Law of Conservation of Energy ambiguity in Giancoli textbook

Question

In my version of the textbook by Giancoli: Physics for Scientists and Engineers, in chapter 8, there is a formulation of the law of conservation of energy that seems unintuitive and correctable to me. It first states the law in words which seems fine, and in a formulation:$$\Delta K+\Delta U+\text{[change in all other forms of energy]}=0,$$where $K$ is kinetic energy and $U$ potential energy. I'm not sure about this, but then it goes on to state the law in this manner:$$W_\text{net}=W_\text{C}+W_\text{NC},$$ where f.e. $W_\text{C}$ means "work done by conservative forces", and equivalently:$$\Delta K+\Delta U=W_\text{NC}.$$Now, you can rewrite this as $$E_2=E_1+W_\text{NC},$$ where $E$ of course means "mechanical energy". But this seems nonsense to me. I mean, why would you define it this way? It would make much more sense to me to define the net work by:$$W_\text{net}=W_\text{C}-W_\text{NC} \Leftrightarrow E_2=E_1-W_\text{NC},$$and total work by:$$W_\text{tot}=W_\text{C}+W_\text{NC}.$$Or am I missing something important here?

Answer 1

The net work is the sum of the work done by conservatives and non-conservatives forces: $$W_{net} = W_C + W_{NC}.$$ And it is equal to the change in the kinetic energy: $$W_{net} = \Delta K.$$ If the force is conservative, the system loses potential energy: $$W_C = -\Delta U.$$ Then, $$\Delta K + \Delta U = W_{net} - W_C = W_{NC}.$$ So, if there are only conservatives forces, we have $\Delta K + \Delta U = 0$. Moreover, if it is not zero, it means it is equal to the work done by non-conservatives forces: $$\Delta E = W_{NC}.$$

If there exists even more kinds of energy, not included in a non-conservatives-forces term, then you need to use Thermodynamics.

Example: If a system loses $5J$ from its potential energy and it gains $2J$ for its kinetic energy, it means: $$\Delta U = -5 = -W_C$$ $$\Delta K = +2 = W_{net}$$ (the net effect of this process is to raise 2J the kinetic energy). So we see that there are $3J$ waisted. For example, they could be lost in a friction process, which is a non-conservative force:

$$W_{NC} = -3$$

Therefore:

$$\Delta U + \Delta K = -5 + 2 = -3 = W_{NC}.$$

Answer 2

The equation $\Delta K+\Delta U=W_\text{NC}$ in the textbook (p168) is in the context of a car being slowed down by frictional (non-conservative) forces with the conservative force (gravitational attraction) doing no work.
It is also pointed out that non-conservative force direction is opposite to the direction of motion of the car and so $W_\text{NC}$ is negative.

But this just seems like total idiocy to me. I mean, why would you define it this way?
Because in the car example given in the textbook it equates the change in the mechanical energy of a system to the work done by non-conservative forces when conservative forces are doing no work.

I am also unsure about is what net work is as opposed to total work?
If you add your two final equations together you get $W_{\rm net}+W_{\rm tot} = 2W_{\rm C}$ and what useful information is contained there?