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Work Done by Sliding Friction

This paper utilizes a numerical model of flexible asperities interacting via conservative electric forces to demonstrate that the difference between work and pseudowork during sliding friction accounts for the increase in internal energy (thermal energy), with the distinction being crucial for analyzing both forced and free sliding scenarios.

Original authors: J. David Brown

Published 2026-08-12

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: J. David Brown

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Dance of Rubbing Surfaces

Imagine you are sliding a heavy box across a carpet. You push it, it moves, and eventually, it stops. You know the box gets warm, and the carpet gets warm, but where did that heat come from? In the world of physics, this is the story of friction. For a long time, textbooks taught us that friction is a simple, stubborn force that just "happens" when two things rub together, slowing things down and creating heat. But this simple story leaves a big mystery: if the box moves at a steady speed, the forces pushing it and the friction holding it back seem to cancel each other out perfectly. If they cancel, no net work is done, so where does the extra energy for the heat come from?

To solve this, we need to understand that surfaces aren't actually flat like a sheet of glass. Even a smooth table or a block of wood is covered in tiny, jagged peaks and valleys called "asperities." Think of them like the bristles on two toothbrushes brushing against each other. When they slide, these bristles bend, wiggle, and snap back. This paper dives into that microscopic dance. It uses a clever computer model to show that because these tiny bumps are flexible, the force of friction doesn't act exactly where we think it does. This difference between where the force acts and where the object's center moves is the secret key to understanding how sliding creates heat.

The Secret Life of Bouncing Bumps

In this study, physicist J. David Brown from North Carolina State University built a simple but powerful simulation to watch how energy moves when two objects slide past each other. Instead of looking at a whole block of wood, he zoomed in to model just one tiny "bump" (an asperity) on the block and one on the table. He imagined these bumps as little pendulums attached to bars, connected by springs. As the two pendulums slide past each other, they interact with an invisible electric force—sometimes pushing apart like magnets with the same pole, sometimes pulling together like opposite poles.

The paper explores two main scenarios: "forced sliding," where you keep pushing the object so it moves at a constant speed, and "free sliding," where you let the object slide on its own until friction slows it down. The big question the paper asks is: How does a force that is fundamentally "conservative" (like electricity, which usually just swaps energy back and forth) turn into friction that creates permanent heat?

The answer lies in a concept the author calls the difference between real work and pseudowork.

Pseudowork is what you calculate if you pretend the object is a rigid, unbreakable block. You just multiply the force by the distance the center of the block moved.
Real work is what actually happens. Because the little pendulum bumps are flexible, they bend and wiggle. The point where the friction force actually touches the object moves a different distance than the center of the block does.

Here is the magic trick the simulation reveals: Real work is almost always greater than pseudowork.

When the blocks are forced to slide at a constant speed, the simulation shows that the total real work done on the system is positive, even though the pseudowork is zero. This extra "real work" doesn't make the block go faster; instead, it gets stored as internal energy. In our pendulum model, this energy shows up as the pendulums swinging wildly and the springs stretching. In the real world, this wild swinging is what we feel as heat. The paper confirms that whether the bumps are pushing apart (repulsive) or pulling together (attractive), the flexibility of the bumps ensures that real work is always higher than pseudowork, pumping energy into the system.

When the block is sliding freely without being pushed, the story is slightly different but the result is the same. The friction force does negative pseudowork, which explains why the block slows down (losing its forward speed). However, the real work done by friction is less negative than the pseudowork. Because the real work is "greater" (mathematically closer to zero) than the pseudowork, the difference again goes into the internal energy, making the bumps vibrate and heat up.

The paper also plays a fun game with perspective. If you watch the same sliding event from a different angle—say, sitting on the block while the table rushes underneath you—the numbers change. In this new view, the block might actually speed up, and the pseudowork becomes positive. But here is the cool part: the difference between real work and pseudowork stays exactly the same. That difference is the change in internal energy, and it doesn't care which way you are looking. It's a universal truth of the simulation.

The author is careful to point out that their model is a simulation, not a direct measurement of a real block of wood. They used specific numbers for their model, like setting the mass of the pendulum bob to 1 and the bar to 2, with a spring constant of 2. They found that in their repulsive scenario, the total work was exactly 0.520, which perfectly matched the increase in internal energy. In an attractive scenario, the total work was 0.347. These numbers prove that their model works, but the paper doesn't claim to have solved the mystery of friction for every material in the universe.

One of the most fascinating insights is that these processes are "time-reversible." In the computer, you could run the movie backward, and the physics would still make sense. The pendulums would stop swinging, and the block would speed up. However, the author suggests that in the real world, this backward movie is incredibly unlikely to happen by accident. It would require the tiny bumps to be perfectly synchronized in their wiggles to give energy back to the motion. Since they usually start out calm and still, the energy almost always flows one way: from the motion of sliding into the heat of vibration.

So, the next time you rub your hands together to warm them up, remember: you aren't just fighting a stubborn force. You are watching trillions of tiny, flexible bumps bend and snap, turning the energy of your motion into a chaotic dance that we feel as warmth. The paper suggests that this flexibility is the missing link that explains how a simple slide turns into a hot mess.

