Pseudowork and real work

I have a story to tell about pseudowork, the integral of the net 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 (see here for the history of the PLATO project). 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.

In 1983 I wrote to Halliday and Resnick about these matters, 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. Bill Bernard’s colleagues told him that he must be wrong about these issues, because “Halliday and Resnick don’t treat energy that way.” They were nonplussed when he showed them the letter from Halliday. 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. The 2019 AJP special issue on energy has an article by Ruth Chabay, Aaron Titus, and me that updates our thinking on these and related issues.

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 Matter & Interactions 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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41 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.



  8. Ying-Lun Kao says:

    Hello Professor Sherwood,

    I really appreciate your paper regarding the applications of first law of thermodynamics. Recently, I published these ideas on our country’s (Taiwan) physics education association facebook discussion group. Some people don’t believe me and they think it’s ridiculous to use FLT on the systems which have no “thermodynamic property”, such as pressure, volume, temperate (!!?) and so on. Since I might be the first one to use FLT on senior high level physics problems in Taiwan, I could understand it must be very weird to them.

    Therefore, I’d like to ask you do me a favor. I really need your help. Could you please allow me to translate your “Work and heat transfer in the presence of sliding friction” into Chinese version and let me publish it on my physics website? Of course, I will attach the original paper and introduce you in the article. If you authorize me to translate and let every Taiwanese see that article. I’ll be very grateful. Well, I know it’s about the intellectual right property so it’ll be fine if you think it’s not appropriate.

    Anyway, thank you so much!!!!!

    (PS. I’m a physics tutor in Taiwan.I major in physics and graduated from National Taiwan University)

  9. Ying-Lun Kao says:

    Hello again, Professor Sherwood,

    I forget ask you a question. In your “Choice of system and the energy equation” talk paper, I found you choose the earth and object as a system to explain where the potential energy is. However, why don’t you simply say the potential energy is in the gravitational field? You see, we can say it is the gravitational field that acts on the object, rather than the earth. In Mary B. Hesse’s book, “Forces and Fields”, we know that Faraday think there must be “something” between earth and the object to transfer the energy. Without gravitational field, we can’t say where the potential energy is. As what you mentioned in that talk —- “The rock has no potential energy”, I believe the earth has no potential energy, either. Therefore, potential energy must be in the gravitational field.

    I really want to know your thought. Thank you in advance. Please forgive my bad English and rude expressing ability. I’m not sure how to politely say those ideas in English. Thank you so much!!

  10. BruceSherwood says:

    I would be delighted for you to translate the paper to Chinese. Thank you for offering to do this. It would be good to append a brief summary of the approach taken now in the textbook “Matter & Interactions” by Ruth Chabay and Bruce Sherwood (Wiley, 4th edition, 2015; The key point is to treat two systems (point-particle and extended) rather than two similar-looking equations (CM and FLT). Also, we always refer to “the Energy Principle” rather than to the first law of thermodynamics. The FLT terminology is fundamentally inappropriate, as there is just one Energy Principle, namely that the change in energy of a system is equal to the inputs from the surroundings minus the outputs to the surroundings. (Alternatively, the Energy Principle states that the energy of the chosen system plus the energy of the surroundings cannot change.) One of the many great confusions for students is that the traditional treatment makes it seem as though there are many different energy equations. There are not. There is only one. In addition there is the work-energy theorem, which is actually a momentum equation, derived by integrating the net force along the path of the center of mass. The Energy Principle cannot be derived; it is a summary of a very wide range of experimental observations, together with experimentally motivated evaluation formulas for the many different kinds of energy and energy transfer.

