Work and energy for an accelerating car

Someone wrote to me with questions about work vs pseudowork (real system vs point-particle system). At one point he asked me to analyze a subsystem of an accelerating car, the car minus the wheels, and I learned something in the process that’s worth sharing. It’s yet another example of the need to be careful about the fact that in calculating work it’s critically important to pay close attention to the individual displacements of the points of applications of each force, not simply use the displacement of the center of mass.

A car moves with constant acceleration in the direction, with no tire slippage. Simple model: there’s a forward static friction force applied by the road to the bottom of each tire, where the instantaneous speed is zero, and therefore this force does no work on the bottom of the wheel. Total mass of car is , mass of each wheel is , each wheel is like a bicycle wheel, with all mass at the rim of radius so the moment of inertia of the wheel is . From the Momentum Principle we have . No work is done on the car, so , where , where is the translational kinetic energy of the car and includes kinetic energy of wheels, pistons, camshaft, chemical energy of gasoline, thermal energy of engine block, etc. For a point-particle system subjected to the same forces, , where is the distance the car moves, and this is also the translational kinetic energy of the actual car.

Now consider a system I’ll call “sys” consisting of the car minus the wheels, with mass . In this simple model, suppose the engine pushes on the top of each wheel. When I was a kid there were little electric motors you could mount on the top of the front wheel of a bike. The motor turned a small wheel in contact with the big wheel, to drive the big wheel. Or you could imagine an arm or arms continually pushing the top of the wheel, then being lifted and retracted. What are the energetics of “sys”?

Start by analyzing the wheel alone, which is acted upon by the force of the road, the force of the car axle on the inside of the hub of the wheel, and the force of the engine along the top of the wheel. By determining these forces we get by reciprocity of electric forces the forces the wheel exerts on “sys”. In the direction the engine exerts a force and the axle exerts a force (the engine force pushes the hub of the wheel against the car’s axle). Remember that , so .

Momentum Principle:

Angular Momentum Principle ( direction): , or , and

From the Momentum Principle we have , so , which is twice as large as (which is a bit surprising)

Summary for wheel: The wheel is acted upon by the road, in the direction, the axle, in the direction, and the engine, in the direction. The engine force is only slightly larger than , by the amount which is small compared to , since the wheel mass is very small compared to the large mass of the car. It’s interesting that the force of the axle on the wheel is so large, twice as big as . Of course if the mass of the wheel is distributed differently the values of and will be different, related to not being simply .

Now we can look at “sys”, the system consisting of the car minus the wheels, with mass . The forces in the direction acting on “sys” are the forces due to the hubs of the wheels, , and the forces due to the tops of the wheels on the engine, .

Momentum Principle: , which checks.

What about the Energy Principle for “sys”? Here’s the element in the analysis that I found particularly interesting. The forces of the hubs of the wheels are applied to the axle, which in a small time moves through a small displacement , where is the instantaneous speed of the car. But the forces of the tops of the wheels on the engine act through a distance ! The instantaneous speed of the top of the wheel is (speed of hub is , speed of bottom of wheel is zero).

Energy Principle:

As usual, real work must be calculated by integrating EACH force through ITS point of application, THEN you add up all the contributions to the total net work. Here, as in all cases of deformation or rotation, there is the possibility that different forces act through different distances. The case here is particular striking, because the force at the top of the wheel acts through twice the distance of the force at the hub.

It may at first glance seem odd that the work done by the wheels on “sys” is negative. However, note that “sys” increases the wheels’ translational and rotational motions, thereby increasing the energy of the wheels, so there must be a small decrease in the energy of “sys” associated with giving energy to the wheels. There is of course a much larger decrease in chemical energy associated with accelerating the mass of the “sys” system.

Compare with car: , so , which is negative, with

The energy of one wheel is

The rate of energy change is , and for 4 wheels we have . In a time the amount of work done on the wheels is , which is indeed what we found above.