Technical Summary: Work Done by Sliding Friction

Problem Statement
Textbook treatments of sliding friction often rely on empirical laws (da Vinci, Amontons, Coulomb) that describe friction as a force proportional to the normal force but fail to explain the mechanism of energy transfer. A central paradox arises in the analysis of "forced sliding" (a block moving at constant velocity under an applied force): if the net force is zero, the net work on the block should be zero, yet the block's temperature rises, indicating an increase in internal (thermal) energy. Similarly, in "free sliding" (a block slowing down on a stationary surface), treating the friction force as fundamentally conservative (derived from atomic electric forces) suggests the net work over a closed path should be zero, yet the block loses translational kinetic energy. The paper addresses the discrepancy between the work done by forces and the change in translational kinetic energy, specifically investigating how conservative interatomic forces result in non-conservative macroscopic friction and thermal energy generation.

Methodology
The author employs a simple numerical model consisting of two flexible "asperities" (one representing the block, one the table) sliding past one another.

Model Structure: Each asperity is modeled as a pendulum bob attached to a massless rod, which is connected to a uniform bar via a torsion spring. The center of mass of the asperity lies on the pendulum rod.
Interaction: The interaction between the two asperities is modeled as a conservative electric force (either attractive or repulsive) between the pendulum bobs. The author notes this can be extended to more realistic potentials (Morse or Lennard-Jones).
Dynamics: The equations of motion are derived using Newton's second law and the torque-angular momentum principle, accounting for the deformation of the asperities (the pendulum angle $\theta$ and bar orientation $\phi$ ).
Scenarios: The model simulates two distinct regimes:
1. Forced Sliding: An applied force balances the friction force, maintaining constant velocity for the center of mass.
2. Free Sliding: No applied force acts in the direction of motion; the object slows down due to friction.
Analysis Framework: The study distinguishes between real work ( $w$ ), defined as the integral of force over the displacement of the point of application, and pseudowork ( $w_{ps}$ ), defined as the integral of force over the displacement of the system's center of mass. The relationship is governed by the Invariant First Law of Thermodynamics (IFLT): $w - w_{ps} = \Delta e_{int}$ , where $\Delta e_{int}$ is the change in internal energy.

Key Contributions and Results
The paper demonstrates that the deformation of the interacting bodies is the critical factor resolving the energy paradoxes.

Work vs. Pseudowork: Because the asperities are not rigid, the point of application of the friction force moves a different distance than the center of mass.
- In forced sliding, the applied force acts on the bar, while friction acts on the pendulum bob. The paper finds that for both repulsive and attractive interactions, the real work done by the applied force exceeds the pseudowork, and the magnitude of the negative real work done by friction is less than the magnitude of the negative pseudowork. Consequently, the total real work ( $w$ ) is positive, while the total pseudowork ( $w_{ps}$ ) is zero (since the center of mass velocity is constant). This positive net work accounts for the increase in internal energy ( $\Delta e_{int}$ ).
- In free sliding, the real work done by friction is always greater than the pseudowork ( $w > w_{ps}$ ). The difference corresponds to the increase in internal energy. The pseudowork is negative, explaining the decrease in translational kinetic energy.
Mechanism of Energy Transfer: The increase in internal energy arises because the flexible asperities oscillate as they interact. The "approach" and "recede" phases of the interaction involve different displacements of the contact point relative to the center of mass.
- For repulsive forces, the bob lags behind the center of mass during approach and leads during recession.
- For attractive forces, the bob leads the center of mass during approach and lags behind during recession.
- In both cases, the asymmetry in displacement ensures that the real work exceeds the pseudowork, leading to a net increase in internal energy (oscillatory kinetic and potential energy of the asperite).
Reference Frame Dependence: The paper analyzes free sliding in a frame where the block is initially at rest and the table moves. In this frame, the pseudowork is positive, and the block's translational kinetic energy increases. However, the difference between real work and pseudowork (the change in internal energy) remains invariant across reference frames.

Significance and Claims
The paper claims to provide an intuitive, mechanical model for the transfer of energy in sliding friction using fundamentally conservative forces. Its primary significance lies in clarifying the distinction between work and pseudowork:

Resolution of the Paradox: It demonstrates that the apparent contradiction between zero net force (in forced sliding) and rising temperature is resolved by recognizing that the point of application of the force moves differently than the center of mass due to deformation.
Source of Thermal Energy: The model identifies the source of thermal energy as the difference between real work and pseudowork, which manifests as internal energy (vibrations) in the flexible asperities.
Time-Reversibility: The author explicitly notes that the dynamical processes described are time-reversible. The direction of energy transfer (increase vs. decrease in internal energy) depends on initial conditions. While the model shows energy dissipation (increasing internal energy) for typical initial conditions (asperities at rest/equilibrium), the reverse process (cooling and speeding up) is physically possible but statistically unlikely without fine-tuned initial conditions.

The paper does not propose new experimental setups or claim to derive the macroscopic laws of friction (da Vinci/Amontons/Coulomb) from this model. Instead, it offers a conceptual framework to understand how energy is transferred and transformed in the presence of sliding friction, emphasizing the role of flexibility and the distinction between center-of-mass motion and internal degrees of freedom.

The Invisible Dance of Rubbing Surfaces

The Secret Life of Bouncing Bumps

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