Is conservation of energy derived from the work energy theorem?

[hutchphd]

That's really interesting, I just wonder how then we might rationalise the frictional forces between the block and the surface. If we do break it down into electric interactions between the atoms at the interface between the surface, then we could technically attribute all of that work to microscopic internal potential energy.

No because the internal (mobile atoms at surface) degrees of freedom also allow kinetic energy (which we call roughly "temperature") in addition. So both potential and kinetic in "external" Degrees of Freedom. No longer a closed system.
The fact that we can average over the surface DF and get ~simple reproducible results lulls us into thinking it is "just another force". It is not intrinsically so, but the concept has some utility. Pedagogically it is a disaster IMHO..

 

[cianfa72]

Though I might consider a rigid block sliding across a rough surface, and take the system to be the block and the surface. A frictional force acts on both the block and the surface.

In this example, it is fairly clear that the KE of the block is reducing, so the mechanical energy of that system is reducing. The frictional force of the block on the surface also results in the thermal energy of the surface increasing.

Since the displacements of the two forces are equal, it is fairly evident that the total work done by the two frictional forces (which are internal, non-conservative) is zero. And that $\Delta E_{mech} = -\Delta E_{th}$.

As far as I can understand, your claim is the following:

Take the system 'block+surface' as closed (no external work/heat exchanged); mechanical energy of this system is just KE. Frictional forces (internal, non-conservative) act in a couple: the former acts on the block and latter on the surface.

If the total work done by the two frictional forces was zero there shouldn't be any change of KE (assumed to be the total mechanical energy of the 'block+surface' system).

Thus, IMHO, we shouldn't assume the 'block+surface' as a 'closed' system. Said that in other words: considering 'non-conservative' forces acting on parts of a system implicitly assume a 'non-closed' system

 

[etotheipi]

Thus, IMHO, we shouldn't assume the 'block+surface' as a 'closed' system. Said that in other words: considering 'non-conservative' forces acting on parts of a system implicitly assume a 'non-closed' system

I suppose that's one way of viewing it also.

I'd still be inclined to say that the system is still closed, so long as we account for the thermal energy evolved. Such that $T_{block} + E_{th} = \text{constant}$. Since the frictional forces do seem to be internal to that system.

Though I know @hutchphd mentioned in another thread that we can treat energy dissipated due to friction as outside the configuration space. So the interpretation you propose appears to be just as correct.

 

[cianfa72]

I'd still be inclined to say that the system is still closed, so long as we account for the thermal energy evolved. Such that $T_{block} + E_{th} = \text{constant}$. Since the frictional forces do seem to be internal to that system.

ok, thus coming back the to your OP question: saying $T_{block} + E_{th} = \text{constant}$ means to say that 'general' principle of conservation of energy (COE) cannot be 'derived' from work-energy theorem/principle (WEP)...basically including non-conservative forces we are not able to introduce an energy (potential energy) as a 'rescue' for the work-energy theorem to be always true (better to say applicable)

 

[hutchphd]

If the system is closed then energy of the system is conserved.
If the system is not closed, then to track the energy one must characterize the leaks.

This seems sufficient, and I do not understand what more needs to be said.

 

[wrobel]

I have not read the thread carefully, and surely what I am going to say is clear for everybody but I nevertheless want to recall this.
Classical mechanics systems are described by the following equations
$$\frac{d}{dt}\frac{\partial L}{\partial \dot x^i}-\frac{\partial L}{\partial x^i}=Q_i,\quad L=L(t,x,\dot x)\quad i=1,\ldots,m.\qquad (*)$$
The work energy theorem is written as follows
$$\dot H=-\frac{\partial L}{\partial t}+Q_k\dot x^k,\qquad (**)$$ here
$$H=\frac{\partial L}{\partial \dot x^i}\dot x^i-L$$ is the energy of the system.
One may also rewrite formula (**) in the integral form.

And formula (**) is deduced from the equation (*). And it is clear what one should demand for the energy to be conserved: $\dot H=0.$