Pseudowork and real work

I have a story to tell about pseudowork, the integral of a force along the displacement of the center of mass, which is different from the true work done by a force on a system, which must be calculated as the integral of the force along the displacement of the point of application of that force. If the system deforms or rotates, the work done by a force may be different from the pseudowork done by that force. For example, stretch a spring by pulling to the left on the left end and to the right on the right end. The center of mass of the spring does not move, so the pseudowork done by each force is zero, whereas the real work done by each force is positive. Because the total pseudowork is zero (which can also be thought of as the integral of the net force through the displacement of the center of mass), the translational kinetic energy of the spring does not change (more generally, the work-energy theorem for a point particle shows that the change in translational kinetic energy is equal to the total pseudowork). Because the total work done on the spring is positive, the internal energy of the spring increases.

In 1971 in the context of the big PLATO computer-based education project at UIUC I had several physics grad students working with me to develop a PLATO-based mechanics course. They and I each picked an important/difficult mechanics topic and started writing tutorials on the topics. Lynell Cannell was assigned energy and I became concerned that she was the only member of the group not making progress. I was about to have a talk with her about this when she came to me to say that she was hung up on a simple case.

She said, “Suppose you push a block across the floor at constant speed. The net force (your push and the opposing friction force) is zero, so choosing the block as the system no work is done, yet the block’s temperature rises, so the internal energy is increasing. I’m very confused.” I said, “Oh, I can explain this. You just, uh, well, you see, uh…..I have no idea.”

We went and talked to Jim Smith, an older physicist very interested in education, very smart, and a good mentor for my then-young self. Jim had thought it through and explained the facts of life to us, with a micro/meso model of the deformations that occur at the contact points on the underside of the block, such that the work done on the block is different from the pseudowork done on the block.

I got very interested in the matter and fleshed out Jim’s insight in more and more detail, but when I showed my analyses to physics colleagues they weren’t having any. Finally I decided to send my paper to AJP (the American Journal of Physics), and the reviewers rejected it. One reviewer said, “Sherwood applies Newton’s 2nd law to a car, which is illegitimate, because a car isn’t a point particle.” I sent it to The Physics Teacher, and the editor replied that he wouldn’t even send it out to reviewers because the physics was so obviously completely wrong.

I asked AJP for an editorial review, and the reluctant response by an associate editor was, “Well, I guess Sherwood is right….but that’s not how we teach this subject!” Finally, in 1983, AJP did reluctantly print the paper “Pseudowork and real work” which you’ll find here. This was the first half of the original paper. The second half, applying the theory to the case of friction, “Work and heat transfer in the presence of sliding friction” (also available here), was published jointly in 1984 with William Bernard, because AJP had received a related paper from Bernard and put the two of us in contact with each other.

At that time there had been some short articles in AJP on the topic, but there hadn’t been a longer article on all the aspects. In fact, given physicist resistance to the truth, Bernard was engaged in a war of attrition, sending short articles to AJP on various aspects of the problem, trying to build up to the full story. Nor had there been any article on friction.

The grand old man of Physics Education Research (PER), Arnold Arons, was a fan of my first paper and summarized it in his books on how to teach intro physics (A Guide to Introductory Physics Teaching, 1990, and Teaching Introductory Physics, 1997). Even he however was quite skittish about the friction analysis, in large part because he was strenuously opposed to mentioning atoms in the intro physics course, for philosophical reasons. Arons tried to explain the pseudowork issue to his friend Cliff Schwartz, the editor of The Physics Teacher, but he never succeeded; Schwartz remained forever convinced that this was all massively wrong.

After the papers were published, in 1983 I wrote to Halliday and Resnick about the matter, emphasizing that their textbook was certainly not alone in mishandling the energetics of deformable systems. I got a nice letter back from Halliday which said about their book, “Let me say at once that we are well aware of its serious flaws, along precisely the lines that you describe. We have tried several times to patch things up in successive printings but the matter runs too deep for anything but a total rewrite. We have, in fact, such a rewrite at hand, awaiting a possible next edition.” I have the impression that this major rewrite never occurred, as I don’t know of an edition that fully addresses the issues. It is amusing that Ruth Chabay and I were given the 2014 AAPT Halliday and Resnick Award for Excellence in Undergraduate Teaching (here is a video of our talk on the occasion, dealing with thinking iteratively).

Most textbooks make major errors in the energetics of deformable systems, or simply ignore the issues. A few textbooks have a brief section on related matters, but as Halliday discerned, handling the physics correctly requires significant revisions throughout introductory mechanics. Since the early 1980s there have been many good articles about these matters in AJP, with little impact on the teaching of introductory physics. In 2008 John Jewett published a solid five-part tutorial on the subject in The Physics Teacher.