    Because it is a sophisticated advanced topic, we leave the field concept to the second-semester E&M course. We do treat electric force and electric potential energy in the first semester, and spend a lot of time on the atomic nature of matter, so that the only big new concept in E&M is the field concept, whereas at the start of the traditional approach to E&M the student is immediately overwhelmed with the new concepts of charge and field and flux and Gauss’s law and potential. However, in our first-semester treatment of energy we do point out the following in an optional section:

    Consider two stars of equal mass initially at rest far from each other. Choose both stars as the system, so there are no surroundings, and the sum of the kinetic energies of the two stars and the gravitational potential energy of the two-star system is always zero. Next, take as a system star 1, whose surroundings consists solely of star 2. The gravitational force acting on star 1 does work which increases the kinetic energy of star 1. The Energy Principle says that the surroundings must lose energy, but in fact star 2 gains kinetic energy. We then say that to analyze the situation fully one must introduce the abstract concept of “field,” which is the subject of the second-semester course on E&M, and that there can be energy and momentum in the field. We also say that, nevertheless, the concept of gravitational potential energy of two interacting objects gives the correct results. We even point out that despite not knowing anything about “field energy” we can see from the Energy Principle that the field energy must decrease by an amount 2*dK, where dK is the change in kinetic energy of each of the stars. This is precisely the same quantity as obtained from the classic non-field potential energy approach.

    Although I know almost nothing about general relativity, I’ve been told that there are subtle and complex issues about energy in GR.

  11. BruceSherwood says:

    A minor point: Agreed that neither the falling rock nor the Earth “has” potential energy. Potential energy is a property of two interacting objects, not of single objects. A better name might be “configurational energy.”

  12. BruceSherwood says:

    I failed to mention that, as is customary with most scientific journals, the copyright on the American Journal of Physics article rests with the journal. On your behalf I will request the journal’s permission for you to make the translation and place it on your web site.

  13. Ying-Lun Kao says:

    Hello Professor Sherwood :

    I’m so glad that you allow me to translate your paper. What would you do to help me request the journal’s permission? Do I need to give you my personal information like ID card or something like that? Thanks a lot!

    As to the energy problem, in my opinion, I think it’s necessary to use field concept. Otherwise, it would violate the spatio-temporal locality of causality. In your two-stars scenario, if we want to explain why the kinetic energy of start A increases, then we have the following options:

    1. The energy comes from its own hidden and “potential” energy.
    Since energy is transferred by work done on it, and obviously, the work done on star A is not done by “star A” itself. Therefore, I think it is not plausible.

    2. The energy comes from star B.
    This explanation involves the creation of energy. And it’s right, then how could we explain why the energy star B “knows” how much energy star B should decrease in order to increase the kinetic energy of star A. Besides, the direct effect of the decrease of energy of star B is the increase of energy of star A. This cause and effect pair has a spatial distance and it’s very weird. Newton himself also thought action-at-a-distance is very weird, too.

    3. The energy comes from the surrounding environment — gravitational field. I think it’s the only way to explain it. I know it would lead to some strange theoretical result, like infinitely negative energy. However, I think it’s still better than attributing potential energy to “earth-object system” because that would make “system concept” more ambiguous. That is, there’s no potential energy when we consider star A or B system, respectively, and there “is” potential energy when we consider star A and B as a single system.

  14. Ying-Lun Kao says:

    Hello Professor Sherwood, I’ve accomplished the translation of your “Work and heat transfer in the presence of sliding friction” paper. I found that your “Pseudowork and real work” paper is also worthy of being translated. Could I also translate your “Pseudowork and real work” paper? Thank you so much!

    I’m still refining this translation paper. I’ll post here after I finish it! Thank you so much!

  15. Ying-Lun Kao says:

    Oh! By the way, I’m not sure if it’s ok, but I think I should try to ask for your help. I really want to let more people read professor’s paper. I mean “Work and heat transfer in the presence of sliding friction” (and “Pseudowork and real work” if you and AJP allow me to translate it). As a result, I want to publish it on the Journal of Chinese Physics Education(物理教育學刊). Could you please allow me submit this translation version? If it’s not appropriate, then it’s fine 🙂 I’ll just post it on my personal website.

    Here’s the link of translation version 🙂

    • BruceSherwood says:

      1) Recall that permission was granted with the sentence, “Ying-Lun Kao has permission to post the translation if he includes a clear reference to the AJP origin of the paper.” While it is true that the abstract clearly shows the AJP origin, I request that you state (in Chinese) at the top of the article “The American Journal of Physics gave permission to post this translation of the original article.”

      2) Please add a note (in Chinese) to the end of the article, explaining that in the “Articles and talks” section of the article “Bringing atoms into first-year physics” has a section summarizing the important pedagogical changes we made to the teaching of the energetics of deformable systems.