GlowScript: 3D animations in a browser

You are invited to try out GlowScript (“Graphics Library on Web”), an easy to use 3D programming environment inspired by VPython, but which runs in a browser window. The programming language is JavaScript, and it drives the new WebGL 3D graphics library to display in a “canvas” element on a web page. GlowScript has been developed by David Scherer (the originator of VPython) and me.

* At glowscript.org, click Help in the upper right corner of the window and see up-to-date information on which browsers and hardware currently support GlowScript, which requires WebGL. There are also links there for learning JavaScript.

* GlowScript uses the WebGL 3D graphics library that is included in current versions of major web browsers. You must have a modern graphics card with Graphics Processing Units (GPUs). Browser details follow.

Chrome: Runs GlowScript without modification.

Firefox: To activate WebGL in Firefox, go to the fake URL about:config, search for webgl, and set webgl.force-enabled=true (this may not be necessary on Ubuntu Linux). To find out what version of Firefox is installed, use the fake URL about:.

Safari: To activate WebGL in Safari 5.1 (included with OSX Lion), go to the Advanced section of Safari preferences and check “Show Develop menu in menu bar”, then on the Develop menu check “Enable WebGL”. Prior to Safari 5.1, to get WebGL required downloading and installing WebKit from Apple (nightly.webkit.org).

Internet Explorer: Although the basic version of Internet Explorer does not include WebGL, there is a WebGL plug-in available at code.google.com/chrome/chromeframe which lets you run GlowScript.

Tablets and smart phones: Most tablets and smart phones do not yet support WebGL, though this is likely to change in the future.

* At glowscript.org, run the example programs.

* Log in (you’ll be asked for a Google login, such as a gmail account), and try writing some programs yourself.

* Click “Run this program” or press Ctrl-1 to execute your program in the same window, then click “Edit this program” to return to editing.

* Alternatively, while editing press Ctrl-2 to execute your program in a separate window, so that you can view the execution and the program code simultaneously. After making edits, press Ctrl-2 in the editor to run the new program.

* While running a program, click Screenshot to capture a thumbnail image for your program page.

* In the editor, click Share this program to learn how to let other people run your program.

You will see in the Help that some parts of the language are particularly likely to change, so be prepared for that. On the other hand, there is a version system in place that will allow old programs to continue running in the future. The first line of a program you write is automatically created to be “GlowScript X.Y” (where X.Y is the current version number). When a new version comes out, the software for running the older version is retained for use whenever a program with an old version number is encountered. For example, programs written for either GlowScript 0.3 or 0.4 or 0.5 can all run today (the version system was created after GlowScript 0.2).

There is now a user forum connected to glowscript.org, where you can describe your experiences or ask for assistance.

I am committed to maintaining and hopefully extending VPython, as it is obviously much more mature than GlowScript and benefits from the existence of lots of useful modules, especially for scientific programming.

While the graphics in GlowScript today are pretty basic, it is already clear that WebGL with its emphasis on the use of Graphics Processing Units (GPUs) available on modern graphics cards, will make it possible for GlowScript to do very high-quality graphics. VPython was created at a time when that was not an option, but it is possible that the graphics developments for GlowScript will provide a foundation for improving the graphics in VPython.

For users of VPython, note that on the first page of the GlowScript Help there is a document summarizing the main differences between VPython and GlowScript, much of which is just differences between Python and JavaScript. Also available there is a Python program for converting VPython programs to GlowScript programs. It doesn’t do a perfect job, but it does deal with most of the basic changes that are needed and should save quite a bit of time in the conversion.

If you are new to programming, or just new to JavaScript, you’ll find the JavaScript tutorials at www.codecademy.com very helpful.

There are also helpful interactive tutorials on JavaScript available at www.w3schools.com/js. However, these tutorials assume quite a lot of prior knowledge about html and related web design issues.

An excellent textbook is JavaScript: The Definitive Guide by David Flanagan (6th edition, 2011). This book assumes some prior programming experience but has the advantage of being quite complete, including extensive information on how to use JavaScript and jQuery to make dynamic web pages.

JavaScript reference materials are found at developer.mozilla.org/en/JavaScript.

