Work and energy in introductory physics

Some of my Physics Education Research Facebook friends have been questioning the value of some forms of the work-energy theorem in teaching introductory physics for life sciences (IPLS). Since over the past few years I have made this theorem an important component of my teaching of mechanics, I thought I'd take the opportunity to describe how I do it.

I've been teaching a – what? – reformed? reinvented? renewed? – IPLS course.[1] (link to NEXUS paper) I struggle with the adjective since our reformation process went beyond usual course reform. We spent a lot of time communicating and discussing (and arguing) content and approach with biologists and chemists, and a lot of time researching student responses and what they brought to the table. This produced a deep philosophical change in the way we designed and that I teach the class.

We learned that it wasn't just that many life science students didn't know the required math (perhaps because they hadn't used it in previous courses) or that they weren't familiar with physics concepts (perhaps because they hadn't taken high school physics, or if they had, hadn't taken it seriously). Rather, there were some serious barriers in the way many students were thinking about the nature of the scientific knowledge they were learning – epistemological barriers, if you will.

Here are some of the issues that we found:

·      Life science students often saw scientific knowledge as bits and pieces of memorized knowledge, failing to build a coherent picture. Although they had learned some heuristics (usually in chemistry), they had little or no experience with the use of deep and powerful principles such as those that drive even introductory physics.

·      Life science students often were deeply skeptical of highly simplified "toy" models. Since life depends in a critical way on complexity, simplification was seen as "playing irrelevant games." Few had any experience with the concept of modeling and few understood the insight that could be derived from studying simplified systems.

·      Even when life science students knew and were comfortable with the required math, almost all saw math as a way to calculate, rather than as a way to think about physical relationships. They were missing not only the skill of estimation and intuitions of scale, they missed being able to read qualitative implications from equations.

Our course is designed to address these epistemological issues as well as the issues of choosing content relevant for life science students (like doing more fluids and including diffusion and random walk). We try to stress coherence, modeling, and the value of using equations to build understanding and insight. The work-energy theorem plays a pivotal role in this structure.

Newton's three laws form the framework for building understanding of mechanics and building models of physical motion. I treat the three laws [2] as the basic structure. Any analysis of a particular motion requires a model – a choice of what we are going to treat as "objects" and how we are going to model their interactions. We use the method of System Schema [3] as a tool for analyzing systems and building models. This is a pre-requisite to drawing free-body diagrams. Interactions are two-way connecting a pair of objects. When the focus is on one object the interaction is realized as a force. By Newton's third law, the forces on either end of an interaction are equal and opposite. This is the tool that focuses student attention on the modeling character of each system considered.

Newton's second law tells how an object responds to the forces it feels. If the forces are not balanced (cancel out), the object accelerates – changes its velocity according to the rule:

Acceleration = (Sum of the forces)/(mass of the object)

This is a vector law, so forces are requires to change either the object's speed or the direction of its motion.

This naturally leads to the question:

If I only care about the change in an object's speed and not its direction,
what does Newton's second law tell me?

It's pretty easy to figure out how to do this, at least in principle. Forces that are in the same direction as the motion tend to speed it up, forces that are in the opposite direction of the motion tend to slow it down, and forces that are perpendicular to the motion tend to change its direction. So to consider only the speed, we multiply Newton's second law by a small displacement along (or against) the direction of motion. After a little simple algebra (no calculus needed), we get the one-body work-kinetic energy theorem:[4]

The change in an object's kinetic energy = work done on it by the sum of all the forces it feels

Or as an equation, this is written

Δ(1/2 mv²) = F^net.Δr

(Bold here indicates a vector.)This law is not particularly useful by itself unless it is used in connection with the System Schema so one can see that it helps to provide clear and simple answers to two rather subtle questions:

·      Why is there such a thing as "potential energy" but no such thing as "potential momentum"? The impulse-momentum theorem and the work-energy theorem look very similar.

·      Why do we sometimes treat potential energy as belonging to a single object (e.g., the gravitational PE, mgh) but sometimes treat it as belonging to a pair of objects (e.g., the PE between two electric charges, kqQ/r).

