Change in closed system energy with both conservative and non-conservative forces

[zenterix]

Homework Statement

Suppose we have a closed system that undergoes a transformation from an initial state to a final state.

The internal forces in the system consist of both conservative and non-conservative forces.

Relevant Equations

The work done in the system is

$$W=W_c+W_{nc}$$

That, is the sum of work done by conservative forces and non-conservative forces.

Thus

$$\Delta K=-\Delta U + W_{nc}$$

$$W_{nc}=\Delta K+\Delta U=\Delta E_m$$

My question is about the following statement

The total change in energy of the system is zero

$$\Delta E_{system}=\Delta E_m-W_{nc}=0\tag{1}$$

Energy is conserved but some mechanical energy has been transferred into non-recoverable energy $W_{nc}$. We shall refer to processes in which there is non-zero non-recoverable energy as irreversible processes.

The system is closed. $\Delta E_{system}$ does not necessarily have to be zero. Where does (1) come from?

 

[Hill]

Where does (1) come from?

Energy conservation.

 

[Steve4Physics]

Hi @zenterix. Maybe I’m being a bit dense but the question is not clear to me. What exactly do $\Delta E_m$ and $\Delta E_{system}$ represent?

Sometimes a simple physical example can help. E.g. consider a closed system consisting of 2 separated charged (+ and -) objects able to slide (with friction) on a rail.

The objects are released from rest and slide towards one another. After some time:

Work done by conservative (electric) forces = 10J.
Work done by non-conservative forces (friction) = -7J (let's ignore any radiative losses).
Gain in objects' KE = 3J.

Of course the total energy - including thermal energy, - of this closed system is constant.

Maybe it will help to match your various symbols to the above figures.

 

[zenterix]

Hi @zenterix. Maybe I’m being a bit dense but the question is not clear to me. What exactly do $\Delta E_m$ and $\Delta E_{system}$ represent?

Sometimes a simple physical example can help. E.g. consider a closed system consisting of 2 separated charged (+ and -) objects able to slide (with friction) on a rail.

The objects are released from rest and slide towards one another. After some time:

Work done by conservative (electric) forces = 10J.
Work done by non-conservative forces (friction) = -7J (let's ignore any radiative losses).
Gain in objects' KE = 3J.

Of course the total energy - including thermal energy, - of this closed system is constant.

Maybe it will help to match your various symbols to the above figures.

$E_m$ is mechanical energy, ie potential energy $U$ plus kinetic energy $K$.

$E_{system}$ is the total energy of the system.

I think my question is what are all the implicit assumptions in the statement $\Delta E_{system}=\Delta E_m-W_{nc}$.

From what I understand, one type of energy in the system is mechanical energy. Non-conservative forces change this mechanical energy.

In the expression for $\Delta E_{system}$ we have a $W_{nc}$ term. Isn't this already $\Delta E_m$? Why do we write $\Delta E_m-W_{nc}$?

What are other reasons why the mechanical energy can change?

 

[zenterix]

Energy conservation.

Sure, but why this particular expression $\Delta E_m-W_{nc}$? And how do we know $\Delta E_{system}$ is zero? The system is closed but it can exchange energy with the surroundings. It seems there are many implicit assumptions.

 

[kuruman]

First of all it is a mistake to say that when you have a closed system the work done on the system is $$W=W_c+W_{nc}.$$ "Closed" means that no work (or heat) crosses the system boundary. It is completely isolated from whatever is "non-system".

That said, you are disregarding the principle of conservation of total energy. The equation for that is $$\Delta E_{\text{total}}=0.$$ Let me give you a quick example.
You release from rest a book at height $h$ from the ground. The book hits the ground where it comes to rest. Analyze the energy transformations in the closed book + Earth system.
Answer: The mechanical energy change is $$\Delta(ME)= \Delta K+\Delta U=0-mgh.$$Is that all? No. If the system is closed isolated that change in mechanical energy must stay within the system in some other form. That is, if the mechanical energy decreases by a certain number of Joules, something else must increase by the same number of Joules because no Joules are allowed to come in of go out of the closed system.

In this case, the lost mechanical energy goes into various forms of energy, sound as the book hits the ground, vibrations of the molecules on the ground and the air, low intensity seismic waves to name a few. We can lump all these under the umbrella name "thermal energy" and write the total energy conservation equation $$0=\Delta E_{\text{total}}=\Delta E_{\text{thermal}}+\Delta (ME)=\Delta E_{\text{thermal}}-mgh\implies \Delta E_{\text{thermal}}=mgh.$$ This shows that the potential energy decrease of the book+Earth system appears as a rise in the thermal energy of the system.

You can find more about this in the article
https://www.physicsforums.com/insights/is-mechanical-energy-conservation-free-of-ambiguity/

 

[Steve4Physics]

$E_m$ is mechanical energy, ie potential energy $U$ plus kinetic energy $K$.

Thanks for clarifying. It may be worth noting that the symbol $U$ is often in thermodynmics to represent a system’s total internal energy. That's what confused me. But more importantly...

For a ‘closed’ system, matter can’t enter or leave, but energy can. For example heat can pass through the walls, or work can be done on the system by an external force compressing the system.

For an ‘isolated’ system, neither matter nor energy can enter or leave the system.

I think some of the confusion is that you are mistakenly trying to apply equations intended for an isolated system (e.g. $\Delta E_{system}=0)$ to a closed system.

 

[Mister T]

The system is closed. ΔEsystem does not necessarily have to be zero.

Yes it does. That's the definition of a closed system.

 

[Steve4Physics]

Yes it does. That's the definition of a closed system.

Disagree! A closed system is defined as one which allows the transfer of energy (but not matter) in/out. E.g. see https://en.wikipedia.org/wiki/Closed_system

A 'closed' system is not the same as an 'isolated' system. See Post #7.