matter present.”
It’s a nice picture.  It’s just not hard to break.  For instance:

  • What would happen to the temperature if you threw the system?

Bad definition 2: Temperature is a measure of the average translational kinetic energy of the atoms in the rest frame of a system.

Even if you try to correct for this naïvely by specifying that the kinetic energy be measured in the center of mass frame:

  • What would happen to the temperature if you split the system in two pieces and threw them in opposite directions?

Or even worse:

  • What would happen to the temperature if you spun the system?

Bad definition 3: Temperature is a “measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules”.

HyperPhysics has the best bad definition of temperature.  However, I wouldn’t go as far as they do and call it an “operational definition”.  What is “kinetic energy associated with disordered…motion”?  What counts as disordered motion?  Is a sound wave (phonon) disordered?  It’s what we might think of as a vibration (which many texts cite when they talk about temperature).  How do we measure disorder for the purpose of calculating the “associated” kinetic energy?  Why doesn’t rotational energy count?  What do we mean by “measure” (another problem with calling it an operational definition)—even if I account for all the “disordered” kinetic energy, what value do I put down for temperature?

What’s the simplest system we could envision that breaks this definition? Note that to break the measure part, there must be a non-monotonic relationship between temperature and kinetic energy.  Candidates include:

  • photon gas (Many thermal systems include EM radiation.  How do we measure the temperature of light when it isn’t made of atoms?)
  • low-temp quantum solids???
  • diatomic gases???

Good definition 1: Temperature is the rate of change of energy with entropy (TODO: specifics on how this is defined) (with volume and the number of particles fixed)

HyperPhysics also gets credit for this good definition.  However, the authors don’t give any examples where the behavior of temperature differs drastically from the simplistic T\propto U idea.
The standard definition in thermodynamics is:

T=\frac{Q_\textrm{\scriptsize reversible}}{\Delta S}

 
However, the standard definition of temperature in statistical mechanics is:

T=\left(\frac{\partial S}{\partial U}\right)_{N,V}^{-1}

Does this always work?  What does it mean? Without teaching entropy well, this might be hopeless. However, I recently read an interesting paper on a simple model to introduce the need for entropy as a thermodynamic variable:

  • Abreu, Rodrigo de, and Vasco Guerra. “Introducing Thermodynamics Through Energy and Entropy.” American Journal of Physics 80.7 (2012) : 627–637. 4 Jan. 2013. <http://link.aip.org/link/?AJP/80/627/1>.

How could we make the teaching of temperature better?

I envision a tripartite system of teleological, conceptual, and operational definitions, where we scale up the complexity of the conceptual definitions (i.e. develop new models for temperature) as a student progresses through the system. This has to be explicit, or students won’t understand either the nature of science or why their old ideas are not quite right.

Teleological definition: Temperature is a scale that tells us the direction of heating when two systems (both in equilibrium) come into contact with each other.

Even this isn’t obvious and needs some justification. If we have three systems in equilibrium, A, B, and C, why couldn’t A heat B, B heat C, and C heat A if they were brought into pairwise contact?
It does get the point across why we care about temperature, which helps to ground our other definitions and provide continuity in the notion of temperature.

Operational definition 1 (Cf. NGSS K-PS1-b): Temperature can be hotter or colder (than a person). It’s hotter if it feels like it’s a sidewalk on a hot day. It’s colder if it feels like ice.

Operational definition 2 (later elementary): Temperature is what a thermometer measures.

This should be connected to the idea of heating and reference temperatures (from special systems with understood behavior). It can later include the idea of different scales. The idea of absolute zero should come from an extrapolation of gas law data from a student experiment.

Conceptual definition 1: Temperature is a measure of how frenetic and disordered the motion of atoms/molecules in a substance is.

Note that we’re pretty close to a bad definition but that it’s always augmented by our teleological and operational definitions. This is not worth getting to until students understand that matter is made of molecules. Students should see simulations of matter at different temperatures to get a feel for what we’re talking about. It’s simplistic and qualitative.

Conceptual definition 2: Temperature is a measure of the distribution of energies in a substance given by T=\left(\frac{\partial S}{\partial U}\right)_{N,V}^{-1}.

That is, roughly, it’s the ratio of added energy to the change in entropy that results. We’re moving past our bad definitions.

Conceptual definition 3: We should define a new “temperature” as an inverse energy scale \tau=-\beta=-\frac{1}{k_B T}=-\frac{1}{k_B}\frac{\partial S}{\partial E}

Regular absolute temperature fails our teleological definition at negative absolute temperatures. This new definition gets the direction of thermal transfer right when two systems come into contact. See, for instance:

  • Braun, S. et al. “Negative Absolute Temperature for Motional Degrees of Freedom.” Science 339.6115 (2013) : 52–55. <http://arxiv.org/abs/1211.0545>

What happens when physics class is fun?

I’ll admit that I don’t have any evidence to back up my belief that a classroom can be fun-filled and learning-filled, especially if the two are temporally associated. We are learning about conservation of momentum and learning to represent it in the form of an IF (Initial-Final) Momentum Bar Chart. When one of my students started to rap one of the problems in Kelly O’Shea‘s Momentum Transfer Model (MTM) packet, I couldn’t get the phrase/rhythm out of my head, so I’ve completed it for her. What was

A 2 kg melon is balanced on your bald uncle’s head. His son, Throckmorton, shoots a 50 g arrow at it with a speed of 30 m/sec. The arrow passes through the melon and emerges with a speed of 18 m/sec. Find the speed of the melon as it flies off the man’s head.

became

There’s a 2-kg (read: Kay-Gee) melon on your bald uncle’s head.
Cousin Throckmorton’s crossbow looks to shoot him dead,
but his arrows are true and are 50 g (read: Gees)
and will pierce the right melon if his daddy don’t sneeze.
When he pulls the trigger, all the physics lovers scream,
and the arrow’s speed drops from 30 to 18
meters per second!

watermelon balanced on a bald man's head

from the now defunct www.thingsonyourhead.com