To answer the first, let's consider the impulse-momentum theorem in contrast to the work-energy theorem written above:

Δ(mv) = F^netΔt

If we consider a system with two objects interacting, since they interact for the same amount of time, and since the forces they exert on each other are equal and opposite (by Newton 3), they change each other's momenta in equal and opposite ways. This means that if we add together the impulse-momentum theorems for the two objects, their momentum changes will cancel. We can then see easily see what the conditions are for momentum conservation to hold. (All the other forces acting on the two objects have to cancel.)

For the work-energy theorem, things are a bit difference. If we consider a small time interval when the two objects are interacting, their time intervals are the same, but the distances that they move do not have to be. Therefore, if we add together the work-energy theorems for two interacting objects, even if there are no other forces acting on them, the work terms for the two objects do not have to cancel. And we can easily see that the extract term is the force dotted into the change in the relative separation of the two objects.

This extra term is why we introduce a potential energy (but not a potential momentum). And it makes clear that the potential energy belongs to the interaction between the two objects.

It also helps us understand when we can treat the PE as belonging to a single object rather than to a pair of objects. Since the momentum changes of the two objects are the same, its easy to find that the KE change of each object is Δ(p²/2m). If one object is much larger than the other, the KE change (and therefore the PE) can be totally assigned to the lighter object. This is why the gravitational PE of an object on the earth's surface can be assigned to the object, and why, in an atom or molecule, the electric PE if the interaction of an electron and a nucleus can be assigned to the electron and we can talk about "the PE of the electron."

These are nice results, if abstract. But I like the work-energy theorem for more reasons. Here are three:

·      When we have a situation where there one of the interacting objects is much larger than the other, there are a lot of nice examples where one can write energy conservation and create equations relating position and velocity. This gives the students good practice with using manipulating symbolic equations and interpreting the result.

·      It can be used to generate other relations and show the relation between other principles that are often treated as independent.

·      If provides the link between the fundamental concepts of force and energy, building another powerful coherence.

The first doesn't need much elaboration, but I was a bit surprised at the second. I knew in principle the power of the work-energy theorem, but it wasn't until I included a substantial discussion of fluids in my class that I realized how cool it was. The work-energy theorem, when applied to a bit of fluid in a pipe easily reduces to:[5]

·      The dependence of pressure on depth and the related Archimedes' principle (by assuming no motion and only gravitational and pressure forces)

·      Bernoulli's principle (by assuming no resistive forces)

·      The Hagen-Poiseuille equation (by assuming resistive force but no gravitational change)

It can also be used to generate new equations, such as a modified H-P equation for fluids flowing vertically in a tree.

Of course, each of these can be derived from forces as well, but tying everything to work-energy and thereby back to forces and Newton two emphasizes the coherence of the whole structure and the reliance on powerful overarching principles.

I've seen this work with my students. They all come in knowing that "energy is the ability to do work," but for most, these are just words. Once we've gone through the work-energy theorem they begin to be able to translate forces into work.

My favorite specific example of this occurred in an interview done with Carol, a student in the class's second term. We had completed a discussion of free energy and done a recitation analyzing the separation of oil and water and the formation of lipid cell membranes. The result is somewhat counterintuitive, since it is actually pretty easy for students who have taken chemistry to see that the interaction (electric attraction) between water and oil molecules is stronger than the interaction between two oil molecules. So why does oil and water separate? Why do lipid membranes form?

In the interview, Carol answered the question by referring to the equation for the Gibb's free energy:

ΔG = ΔH – TΔS

As all biology and chemistry students know, Gibb's free energy is what drives chemical reactions. Spontaneous reactions go to a lower free energy. (Here, H is the enthalpy, which, for this discussion, is equivalent to the internal energy.)

She said (paraphrasing), "The force between the molecules goes into the work which creates potential energy. That goes into the H term since it's energy. Since it's attractive, that tends to make the H lower for the separated oil molecules. But the other term competes. It comes from the losing of the opportunities for the water molecules to interact. In this case, that term wins."

I've seen many students reason like this and it makes me happy. They are using equations to reason with qualitatively and bringing together the idea of forces and energy, building an overall coherence and reasoning from principle.

The single-particle work-energy theorem is easy to think about and reason qualitatively and quantitatively with. This is why I like it and why I make it a central element of my IPLS class.