      3) I will have have to ask AJP whether they would grant permission for you to post a translation of “Pseudowork and real work”, and also whether they would grant permission for publication of these translations in the Journal of Chinese Physics Education.

    • BruceSherwood says:

      AJP replies with this: “I hereby give permission for Ying-Lun Kao to translate and post Sherwood, B. A. (1983), “Pseudowork and real work,” AJP 51, 597-602, provided he indicate that it is a translation of the AJP paper.” So the permission is the same as AJP gave for the friction paper.

      However, AJP also says that it is not possible to publish in 物理教育學刊: “Since AJP now has a collaboration with the “mainland” Chinese journal DaXu WuLi, I’m afraid that the answer to publishing in 物理教育學刊 has to be no.”

      • Ying-Lun Kao says:

        Wow! Thank you, prof. Sherwood. FYI, so many people in Taiwan like your paper and most of them are just senior high and college students. Maybe I can, on behalf of these people, say thank you so much 🙂

        And, I really happy to see that AJP give permission to me to translate your another paper. It’s really my honor to have your permission to translate this paper. Trust me, you have help Taiwan a lot. Thank you so much 🙂

        Similarly, I will tell you when I finish the translation. And also, the most important thing is, I will add in the abstract that it’s you and AJP give me permission to translate and publish this on my website in Taiwan. And also, I will add a section to talk about your, explaining the follow-up progress about pseudo-work and real work.

        Oh and btw, is it ok to translate this blog article (pseudowork and real work) into Chinese and let me publish it on my website (in the article of the translation of “Work and heat transfer ……”)? I want to let everyone know your effort on it.

        Anyway, thank you so much 🙂

        Yours sincerely,

        Ying Lun

  16. Apoorv Potnis says:

    Hello Professor Sherwood,

    I would like to tell you that this issue is mentioned in _Physics_ by Halliday, Resnick and Krane, 5th ed., Vol. 1 by Wiley, 2001. In the Indian Student edition, it is given on pg. no. 282 under the sub-topic 13-3 “Frictional Work” in Chapter 13 “Energy 3: Conservation of Energy”. It also mentions your article with W. H. Bernard in the AJP.

    • BruceSherwood says:

      That’s interesting. Thanks for the reference. I don’t have a copy of the Krane version of Halliday & Resnick, which is intended for honors courses. However, if the issues are only mentioned briefly, that doesn’t fully address the problem. Are there any sections of the book that address the larger pseudowork/work issues, beyond their application to friction situations?

      • Apoorv Potnis says:

        Sorry for replying late. Yes. Along with the friction example, there are around 5 instances where this concept is used. But instead of using the terms _pseudowork_ or _center-of-mass work_, they refer to equations as _center-of-mass (COM) energy equation_ and _conservation-of-energy (COE) equation._ The discussed examples include a skater pushing away from a railing, a sliding block, pushing a meter stick, a ball rolling down an incline and a jumping athlete. In my opinion, the Krane _Physics_ 5th edition is much better than the _Fundamentals of Physics_ editions by Walker. HRK is more rigorous and detailed than HRW.

  17. Apoorv Potnis says:

    The 5th edition has a major change in the presentation of introductory mechanics topics. They introduce Work and Energy concepts after the end of rotational dynamics.

  18. BruceSherwood says:

    That’s very interesting. Thanks for the summary.

    • Ethan says:

      Hello, prof. Sherwood! I’ve finished the translation of your “Pseudowork and read work” paper. Here is the download link and I’ve not published it :

      I’ve mentioned the permission from you and AJP. But I’ve not add in the end of the paper the talk between you and Apoorv Potnis. Should I mention this thing? I mean Apoorv Potnis has shared the Halliday 5th edition information with you recently. Or maybe all I need to add is the following:

      “2) Please add a note (in Chinese) to the end of the article, explaining that in the “Articles and talks” section of the article “Bringing atoms into first-year physics” has a section summarizing the important pedagogical changes we made to the teaching of the energetics of deformable systems.”

      Thank you very much, prof. Sherwood 🙂

      Happy New Year!!!!!