The speed of light in a material

This is in response to the question, “So the speed of light differs depending on medium, right? Is this also true for neutrinos?” This question from a friend was prompted by the recent measurements at CERN suggesting that neutrinos might travel very slightly faster than light.

Actually, there is an important sense in which one can (and should) say that the speed of light does NOT depend on the medium! In the Articles and talks section of matterandinteractions.org, see my article “Refraction and the speed of light”. If you accelerate charges, they radiate light. Light consists of traveling waves of electric and magnetic fields, and on that same web page you can see a video about electromagnetic waves titled “Electric Fields, Cell Towers, and Wi-Fi”, a presentation I made to Santa Fe city government staff.

There is an extremely important though underrated property of charges and fields called the “superposition principle”: The value of the electric or magnetic field at a location in space is the vector sum of all the fields contributed by all the charges in the Universe, AND THE CONTRIBUTION OF ANY PARTICULAR CHARGE IS UNAFFECTED BY THE PRESENCE OF OTHER CHARGES.

It is the capitalized portion of the principle that despite its innocent-sounding content leads to quite counterintuitive consequences. For example, you’ve probably heard that a metal container shields out electric fields made by charges outside the container. False! There is no such thing as “shielding”. By the well validated superposition principle, the field at any location inside the metal container includes the field contributed by external charges. However, it LOOKS as though the metal prevents the field from getting in, because the external charges “polarize” the metal by shifting the mobile electrons in the metal, and the polarized metal contributes an additional electric field inside the container that is equal in magnitude but opposite in direction to the field contributed by the external charges. The effect is indeed as though the metal “shielded” the interior, but the actual mechanism has nothing to do with “shielding”, and the field due to the external charges is most definitely present inside the container.

Consider a cubical box with metal walls, and there’s a positive charge to the right of the box. That positive charge makes an electric field through the region, and that field causes (negatively charged) mobile electrons in the metal to move to the right, toward the external positive charge. That makes the right side of the box have an excess negative charge, and it leaves the left side with a deficiency of electrons, hence a positive charge.

By convention, the direction of electric field is said to be in the direction that a positive charge would be pushed, so the electric field inside the box due to the external positive charge is to the left. Note that the “polarization” charges, negative on the right side of the box and positive on the left side of the box, contribute a field inside the box to the right. The 1/r squared character of the electric field of point charges leads to the surprising result that the field inside the box contributed by the polarization charges is exactly equal in magnitude and opposite in direction to the field contributed by the external charge, so the vector sum of the field contributions of all the charges is in fact zero inside the box, as though the metal “shielded” the interior.

Back to the case of light, which is produced by accelerated charges. If you accelerate charges for a short time, they radiate a short pulse of light. Let’s accelerate some charges somewhere off to the left, for a short time. Light (electric and magnetic fields) propagates in all directions, but we’re interested in the light traveling to the right, toward a detector (which could be a camera) some known distance from the “source” (the accelerated charges). We measure the time from when we briefly accelerated the charges to when we detect the light a known distance away. Divide distance by time and get the speed of light in air, 3e8 m/s.

Now let’s repeat the experiment, except that there’s a thick slab of glass between the source and the detector. You’ve surely heard that “light travels much slower in glass than in air”, so you would expect the light to take significantly longer to reach the detector now that the glass is in place. But that’s not what happens! You find the same time interval between the emission and the first light reaching the detector, and you determine the same 3e8 m/s speed as before! And you must, because the field at any location in space is the vector sum of the field contributions of all the charges in the Universe, unaffected by the presence of other charges (in this case, the electrons and protons in the glass). The fields radiated by the accelerated charges are unaffected and reach the detector in the same amount of time as before.

However, there is an effect. As the electric field passes through the glass, it accelerates the electrons and protons (it accelerates the electrons much more than the protons, due to their very low mass). These accelerated electrons radiate electromagnetic radiation, like any accelerated charges. The traveling fields of this re-radiation also come to our detector, so that the shape of the pulse we receive is altered from what we saw without the glass, because there are now additional field contributions that were not present in the absence of the electron-containing glass. The first bit of light shows up on time, but then the situation becomes quite complicated.

