Modelling assumptions when friction is involved

[etotheipi]

I'm aware we've smashed the 100 post mark, and don't want to prolong this thread much longer, though I just wondered if there was a concrete way of defining this interface we speak of? There must be a boundary of some sort, since we've applied the FLT to it without trouble.

A few posts back I suggested the top layer of particles on each interacting body, though I suspect it is actually a more abstract concept.

Since really, the wedge is doing work on the block and the block doing work on the wedge. We can "infer" the work done on this "interface" though I'm not entirely sure what constitutes the interface.

 

[jbriggs444]

A few posts back I suggested the top layer of particles on each interacting body, though I suspect it is actually a more abstract concept.

Abstract is how I like to think of it. We know what happens on the one side of the interface. Just a mechanical force. We know what happens on the other side of the interface. Just a mechanical force. We know there is a relative velocity. Conservation of energy tells us that if we have mechanical work going in, some other energy flow must come out. Experience tells us that there is thermal energy coming out [in principle, the outbound flow could be otherwise].

What more do we need to know? We can ignore the details and concentrate on the results.

 

[Dale]

I just wondered if there was a concrete way of defining this interface we speak of?

At the scale of interest it is simply the two dimensional surface where the two objects are touching. Note the phrase “at the scale of interest”. Obviously at small enough scales it will no longer be two dimensional, but at those scales you don’t have simple friction, you have surface irregularities, elastic and plastic deformation, contact welds, and so forth.

 

[A.T.]

I suspect it is actually a more abstract concept.

More abstract means more general, more widely applicable and thus more useful. The concept of using P = F * v at an interface doesn't just apply to friction:

Instead of surface irregularities that oppose relative motion, you could have many little ants or linear motors propelling the block, which you also don't want to model in detail. Here the sum of the work done by the force pair would be positive. This represents mechanical energy generated at the interface (converted from chemical or electrical).

The whole issue becomes even more fun if an object has two interfaces, with two other object that are in relative motion to each other.

Page 10-11:
https://www.aapt.org/physicsteam/2019/upload/USAPhO-2013-Solutions.pdf

Chapter 2.1:
https://backend.orbit.dtu.dk/ws/portalfiles/portal/3748519/2009_28.pdf

 

[cianfa72]

No. That's double counting. The relative motion is what counts. Not the sum of the motion of the one surface in the rest frame of the other plus the motion of the other surface in the rest frame of the one.

Pick a frame and use it. You get nonsense results when you combine figures drawn from different reference frames willy nilly.

I was wondering about the frame to use to do the calculation. I believe it has to be an inertial one thus avoiding to account for work done by fictitious (not real) forces existing there

In the problem of 'block + wedge' system assuming $M_w \gg M_b$ we've actually no doubt: we can definitely assume the frame at rest with wedge as inertial doing the work calculation there.

In the general case assuming the 'system' considered as closed (no external work/heat) we can select the frame at rest with system CM doing the calculation there

In any case, as highlighted before in the thread, the displacement to account for in work calculation is actually the 'relative' displacement at the interface

 

[etotheipi]

More abstract means more general, more widely applicable and thus more useful. The concept of using P = F * v at an interface doesn't just apply to friction:

Instead of surface irregularities that oppose relative motion, you could have many little ants or linear motors propelling the block, which you also don't want to model in detail. Here the sum of the work done by the force pair would be positive. This represents mechanical energy generated at the interface (converted from chemical or electrical).

The whole issue becomes even more fun if an object has two interfaces, with two other object that are in relative motion to each other.

That Blackbird is a pretty crazy machine! The one other thing I got from it is that we can use either real work or centre-of-mass work at the interface, but to compute different quantities. If we choose centre-of-mass work,

$P_{CM, tot} = P_{CM, 12} + P_{CM, 21}$ and then also $P_{CM, tot} + \dot{E}_{int} = 0$ if the whole thing is isolated; the total centre-of-mass power equals the negative rate of change of internal energy (which in this case, means everything except translational kinetic) right off the bat.

If we choose to analyse real work, it's a little bit more convoluted.

$P_{RE, tot} = P_{RE, 12} + P_{RE, 21}$

on each body,

$P_{RE, 12} + \dot{Q}_{1} = \dot{E}_{1} = \dot{E}_{int, 1} + \dot{E}_{CM, 1}$
$P_{RE, 21} + \dot{Q}_{2} = \dot{E}_{2} = \dot{E}_{int, 2} + \dot{E}_{CM, 2}$

Of course since the system is isolated, $P_{RE, tot} = -\dot{Q}_{tot}$. I don't know how to derive it, but I assume it is the case (from the previous discussion) that the total real work done equals negative the change in thermal energy only, and not just internal as was the case when we considered COM work. In essence, the total real work done at the interface is the change in mechanical energy, whilst the total COM work done at the interface is the change in translational KE.

In the case of the block and wedge, the centre-of-mass work and real work turn out to be identical, and this is fine since there is only translational KE involved. For the Blackbird, there is rotational KE involved in the turbine, so we need to be a little more careful.

 

[jbriggs444]

In the problem of 'block + wedge' system assuming $M_w \gg M_b$ we've actually no doubt: we can definitely assume the frame at rest with wedge as inertial doing the work calculation there.

The relevant point is that every inertial frame gives the same result. We do not have to choose the right one. We just have to choose.

 

[cianfa72]

The relevant point is that every inertial frame gives the same result. We do not have to choose the right one. We just have to choose.

Sure, the fundamental point anyway is that the frame of reference chosen has to be inertial

 

[Dale]

I believe it has to be an inertial one thus avoiding to account for work done by fictitious (not real) forces existing there

I don’t think that is a requirement. Since we can do this analysis in the presence of gravity then by the equivalence principle we should be able to do it in a uniformly accelerating non inertial frame also.

 

[jbriggs444]

I don’t think that is a requirement. Since we can do this analysis in the presence of gravity then by the equivalence principle we should be able to do it in a uniformly accelerating non inertial frame also.

It would be nice for the non-inertial frame to at least have a definable potential. [Which a uniformly accelerating or rotating frame would have, of course].

 

[Dale]

It would be nice for the non-inertial frame to at least have a definable potential.

Yes, otherwise the conservation of energy wouldn’t work, but I don’t think that would change the friction part of the analysis.

 

[cianfa72]

Yes, otherwise the conservation of energy wouldn’t work, but I don’t think that would change the friction part of the analysis.

Here the system taken into account for the analysis is 'block+wedge' in the external gravity field. This way gravity, basically, is accounted as an external force doing external work on a system component (the block).

If we choose a non-inertial frame to do the analysis, as far as I can understand, we have to add the work done by inertial force as measured by the potential defined in that (non-inertial) frame

 

[jbriggs444]

Here the system taken into account for the analysis is 'block+wedge' in the external gravity field. This way gravity, basically, is accounted as an external force doing external work on a system component (the block).

If we choose a non-inertial frame to do the analysis, as far as I can understand, we have to add the work done by inertial force as measured by the potential defined in that (non-inertial) frame

Alternately, we might focus our attention on a region very close to the block+wedge interface so that the mass of the system under consideration is negligible and the choice of frame becomes irrelevant.