  19. BruceSherwood says:

    No, there’s no need to mention the comments of Potnis. Incidentally, two days ago at a physics education conference I had the opportunity to see the Halliday, Resnick, and Krane 5th edition. We know Krane and know that he had followed our work but didn’t know that he included a few pages on the issues in his own book. That section includes a footnote to the Sherwood and Bernard friction paper, and another footnote to the Arons book, which I had mentioned.

    • Ethan says:

      Thank you, prof. Sherwood. 🙂

      Ok, then I will just paste the same Postscript section. And I’ll add the translation of this blog page content in my blog article where I share your paper’s translation.

      I will let you know when I finish all of it 🙂

      Thank you!

    • Ethan says:

      Hello, professor, I’ve finished the translation. I’ve also added the translation of this blog post in my WordPress’s article. Here is the link :

      Thank you very much 🙂

      • BruceSherwood says:

        A colleague pointed out a mistake in the chronology. The sentence about writing to Halliday and Resnick should read like this (the 1984 paper had not yet been published): “In 1983 I wrote to Halliday and Resnick about these matters, emphasizing that their textbook was certainly not alone in mishandling the energetics of deformable systems.” I’ve made this correction in my blog article.

        • Ethan says:

          Got it! I will edit the Chinese version of professor’s papers. Thank you professor 🙂

          By the way, I’d like to know how you think about the reality of gravitational field. I’m not sure whether it is appropriate to ask this kind of philosophical question.

          I know it’s not accepted in the mainstream of physics because it has something to do with general relativity. However, in the framework of physics education, I think it might still be a helpful concept, right? Otherwise, in my opinion, it’s difficult to explain the energy transfer aspect of the work done on object by the gravitational field.

  20. BruceSherwood says:

    In Chapter 3 of our textbook we introduce Newton’s gravitational force law and show that near the surface of the Earth a convenient approximation is F = mg, where g = +9.8 N/m, not -9.8 m/s^2. We call g “the magnitude of the gravitational field” and explain that we will go much deeper into the field concept in the second part of the textbook, on E&M. In Chapter 6 on energy we have an optional section in which we consider two identical stars far apart initially at rest. The energy equation for the system of the two stars is DeltaK1 + DeltaK2 + DeltaU = 0 because there are no external forces on the two-star system. Next we consider the system of just star 1. DeltaK1 + DeltaE_surroundings = 0, so the energy of the surroundings must decrease as DeltaK1 becomes more and more positive, yet DeltaK2 actually increases, so DeltaE_surroundings must involve some negative quantity in addition to the positive DeltaK2, and this is energy in the gravitational field. We then comment that because our use of potential enegy (DeltaU) gives the correct result, we don’t have to consider field energy in our calculations, but the difficulty in accounting for the energy in the surroundings of a star suggests that eventually we need to include fields in our models of the world.

    • Ethan says:

      Thank you so much. I’m sorry for replying late. I’m a little busy these days.

      You’re the first one professor who agree with me on these ideas. In fact, I tried a lot to find someone who has the same thoughts as mine. I found Dr. Bauman share the same thoughts in his paper “Physics that textbook writers usually get wrong”.

      He said : “if we attribute potential energy the ball at the top, we cannot properly calculate the work done on, and hence energy transferred to, the ball in its descent. It is, of course, convenient to think of objects in gravitational (or electric) fields as having potential energy. We can do that by redefining the system as ball plus field. Then the field can do no work on our system, as the ball goes up or down.”

      I think he implicitly argue that it is the gravitational field that has potential energy. I know maybe the main reason why people don’t think there’s indeed a gravitational field as electromagnetic field is that there is no gravitational waves, just like electromagnetic waves, in the paradigm of Newtonian mechanics.

      Since I want to persuade Taiwan’s physics teachers to use the concept of gravitational field to explain the transfer of energy, if you know there’s other professor who also think in this way, then I’ll be very grateful. I’m trying to contact Dr. Bauman recently. I just want to make sure whether he has the same thought as ours.

      Thank you so much, professor Sherwood. I know my English is not well enough. Thanks for your patience to read this such long reply. Thank you!!

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