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