In my original articles the analysis is couched in terms of the two different integrals, for work and for pseudowork. We found that even strong Carnegie Mellon students had difficulty distinguishing between these two very similar-looking integrals. So eventually we changed our textbook to emphasize two different systems (point-particle and extended) instead of two different integrals. The distinction between the two systems is more vivid than the subtle distinction between the two integrals.

The point-particle model of a system has the mass of the system that is modeled as an extended system and that moves along the same path as the center of mass of the extended system. The change in kinetic energy of the point-particle model is given by the integral of the net force acting at the location of the point particle, and this is equal to the change in the translational kinetic energy of the extended system. The change in the total energy of the extended system is equal to the sum of the integrals of each force along the path of its point of attachment to the system.

Here is a video of an apparatus that shows the effects. Two pucks are pulled with the same net force, but one is pulled from the center and doesn’t rotate, whereas the other puck has the string wound around the disk, and it rotates. Somewhat surprisingly, the two pucks move together, but in fact the Momentum Principle guarantees that the centers of mass of the two pucks must move in the same way if the same net force is applied. Here is a computer visualization of the situation.

Bruce Sherwood

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7 Responses to Pseudowork and real work

  1. Anthony G. says:

    Hello Professor Sherwood,

    I am a High School Physics teacher with a background in Chemical Engineering. I had been trying to put together the sort of First Law Conservation of Energy concepts we learned in engineering, incorporating internal energy, and what I now see is the “Center of Mass” Energy equation that contains “pseudo-work”. My aim was to come up with an equation that could be used for any situation. I too was hung up on the “object being pushed across the floor problem”. I was also hung up on reconciling dW=PdV for a gas with the center of mass approach. I found your papers from 1982 and 1984. I had been going back and forth as to how work was defined, the displacement of the center-of-mass or the displacement of the point of force application. After reading your papers I now understand the distinction between the CM approach and the FLT approach. I also very much appreciate your derivation of the IFLT. So I wanted to thank you for clearing up a disconnect in my Physics knowledge that has plagued me for years! Now I feel like I understand how to handle a problem like “a spoon being pulled through a jar of honey”.

  2. BruceSherwood says:

    Glad you liked it! It is curious that many introductory physics textbooks still pay little or no attention to these issues, and even continue to handle the energetics of deformable systems incorrectly.

  3. Anthony G. says:

    Hello Professor Sherwood,

    As I mentioned I have read your papers on “pseudo-work” and I they have helped me understand some seeming simple yet problematic physics scenarios. There is a particular simple problem that still bothers me. Imagine a wheel held fast in space by its axis. We take a flat board and roll it across the surface of the wheel at a particular point in space on the wheel without slip. The point of contact between the wheel and the board is always at a particular point in space during this process. We know the wheel’s velocity will change, so there must be work being done on the wheel. However, the contact point is not moving in space, which would imply no work is done on the wheel. Might you have some insight into this problem?



  4. BruceSherwood says:

    Don’t go too soon to the limiting case of a point contact. That is, let the mesoscopically rough surfaces interlock briefly. For concreteness, you could imagine that the wheel and the board have gear-like teeth. Then in a time dt the board does an amount of work on the wheel Fvdt, where F is the force that the board exerts on the wheel and v is the speed of the board. Note that when a car moves at constant speed v an atom in the wheel in (momentary) contact with the ground is at rest, which can be visualized by seeing that the axle moves forward with speed v, and the speed of an atom on the outer edge of the wheel, relative to the (moving) axle is v, so at the bottom of the wheel an atom has speed zero and at the top of the wheel an atom has speed 2v.

  5. Anthony G. says:

    Hello Professor Sheerwood,

    Thank you for your response. I understand and appreciate your point. I was entertaining some thought along the similar lines. I was thinking that maybe we have to allow the wheel to flatten a bit, so that the wheel and board contact over a small finite length (maybe a differentially small length…). Then the contact points could move through space over that length.



  6. BruceSherwood says:

    A better and simpler explanation is this: In the case of the car, the speed of the atom at the bottom of the wheel is zero, and the speed of an atom in the road just under the wheel also has a speed of zero; the two speeds are equal. In contrast, if the axle is stationary and the board moves, the speed of the atom at the bottom of the wheel is the same as the speed of the board (if there is no slipping); the two speeds are equal.

  7. Anthony G. says:

    Hello Professor,

    I understand. When I said the “wheel” I was thinking of a wheel rotating around a stationary axis with a board sliding underneath, not a wheel rolling along a road.



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