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Is it correct to say that mechanical energy will always be conserved in any conservative system (a system with no non-conservative forces), no matter if it's isolated or not? Are there any examples of isolated systems that have non-conservative forces, and if so, can the friction force be one of them? Or is a system with friction force always non-isolated?

I also don't fully understand this example I read about a non-isolated system:

"Hans Full is doing the annual vacuuming. Hans is pushing the Hoover vacuum cleaner across the living room carpet."

The explanation is that it's a non-isolated system because "the friction between the cleaner and the floor and the applied force exerted by Hans are both external forces. These forces contribute to a change in total momentum of the system."

I'm confused because we were never told which elements are part of the system and which are not. If we consider the system to be the floor/carpet and the cleaner, then why is the friction force between them external? I would have thought the friction force is coming from the floor, which is part of the system, therefore it's internal. What if we considered Hans to be part of the system too? Would it then become an isolated system (supposing that the friction force is also considered internal)? All the examples I read about non-isolated systems are systems in which there is friction force. But I don't understand why that force HAS to be external. Is it always so, or is it just that I'm reading too many similar examples?

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    $\begingroup$ Internal and external are subjective labels. You can never say, for any force, if it is always external or always internal. It all just depends on the subjective definition of the system. So, you are right in saying that there isn't enough information in the example. The system must be clearly defined before one can label a force as internal or external $\endgroup$ – Aaron Stevens Jun 26 at 18:52
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Is it correct to say that mechanical energy will always be conserved in any conservative system (a system with no non-conservative forces)

Yes. But you need to be careful as to how you define the "system" and what you mean by the "mechanical" energy of the system.

The system can be anything you define it to be. Once defined, by default everything else becomes the surroundings.

The mechanical energy of a system is normally associated with its macroscopic motion and position with respect to an external frame of reference (its macroscopic kinetic and potential energy). This is sometimes referred to as the systems "external" energy. An example is a container of gas moving in a room with a velocity $v$ at a height $h$ with respect to the reference frame of the floor of the room.

But a system also possesses microscopic kinetic and potential energy, that is the kinetic and potential energy at the molecular level. This is the systems "internal" energy with respect to the frame of reference of the system. An example would be the kinetic and potential energy of the molecules of the gas within the container.

The mechanical energy, as defined above, of a system is conserved if the system is only subject to conservative forces. This would apply to our container of gas if the room was evacuated of air to eliminate the friction of air drag.

no matter if it's isolated or not?

That would depend on what your definition of an "isolated" system is.

The most general definition of an an isolated system is one that cannot exchange mass nor any form of energy (heat or work) with its surroundings. Of course no system can be isolated from gravity. But at least gravity is a conservative force.

Suppose our container of gas is moving in a room with no air, so there is no air friction. The total mechanical energy of the container of gas is the sum of its kinetic and gravitational potential energies with respect to the frame of reference of the room. Since there are no non-conservative forces acting on the container, mechanical energy is conserved.

Since the gas is in a closed container, there is no mass transfer between our system and the room. If the container walls are rigid so that they cannot expand or contract, no energy transfer in the form of boundary work is possible. But if the container of gas is not perfectly thermally insulated, there is the possibility of radiant heat transfer between the gas and the room if there is a temperature difference. Therefore, our system is not isolated. So the question is, would heat transfer between the gas and the room contents necessarily cause its external mechanical energy to not be conserved? (In this case I am talking about heat transfer that is not the result of friction.) I cannot think of any example. I would be interested if anyone can provide any examples.

If this is correct, then an isolated system subject only to conservative forces is a sufficient but not necessary condition for conservation of mechanical energy.

Are there any examples of isolated systems that have non-conservative forces, and if so, can the friction force be one of them?

Yes, if we define an isolated system in such a way that it includes non-conservative forces.

As an example, let's say our system is a simple pendulum and includes the bob, string connected to the bob, pivot connected to the string, in motion and located in a rigid, perfectly thermally insulated container. The system is located in a room which we will consider to be its surroundings. We consider the system isolated from its surroundings (with the exception of course of gravity).

Now let there be friction at the pivot and let the bob encounter air in the container. The system is therefore subject to non conservative forces, yet it is an isolated system. Mechanical energy will not be conserved.

Or is a system with friction force always non-isolated?

No. The pendulum example system just described is by definition an isolated system, and yet it is subject to non conservative (friction) forces.

I'm confused because we were never told which elements are part of the system and which are not.

Your confusion about the vacuum cleaner example is well justified. It is clear that in the example the vacuum cleaner alone (without Hans or the floor) must constitute the system and that the system is not isolated because there is both heat and work transfer between the vacuum and its surroundings (Hans and the floor). But from what you have described, the "system" was not explicitly defined. Very poor example, in my opinion.

Hope this helps.

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Energy is only conserved if the system is isolated. If two systems are in contact and only involve conservative forces, then the energy is conserved if you look at both systems together but the energy of each system individually need not be conserved since energy can be exchanged between the systems.

Conservation of mechanical energy also requires that only conservative forces be involved since mechanical energy is defined as the sum of kinetic and potential energies. Potential energy is only defined for conservative forces, so if energy is transferred via work done by a non-conservative force the mechanical energy will not be conserved. Essentially, mechanical energy is kinetic energy or energy "stored" such that the energy is available to do work (i.e. to give a body kinetic energy).

What the system in question is is really up to you. But the examples you give have chosen the system for you. Friction is usually just treated as an external force, because (although the total energy is conserved) the energy lost to friction has been dissipated in the form of sound and heat and cannot be immediately used to do work nor has it been turned kinetic energy (of the rug). So the mechanical energy of the system has decreased.

There are also examples of closed, isolated systems with non-conservative forces. Other than friction, any time-dependent or velocity-dependent force will be non-conservative. Along those lines, electromagnetism provides some important examples. For instance, the closed electric field lines generated by a changing magnetic flux (Faraday's Law) are non-conservative. Remember that conservative forces have the very specific definition that the net work they do along a closed path is zero.

Keeping track of both the driving system and the system being driven self-consistently can be difficult to do, so often we approximate the driving system as unaffected by the system it drives. Similarly, when dissipation is involved we assume the system into which energy is dissipated is unaffected by that dissipation. Essentially, when non-conservative forces are involved the question is simplified by approximating the external system as an energy source or energy sink.

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I couldn't provide any intuitive reasoning for your question but I try to convert each and every aspect of classical mechanics into some wonderful equations.For any system of particles we have from work energy theorem:-$$dW_{total}=dK_{system}$$ $$dW_{int,con}+dW_{int,non-con}+dW_{ext}=dK_{system}$$ $$-dW_{int,con}=dU_{system}$$ $$dW_{int,non-con}+dW_{ext}=dU_{system}+dK_{system}$$ $$dU_{system}+dK_{system}=dE_{mech,system}$$ $$dE_{mech,system}=dW_{int,non-con}+dW_{ext}$$ I think this last equation may satisfy your every doubt as it is general for any system whether it is conservative or non-conservative.If you have any doubts on notations then please comment.

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