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Consider a rotating disk and a fixed part that "abrades" something. Under normal conditions the disk and the fixed part should not touch. However, they slightly touch due to misalignment. When the disk rotates, this results in friction which ultimately causes heat.

Very intuitively, the higher the angular velocity, the higher the heat generated. A match only ignites when it is moved fast, an angular grinder causes heat when it is fast, moving my hand over a carpet feels only hot/hurts if I do it fast, etc.

The thermal energy released is given by the path integral over the friction force $F$. And the friction force is directly proportional to the normal force $F_N$. The proportionality is a constant ($F=\mu F_N$) and in my case, I think, the normal force is also constant ($F_N=p/A$ where $p$ is constant pressure between the disk and the fixed part and $A$ the contact area). And "Coulomb's Law of Friction" states that the friction is independent of sliding velocity. How can this be?

My actual problem at hand is: Consider the disk as the brake disk of a car and the fixed part the brake jaw (or similar). Assuming I need to drive the car a distance $x$, I would like to find the optimum speed (which is directly related to the angular velocity of the disk) that causes minimum harm. I assume minimum harm is achieved when the heat due to friction is lowest. Intuitively, it is better to drive the car slower but have the friction applied longer than driving fast but have the friction applied for a shorter time. I would like to see this in an equation.

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Good question! The answer is that temperature rise is not the same thing as generated heat energy. You would be forgiven for thinking so since $$ \Delta T = \frac{\Delta E}{C},$$ Where $\Delta T$ is the temperature change, $\Delta E$ is the heat energy change, and C is the heat capacity.

But here’s the thing: $\Delta E$ is not just the added heat energy from all resistive processes over time, it’s the net energy change at any given time. What’s the difference? Well, every system is connected to the environment in varying degrees. Excess heat energy is constantly being redistributed out to the environment through thermal conduction.

So to increase the temperature, you need to add energy faster than it is subtracted. How does that happen? By converting the motion into heat energy in a short amount of time! Sliding your hands together quickly or slowly will generate the same total energy. But sliding them quickly will generate that energy faster than it dissipates away, giving a noticeable temperature rise.

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  • $\begingroup$ Oh, that's a very good explanation! Any chance you could also elaborate on how you would approach the "actual problem at hand"? How could I determine the optimum speed with which I need to drive to cause the least harm? (I guess this would somehow be related to the heat capacity of the disk/fixed part and the thermal resistances) $\endgroup$
    – divB
    Commented May 27, 2020 at 3:01
  • $\begingroup$ @divB Yeah, for that, you’ll need some handle on the thermal conduction or “cooling power” of any heat mitigation system there is. And moreover, damage due to heat will not be linear: for example, heating a few degrees over a long period of time is not the same as heating 1000 degrees over a short period of time. Probably, you’d rather just set a temperature limit that you’d attempt to stay below to avoid damage, then figure out the driving parameters to achieve that. But ultimately, yours is not a simple question to answer. $\endgroup$
    – Gilbert
    Commented May 27, 2020 at 16:33
  • $\begingroup$ @divB also, I’ll point out, that it is theoretically possible to drive a car a distance $x$ without using the brakes at all, and relying solely on the internal/rolling resistance. So, you could get where you want to go with no heat delivered to the brake disks. $\endgroup$
    – Gilbert
    Commented May 27, 2020 at 16:36
  • $\begingroup$ First off, I think you misunderstood the second part of the question: It is not about breaking! I do not want to break. When the car is moving, a part is rubbing at the brake disk which is not supposed to rub. For example the brake jaw. That's the whole point. $\endgroup$
    – divB
    Commented May 27, 2020 at 19:59
  • $\begingroup$ Second, I really found your answer very useful but unfortunately I am still confused about the underlying problem. I do have "some" handle on thermal curcits but I do not know how to connect the dots. We can assume the parts have a heat capacity of $C$ and a thermal resistance of $R$ (all heat migration lumped into these two). I know that $q=\Delta T/R$ and $q=C \Delta T/\Delta t$. I know that this is not a simple question to answer but it can be made simple with such simplifying assumptions. $\endgroup$
    – divB
    Commented May 27, 2020 at 20:02

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