Gracy, the best thing for you to remember is that, by definition, the work done by a conservative force equals ##-ΔPE##. What follows may only muddy the waters, but you need to see it and if you remember nothing else remember that, by definition, the work done by a conservative force equals ##-ΔPE##.
gracy said:
##Wnonconservative##=##ΔKE##+##ΔPE##
This is a perfectly valid dynamical relation and will serve you well when calculating answers to chapter-end problems and problems on tests and homework for those portions of your introductory physics courses. It is not, however, a valid energy relation. For that reason many textbook authors are abandoning this approach. It follows from a definition of work that cannot be generalized to include energy transfers in thermodynamic effects.
Here's a simple example. You exert a force of ##10.0## N on a block pushing it a distance of ##1.0## m across the surface of a level table top at a steady speed. There is no change in kinetic energy during the process. Neither is there a change in potential energy. You do ##10## J of work on the block. If we say the net work done is zero because the work done by friction is -##10## J, then we satisfy the above quoted relation, but we get in trouble when we try to apply the First Law of Thermodynamics:
##ΔE##
int = ##Q## + ##W##
where ##ΔE##
int is the change in internal energy of a system, ##Q## is the heat energy transferred to that system, and ##W## is the work done on that system. Heat energy ##Q## is defined as energy transferred to a system as a result of a temperature difference, so if we insulate the block and table from its environment ##Q = 0## and we have
##ΔE##
int = ##W##.
There is an increase in the internal energy of the block (it warms up) and the table top (it warms up) totaling ##10## J. (Assuming the block and table are isolated from their environment.)
How much work ##W## is done on the system? We don't want to choose the block alone to be our system because it won't help us verify the validity of the relation ##ΔE##
int = ##Q## + ##W##. This is because we cannot say ##ΔE##
int = +##10## J. The block's internal increases by less than ##10## J. It was the combined block-table system that gained ##10## J of internal energy. (Both the block and the table top warmed up).
So, let's consider the system to be the block and table combined. Now we can write ##ΔE##
int = +##10## J. Now ##W = +10## J, the work you did on the block, and we satisfy the relation ##ΔE##
int = ##Q## + ##W##. We do not say the work done by friction is -##10## J. To understand why we have to understand the correct way to define ##W##.
We can define ##W## in a way that makes it consistent with the First Law of Thermodynamics. Some popular introductory physics textbooks started doing this about 20 years ago, for example Halliday and Resnick. Others have followed suite, but it's slow to catch on in the classrooms. An alternative approach is to continue to define ##W## in the same way, but call it something else, such pseudo-work. That approach doesn't seem to be favorable as we do not see it appearing in the textbooks.
Here is the abstract of a paper written by Arnold B. Arons. It appeared in the American Journal of Physics in 1999. Citation: Am. J. Phys. 67, 1063 (1999).
The work-energy theorem, derived from Newton’s second law, applies to the displacement of a particle or the center of mass of an extended body treated as a particle. Because work, as a quantity of energy transferred in accordance with the First Law of Thermodynamics, cannot be calculated in general as an applied force times the displacement of center of mass, the work-energy theorem is not a valid statement about energy transformations when work is done against a frictional force or actions on or by deformable bodies. To use work in conservation of energy calculations, work must be calculated as the sum of the products of forces and their corresponding displacements at locations where the forces are applied at the periphery of the system under consideration. Failure to make this conceptual distinction results in various errors and misleading statements widely prevalent in textbooks, thus reinforcing confusion about energy transformations associated with the action in everyday experience of zero-work forces such as those present in walking, running, jumping, or accelerating a car. Without a thermodynamically valid definition of work, it is also impossible to give a correct description of the connection between mechanical and thermal energy changes and of dissipative effects. The situation can be simply corrected and student understanding of the energy concepts greatly enhanced by introducing and using the concept of internal energy, that is, articulating the First Law of Thermodynamics in a simple, phenomenological form without unnecessary mathematical encumbrances.
So the issue here is the application of the friction force to the underside of the block. The underside of the block is not undergoing a displacement of ##1.0## m. The displacements are microscopic and there is no way, or at least no direct or easy way, to calculate them. Thus we cannot know how much of the ##10## J of energy was received by the block, with the remainder of it being received by the table top.
