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:
\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.
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?

Problem: Marble and tube

My family recently visited Moon Marble Company, where my daughter picked up a bunch of marbles.  We’ve been having fun concocting ways to entertain ourselves, one of which was to roll the marbles down a cardboard tube.

The tube is at an angle theta with the floor.

A marble is rolled down a cardboard tube. What can we calculate?

To calculate how far a marble will roll along the floor after being dropped down a tube leaves the comfortable world of school physics and enters the real world of rolling friction on cardboard and carpet.  If the tube is close to vertical, the marble will be going quite fast as it exits the tube but won’t go very far horizontally.  If the tube is close to horizontal, the marble will be going quite slowly as it exits the tube.  Our intuition tells us that neither will work as well as somewhere in the middle, so we can search for the angle \theta that maximizes the horizontal distance of the marble along the floor.  This is a case for which the rolling resistances in the tube and on the floor will partly “cancel”, and my daughter’s play tests show that we can get pretty good results just by tracking the proportionality of the velocity to a function of the angle \theta.

What physics models do you have to use to solve this problem?