[1] NEXUS/Physics: An interdisciplinary repurposing of physics for biologists, E. F. Redish, C. Bauer, K. L. Carleton, T. J. Cooke, M. Cooper, C. H. Crouch, B. W. Dreyfus, B. Geller, J. Giannini, J. Svoboda Gouvea, M. W. Klymkowsky, W. Losert, K. Moore, J. Presson, V. Sawtelle, K. V. Thompson, C. Turpen, and R. Zia, Am. J. Phys. 82:5 (2014) 368-377.

[2] I actually introduce a "zeroth law" – that every object responds only to forces it feels and only at the instant it feels them. While this might seem trivial to an experienced physicist, a significant fraction of the errors that introductory students make are a violation of this law.

[3] V. Sawtelle & E. Brewe, System Schema Introduction, NEXUS/Physics; L. Turner, System Schemas, Am. J. Phys. 41:9 (2003) 404.

[4] E. Redish, Kinetic Energy and the Work-Energy Theorem, NEXUS/Physics

[5] E. Redish, The Work-Energy Theorem in Fluids, NEXUS/Physics

Could dark matter be super cold neutrinos?

Probably the greatest physics problems of the current generation are the cosmological questions. Thanks to the development of powerful new telescopes (many of them in space) in the last years of the twentieth century, startling new and unexpected results have pointed the way to new physics. These currently go under the names of "dark matter" and "dark energy", but those aren't real descriptions; rather they are suggestions for what might provide theoretical solutions to experimental anomalies. And, as naming often does, they guide our thinking into explorations of how to come up with new physics.

The problem that "dark matter" is supposed to resolve began in the 1970s with the observations of Vera Rubin. By making a careful analysis of the motion of stars in galaxies, she found an unexpected anomaly. As any first year physics student can tell you, Newton's law of gravitation tells you how planets orbit around the sun. The mass of the sun draws the planets towards it, bending their velocities ever inward in (nearly) circular orbits. The mathematical form of the law produces a connection between the distance the planets are from the sun and the speed (and therefore the period) of the planets.

That connection was known empirically before Newton to Kepler (Kepler'sthird law of planetary motion: the cube of the distance from the sun is proportional to the square of the planet's period). The fact that Newton's laws of motion together with his law of gravity explained that result was considered a convincing proof of Newton's theories.

A galaxy has a structure somewhat like that of a solar system. There is a heavy object in the center – a massive black hole – that is responsible for most of the motion of the stars in the galaxy. Rubin found that the speed of the stars around the center didn't follow Kepler's law. The far out stars were going too fast. This suggested that there was an unseen distributed mass that we didn't know about (or that Newton's law of gravity perhaps failed at long distances; In my opinion this option has not received enough attention, though that's for another post.).

Observations in the past thirty years have increasingly supported the idea that there is some extra matter that we can't see – and a lot of it. More than the matter that we do see. As a result, a growing number of physicists are exploring what might be causing this.

I saw a lovely colloquium yesterday about one such search. Carter Hall, one of my colleagues in the University of Maryland Physics Department, spoke about the LUX experiment. This explores the possibility that there is a weakly interactive massive particle (a "WIMP") that we don't know about – one that doesn't interact with other particles electromagnetically so it doesn't give off or absorb light, and it doesn't interact strongly (with the nuclear force) so it doesn't create pions or other particles that would be easily detectable in one of our accelerators. This would make it very difficult to detect. The experiment was a tour de force, looking for possible interactions of a WIMP with a heavy nucleus – Xenon. (The interaction probability goes up like the square of the nuclear mass so a heavy nucleus is much more likely to show a result.) The experiment was incredibly careful, ruling out all possible known signals. It found no results but was able to rule out many possible theories and a broad swath of the parameter space – eliminating many possible masses and interaction strengths. An excellent experiment.

But as I listened to this beautiful lecture, I wondered whether the whole community exploring this problem hadn't made the mistake of looking under the lamppost for our lost car keys. It's sort of wishful thinking to assume that the solution to our problem might be exactly the kind of particle that would be detectable with the incredibly large, powerful, and expensive tools that we have built – particle accelerators. These are designed to allow us to find new physics – in the paradigm we have been exploring for nearly a century: finding new sub-nuclear particles and determining their interactions in the framework of quantum field theory.

