Modelling assumptions when friction is involved

[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.