An important special case is that where the source charges off to the left are accelerated not for a short time, but continuously, sinusoidally up and down (which involves accelerations as the charges move faster and slower and turn around). If you turn on this sinusoidal radiation abruptly, of course you’ll first see some light at the detector on time, with or without the glass being present. But let the sinusoidal acceleration of those source charges continue for a long long time. It can be shown that the vector sum of this radiation and the re-radiation from electrons accelerated in the glass leads to a detection of sinusoidal radiation, and that sinusoidal radiation has a phase which is shifted. That is, the peaks come at a different time than they did without the glass. In fact, in the “steady state”, the peaks come later than they used to, and the lateness is proportional to how thick the glass is. It is a useful shorthand to say that the “light travels more slowly in the glass”, as that description is consistent with the phase delay of peaks in the sinusoid, in the steady state, even though the speed of light in the glass is the usual 3e8 m/s. (The initial transient is messy, and not a simple sinusoid.)

Richard Feynman in the famous Feynman Lectures on Physics discusses this quantitatively in Chapter 31 on “The Origin of the Refractive Index”. The “refractive index” is usually denoted by n, and it is common practice to say that “the speed of light in a medium with refractive index n is 3e8/n m/s”. But in fact the speed of light is a universal quantity. Although it is very often convenient to pretend that the speed of light is slower in glass, that’s just a calculational convenience — it’s a misleading description of what’s really going on. In fact, the refractive index and “speed of light” in glass is different for different frequencies of the sinusoidal radiation, because different frequencies of electric field affect the motion of the electrons differently in the glass.

The interaction of the electric field of the light with the matter (glass or whatever) can be (for nonobvious reasons) well modeled by the electric field exerting a force on an outer electron in an atom in an insulator such as glass as though the electron were bound to the atom by a spring-like force, with damping. The details of the spring stiffness and damping depend on the material and on the frequency of the electric field. In some materials this works out in such a way that in the downstream electric field (the sum of the field contributed by the accelerated source charges and the re-radiation by the accelerated electrons in the material) the peaks can actually be earlier than in the absence of the intervening material, in which case it looks as though the speed of transmission is actually faster than 3e8 m/s. But it is of course still the case that the first detection downstream occurs at 3e8 m/s.

Incidentally, when in the steady state light is traveling through glass, the frequency of the light in the glass (how many cycles of the sine function occur per second) is the same as the frequency of the light in the air. The speed with which a crest of the sine wave advances (the phase speed) is the distance between crests (the wavelength) divided by the time for one cycle, which is 1/frequency. Because the phase speed is slower in the glass, the wavelength is shorter in the glass than in the air: the crests are pushed closer together.

As to whether the (apparent) speed of propagation of neutrinos would differ in different materials, I think not. The change in phase speed for light is due to the rather strong interaction of light with matter, leading to re-radiation. Neutrinos have an amazingly small probability of interacting with matter, which is why one can detect them after they’ve traveled hundreds of kilometers through solid rock. So I wouldn’t expect matter to have any effect on the speed of neutrinos.

Good popular physics books

I’ve recently read three excellent popular physics books that I strongly recommend.

Quantum Man: Richard Feynman’s Life in Science (available in Kindle)
Lawrence Krauss (author of The Physics of Star Trek)

None other than Freeman Dyson wrote a laudatory review of this book in the New York Review of Books, in which he emphasized what a wonderful job Krauss has done in describing and explaining Feynman’s physics, in a way that’s probably accessible to any intelligent layman, and certainly to any physicist. I’ve read and enjoyed lots of popular science books, especially those written by scientists, and I’ve read a number of scientific biographies. This is the first time I’ve seen the science and the scientific thinking explained so well in a popular science book. Krauss’ book does not spend a lot of time on Feynman’s personal life, which has been widely written about. He focuses on the physics. I learned a lot. I had not fully realized how broad Feynman’s contributions were.

