inconsistent concept of "work"

ericmazur / PnPbook

Tracking of typos, errors, and improvements for "The Principles and Practice of Physics"

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Short version: In a situation where Q=0, the equation ΔE = W + Q says that work changes the energy. Meanwhile, the work/KE theorem says that work changes the _kinetic_ energy. You can't have it both ways.

Longer version: Consider the "categories of energy" equation as discussed in item #185, namely the un-numbered equation inside the box inside figure 7.12 in section 7.4 on page 156, i.e.

E = K + U + E_s + E_th

Also consider the energy law as set forth in equation 20.1 in section 20.5 on page 547, i.e.

ΔE = W + Q

Also consider the definition of work, as set forth in equation 9.22 in section 9.7 on page 220, i.e.

W = ∫ F dx which is restated as an un-numbered equation on page 547.

My point is, those equations are not consistent ... even though they are used together. One is substituted into another on page 547, so we cannot escape the inconsistency by saying they are valid under distinct assumptions.

To see the inconsistency, consider using a thermally-insulating pushrod to exert an upward force on a box of gas, raising it slowly and reversibly against the force of gravity. Since the F in the definition of W in equation 9.22 is the total force (aka net force), the work done in this scenario is zero. This is consistent with the work/KE theorem; the KE is negligible before, during, and after the operation. However, the potential energy U has increased. The other terms on the RHS of the "categories" equation are unchanged, so E has increased. This is alas inconsistent with the ΔE= W+Q law.

Some authors try to get around this by writing ΔU=W+Q, where U is the «internal» energy, but that has its own problems.

Overall I'm not sure what is the best way to fully resolve the inconsistency ... but "something" needs to be done.

This is yet another reason why it is not humanly possible to make sense of the discussion on page 547, as discussed in item #36.

Suggestion:

The first step in fixing this problem is to get rid of the "categories" equation. Additional arguments leading to the same conclusion can be found in item #185 and #35.

Also:

Note that I am considering the gravitational potential energy to be internal to "the system" while whatever is driving the pushrod is external. One could define the gravitational potential energy to be external, and there would be some logic in that ... but it would be unconventional. It would require an elaborate explanation and frequent reminders.

Here's another manifestation of the same misconception:

In section 9.1 on page 203 there is a "conceptual" yet very specific definition. It says

Work is the change in the energy of a system due to external forces.

This is however inconsistent with the "quantitative" development, starting with equation 9.1 on page 213 in section 9.5 and including endless consequences.

Suggestions:

Warn students that there are multiple inconsistent definitions of «work» in the technical literature ... not to mention all of the non-technical and metaphorical meanings.
For the purposes of this book, pick one definition and stick with it. I suggest the F•dx definition.
Make it clear to students that neither ΔE= W nor ΔE= W+Q is the whole story; not even close. A change in potential energy is an obvious counterexample, as discussed at e.g. https://www.av8n.com/physics/thermo/state-func.html#sec-e-other-variables
Make it clear to students that E is a function of state and hence ΔE is a function of two states (initial and final) ... whereas W is not any of those things. In general, W is a functional of the entire path connecting the initial and final states. Similar words apply to Q. Contrary to what it says on page 156, it is not generally possible to identify the «thermal energy» content of a system. There may be special cases where this is possible, but they tend to be spherical-cow-in-the-ivory-tower examples. Similarly, contrary to what is assumed in the first sentence on page 547, it is generally not possible to identify a «mechanical energy» that can be transferred to or from a system.
Separate issue: Don't use the words "due to ... forces". This conveys the idea of causation. I insist that force does not «cause» energy change just as force does not «cause» mass times acceleration. This is discussed in more detail in item #79 and at https://www.av8n.com/physics/causation.htm There is already a big problem due to conflicts between the physics meaning of "force" and the vernacular meaning, as used e.g. on page 8 of the book. You don't want to exacerbate this conflict.

ericmazur / PnPbook