This reflects a discussion my friend Royce Zia and I have been having for five decades. Royce an I met in undergraduate school (at Princeton) and then became fast friends in grad school (at MIT). We spent many hours there (and since) arguing about deep issues in physics. We both started out assuming we wanted to be elementary particle theorists. That, after all, was where the action was. Quarks had just been proposed and there was lots of interest in the nuclear force and how to make sense of all the particles that were being produced in accelerators. But we were both transformed by a class in Many Body Quantum Theory given by Petros Argyres, a condensed matter theorist. In this class we saw many (non-relativistic) examples of emergent phenomena – places where you knew the basic laws and particles, but couldn't easily see important results and structures from those basic laws. It took deep theoretical creativity and insight to find a new way of looking at and rearranging those laws so that the phenomena emerged in a natural way.

There are many such examples. The basic laws and particles of atomic and molecular physics were well known at the time. Atoms and molecules are made up of electrons and nuclei (the structure of the nuclei is irrelevant for this physics – only their charge and mass matters) and they are well described by the non-relativistic Schrödinger equation. But once you had many particles – like in a large atom, or a crystal of a metal – there were far too many equations to do anything useful with. Some insight was needed as to how to rearrange those equations so that there was a much simpler starting point.

Three examples of this are the shell model of the atom (the basis of all of chemistry), plasmon oscillations in a metal (coherent vibrations of all the valence electrons in a metal together), and superconductivity (the vanishing of electrical resistance in metals at very low temperatures). Each of these were well described by little pieces of the known theory arranged in clever and insightful ways – ways that the original equations gave no obvious hint of in their structure.

I was deeply impressed by this insight and decided that this extracting or explaining phenomena from new treatments of known physics was just as important – as just as fundamental – as the discovery of new particles or new physical laws. Royce and I argued this for many hours and finally decided to grant both approaches the title of "fundamental physics" – but we decided they were different enough to separate them. So we called the particle physics approach "fundamental-sub-one" and the many-body physics approach "fundamental-sub-two". (Interestingly, both Royce and I went on to pursue physics careers in the f2 area, he in statistical physics, me in nuclear reaction theory.) In the decades since we had these arguments, physics has made huge progress in f2 physics – from phase transition theory to the understanding and creation of exotic (and commercially important) excitations of many body systems.

So yesterday, I brought my f2 perspective to listening to Carter talk about dark matter and I wondered: He was talking all about f1 type solutions. Interesting and important, but could there also be an f2 type solution? We already know about weakly interacting massive particles: neutrinos. They only interact via gravity and the weak nuclear force, not electromagnetically or strongly. 

Could dark matter simply be a lot of cold neutrinos? They would have to be very cold – travelling at a slow speed – or else they would evaporate. When we make them in nuclear reactions in accelerators they are typically highly relativistic – travelling at essentially the speed of light. The gravity of the galaxy wouldn't be strong enough to hold them.

That leads to a potential problem for this model. Whatever dark matter is, it has to have been made fairly soon after the big bang – when the universe was very dense, very uniform, and very hot -- hot enough to generate lots of particles (mass) from energy. (Why we believe this is too long a story to go into here.) So you would expect that any neutrinos that were made then would be hot – going too fast to become cold dark matter.

But suppose there were some unknown emergent mechanism in that hot dense universe -- a phase transition -- that squeezed out a cold cloud of neutrinos. Neutrinos interact with matter very weakly – and their interaction strength is proportional to their energy so cold neutrinos interact even more weakly than fast neutrinos. If there were a mechanism that spewed out lots of cold neutrinos, I expect they would interact too weakly with the rest of the matter to come to thermal equilibrium. If the equilibration time were, say, a trillion years, they would stay cold and, if their density were right, could serve as our "dark matter".

Most of the experimental dark matter searches wouldn't find these cold neutrinos. Searching for them at this point would have to be a theoretical exploration: Can we find a mechanism in hot baryonic matter that will produce a phase transition that spews out lots of cold neutrinos? I don't know of any such mechanism or where to start, but wouldn't it be fun to consider?