The Quantum Story: A history in 40 moments (available in Kindle)
Jim Baggott

Many histories of quantum mechanics deal solely with the period 1905-1935, approximately. This one continues to the present day. Almost all of it is accessible to the intelligent reader, except for a chapter where he gets carried away with SU(3) etc. Although much of the story was familiar, I nevertheless learned quite a lot. There’s an almost eerie episode involving Bohr and Rutherford, about which I’d not heard. Bohr had his great idea for explaining the hydrogen atom, based on Rutherford’s discovery of the nucleus. Before publishing, Bohr went to talk with Rutherford about his ideas (Bohr had previously spent some time in Rutherford’s lab). As I understand it, Rutherford liked to portray himself as just a New Zealand country bumpkin, but wow…. Rutherford said, “If the atom is in a multiply excited state, and it can decay to one of several lower-energy states, what about causality? How does it choose?” I was just blown away to read this earliest (and quick) realization that the atomic world is probabilistic.

The Dance of the Photons
Anton Zeilinger

Zeilinger heads a powerful experimental quantum mechanics group in Vienna that has made stunning advances in our understanding of the nature of reality in the context of quantum mechanics. In this book he makes the ideas come alive. The book includes detailed discussions of Bell’s inequality and much else. It seems highly likely that Zeilinger will get the Nobel Prize for the work he and his group have done. A charming feature of the book is that Zeilinger is very generous in giving credit to many others working in this fascinating field. (Incidentally, there is some movement in the physics community to bring contemporary quantum mechanics into the physics major’s curriculum, which in the past has been dominated by stuff from the 1920s.)

History of Matter & Interactions

People often ask how we came to develop the Matter & Interactions curriculum, and particularly why we felt it essential to include computational modeling.

Starting in the late 1980s at Carnegie Mellon University, Ruth Chabay and I worked first on E&M.  We chose E&M as a domain for study partly because most Physics Education Research (PER) work was on mechanics, and E&M was significantly less studied. Because E&M is significantly more abstract than mechanics, we thought it likely that students would benefit from the infusion of a healthy slug of qualitative and conceptual reasoning, and from simple experiments designed to let students observe some of the basic phenomena that E&M tries to explain.  To do this, we developed the E&M experiment kits now distributed by PASCO, and a variety of computer visualizations (Ruth’s “Electric Field Hockey” game came from that period).

We quickly came to see that the traditional sequence of E&M topics lacked coherence and made it difficult for students to develop any conceptual understanding.  A large number of abstract and highly mathematical concepts (charge, force, field, flux, Gauss’s Law) were introduced very rapidly at the beginning of the course.  Before many students had begun to understand the difference between charge and field, all of these concepts were abandoned, and students were required to deal with potential and conventional current, which seemed to them unconnected to any previous concepts.  At the end of a semester, even the best students had trouble remembering anything about the electric field of a point charge!

To address some of the difficulties students were having,  we started writing supplements to a traditional textbook, and by 1995, with the encouragement of our students, we had backed into writing a full textbook on E&M, published by Wiley.

With the publication of our E&M textbook it became very clear that people interested in this treatment felt a strong need for a compatible prequel on mechanics. At that point, it wasn’t clear to us how to proceed, but when we reflected on which elements of our E&M we liked best, we realized that it was those problems and experiments that involved modeling complicated real-world situations: idealization, approximation, refining the model, etc. (at CMU we were working with unusually strong students). An example is problem 16.P.49 on page 658 of the 3rd edition of our textbook (rub a plastic pen, pick up a small piece of aluminum foil).  We were encouraged in this view by David Hestenes and others in the Arizona State modeling community.  Given an emphasis on modeling the real world, Ruth reasoned that computational modeling should be a prominent component of what would be M&I.  In part this was because of the centrality of computer modeling to contemporary physics, and in part because it was the only way to allow beginning students to see how fundamental principles applied to systems other than simple toy systems.  It is easy to refine a computational model and very difficult to refine an analytical model, especially at this level — classical perturbation theory is not accessible to intro physics students.

