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I am worried this will be deleted as a duplicate question, so I will try to be extra clear what I am asking:

In some reference frames, static friction can do positive work. If you have a crate in the back of an accelerating truck, static friction does work on the crate, when everything is observed from someone in an inertial reference frame on the side of the street.

Can we say something similar, in an even remotely useful sense, for the friction between a wheel and the road (for an accelerating car with no slipping)? Is there a reference frame in which the static friction from rolling does positive work?

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  • $\begingroup$ "If you have a crate in the back of an accelerating truck, static friction does work on the crate, when everything is observed from someone in an inertial reference frame on the side of the street." Only if the create moves wrt the surface it rests on. $\endgroup$
    – Gert
    Commented Apr 26, 2019 at 15:43
  • $\begingroup$ If it moves relative to the surface it is not static friction anymore. You don't need this condition to have the work done by static friction. $\endgroup$
    – nasu
    Commented Apr 26, 2019 at 15:53
  • $\begingroup$ I am not talking about motion relative to the surface. If you have a crate on the back of an accelerating truck, and the crate is not slipping, it accelerates with the truck, because of friction. Friction acts in the same direction as its displacement, it does positive work. I wouldn't think I am saying anything controversial in this. $\endgroup$
    – D Harbane
    Commented Apr 26, 2019 at 16:13
  • $\begingroup$ Could you clarify your terminology? Rolling friction (a better term is rolling drag) is a dissipative force due to non-elastic effects; see en.wikipedia.org/wiki/Rolling_resistance?wprov=sfti1 when you say “static friction from rolling” do you mean the rolling drag described above? $\endgroup$
    – cms
    Commented Apr 26, 2019 at 16:36
  • $\begingroup$ This is why I am using "static friction from rolling." In other words, I am interested in the non-dissipative force. As I say, with static friction, the famous example is that it can do positive work on an object, e.g. a crate on a truck or a book on a conveyer belt, if you look at it from the right reference frame. I want to know if something similar can be said for the static friction between the wheel and the road, which is the external force that moves the car. $\endgroup$
    – D Harbane
    Commented Apr 26, 2019 at 16:39

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Based on what I can understand from the question including comments, I think one example might be the work associated with turning a corner, especially on flat ground.

It's only friction between the tire and road, as that tire is turning, which allows the car to change its direction (a mass-moving force causing acceleration in the direction which is perpendicular to both g and prior "straight-ahead" travel). Without sufficient friction (e.g. on ice), the car will not turn.

You might also be referring to a form of friction which is similar to the static friction you are describing. If you imagine that crate in the back of an accelerating truck, imagine now that it's an older suitcase with wheels. It might move relative to the truck more than the crate does, but accelerate less than a ball bearing in an accelerated truck bed previously coated with a super-slippery surface, due to rolling friction of the wheels of the suitcase on the truck bed.

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The crate's example you present, with the peculiar reference frame, is very neat.

I'm not sure I understand your question. What does classify as "useful"? Regarding the question on positive (vs negative?) work, the answer is simple: at a chosen time instant and for a particular point of a body at that time, on which a force is exerted (surface force or bulk force), we can always choose a reference frame (and inertial at that) in which the power of the force is negative, positive, or zero, as we please.

The reason is that every force $\pmb{F}$ is a frame-invariant quantity (this is also true of inertial forces, if we add them together with "$-m\pmb{a}$" considered as one more inertial force), whereas the instantaneous velocity $\pmb{v}_{\text{frame}}$ of a point depends on the reference frame – so we can choose it to be parallel, anti-parallel, or orthogonal to $\pmb{F}$, or zero. The power $\pmb{F}\cdot\pmb{v}$ can therefore be made positive, negative, or zero as we please – for a specific point and a specific time (but generally not globally).

But this was probably already clear to you, so I think I haven't quite understood your question.

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Can we say something similar, in an even remotely useful sense, for the friction between a wheel and the road (for an accelerating car with no slipping)? Is there a reference frame in which the static friction from rolling does positive work?

Yes. In any reference frame where the ground is moving in the direction of the friction force the friction force will do positive work on the car. For example, in a non-rotating frame at rest with respect to the center of the earth the frictional force does positive work on eastward moving cars that are accelerating

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