Sometime in 1996 or early 1997 we met with CMU colleagues and outlined our thoughts on “Modern Mechanics”, integrating macro and micro, mechanics and thermal physics, and including having students write computer programs to model physical systems. One of our colleagues said, “You can’t possibly do what you’re proposing to do, but if you do, I very much want to teach it!”

We were pondering how best to deal with the thermal aspects of modern mechanics when we read the excellent January 1997 AJP article by Tom Moore and Dan Schroeder, on how to use the Einstein model (isolated-atom version of the ball-and-spring model of a solid) to do quantum stat mech in a way that is easily accessible to intro physics students. The key point is that the simple evenly spaced energy levels of the quantized harmonic oscillator, and the lack of spatial contributions to the entropy, make the central issues highly salient and easy to compute.

In the late 1980s at CMU I had created the cT programming language, a descendant of the TUTOR programming language of the PLATO computer-based education system that Ruth and I had helped develop at the University of Illinois at Urbana-Champaign (see videos of the 2010 conference on the history of PLATO). cT can be thought of as being somewhat similar to a good BASIC, with easy 2D graphics built-in, and cT programs ran in windows without modification on PCs, Macs, and Unix workstations. In our first offering of modern mechanics, in the fall of 1997, we taught students a minimal set of cT features and they were able to write computational models of physical systems, with 2D animations and graphs. An example of a computational problem from that time is problem 3.P.80 on p. 136 (Ranger mission to the Moon).

In the second year of offering both mechanics and E&M, the 1998-1999 academic year, we had an extraordinary student, David Scherer, who while in high school had led a team of friends to create a sophisticated 3D PC game that later won a national prize. After taking our course, Scherer said he thought he saw a better scheme than cT, which would be 3D. I said, “Well, what we have is adequate; I don’t think we need something else.” But Ruth, who had already done a lot of 3D work at great effort, using hard-to-use tools, said, “3D? I will certainly use it!” In the spring of 2000 Scherer created VPython, with Ruth and me spending many hours per day with him on design and testing, and it was fully deployed in our course at CMU in fall 2000. We dropped development of cT, as VPython was far superior. (You can read about cT and even download it from the “cT archives” at vpython.org.)

For at least ten years there has been a lot of talk about the need to increase greatly the role of computation in undergraduate STEM education. Despite all this talk, with various national conferences and financial support from NSF, even today computational modeling doesn’t really have a central role in physics departments. In many physics departments a student can major in physics without ever having done any computational modeling. Even at places with a long history of computational physics courses, computation may be relegated to one course, with no impact on other physics courses, and if the local enthusiast is away for a year, the computational course isn’t taught, in contrast to intermediate E&M, say. And yet computation in the discipline of physics is now co-equal with theory and experiment, which means that here is another way in which the undergraduate curriculum is not representative of what the contemporary discipline of physics is all about.

The wheels grind very slowly. But little by little more and more physicists are getting uncomfortable about the absence of computation in the formal education of physicists, so finally people wanting to bring computational modeling into the introductory course are starting to catch up to what we were doing in 1997. And when they look around for curricula that incorporate computational modeling carried out by students in the introductory course, they find Matter & Interactions. It takes a very long time to develop curricula, so it’s a really good thing that Ruth got us started on this so long ago.

Later developments

We moved to NCSU in 2002, where students are bright and willing to work but on average are less well prepared than students at CMU. We were able to make Matter & Interactions accessible to these students by providing more support in the textbook, and this work was incorporated into the 2nd edition (2007), with further improvements in the 3rd edition (2010). We also designed and coded a suite of questions for the WebAssign computer homework system to accompany the textbook, which is an important resource for students in large classes.

With respect to the computational modeling activities, in the last couple of years Ruth and some of her PER graduate students at NCSU carrried out a lot of research and development aimed at improving the computational modeling activities students carry out as part of their work in labs (which also includes experiments and group problem-solving). The new materials have proved to be a significant improvement. They include helpful videos on basic aspects of the VPython 3D programming environment used by students to model physical systems. They are included in the instructor resources available to adopters of the curriculum and are also available to other physics instructors here.