principled and elegant approach to surface tension : PVW

[JohnDenker]

In section 18.8 on pages 498 and 499 there is nearly a full page of derivation leading to one form of Laplace's law i.e. equation 18.51. Effectively it is more than a page, because it delegates some of the derivation to guided example 18.4 on page II.325.

Then on page 499 it says it can also be shown that equation 18.51 takes the form ... namely equation 18.52.

Very minor point: Technically speaking, 18.51 does not take the form of 18.52; the underlying concept of Laplace's law might take a new form, but not the equation per se.

Much more importantly: The entire page-and-a-half discussion can be collapsed into a few lines using the principle of virtual work. P dV = F dx. In other words, d(interior energy) = d(surface energy).

See #149 for a directory of PVW issues.

See also my comment attached to this issue.

[JohnDenker]

Comment: This is a style issue, and I don't want to argue questions of style, but I think the intended style is clear enough that I'm on safe ground here.

The book emphasizes a principled approach to the subject. To say the same thing in other words, all good teachers at this level put a lot of effort into emphasizing _understanding_ as opposed to equation-hunting and plug-and-chug.

In this light, the discussion of Laplace's law seems like a missed opportunity. The chapter glossary on page 500 mentions Laplace's law as if it were an important result. In my opinion, it's not. Maybe I have seen this law before, and maybe I haven't ... but it doesn't matter, because it's not the sort of thing I would remember. _I do not want to remember_ such things, and I probably couldn't remember them if I tried. Instead I remember the principle of virtual work. I can rederive the surface-tension results in less time than it takes to tell about it.

Minor suggestion:

I would be tempted to leave the existing derivation somewhere, perhaps in an appendix, perhaps on a web site, as a cautionary tale.

I would then offer students a choice of:

• a laborious, error-prone, messy calculation involving integrals over force components, cosines, and whatnot, or
• a clear and concise PVW calculation.

Important suggestion:

I would offer students another choice:

• You can rote-memorize Laplace's law in the form 18.51, and also rote-memorize Laplace's law in the form 18.52. This is nasty, because they differ by a factor of 2, and you'll never be able to remember which is which.
• You can simply remember the principle of virtual work. Then you can safely forget about 18.51 and 18.52, and forget about Laplace. You can rederive the results when needed, in less time than it takes to tell about it ... and get the factor of 2 right every time.

Furthermore, as I like to say: _Utility is the best mnemonic._ In other words: Stuff that gets used gets remembered. Situations where PVW is useful outnumber situations where Laplace's law is useful, by a factor of 100 or more. So your chances of being able to remember PVW when needed are vastly better.

• Words like it can be shown seem like an incitement to rote-memorization. Emphasizing Laplace's law per se seems like an incitement to equation-hunting.
• In contrast, emphasizing PVW shows the power, simplicity, unity, usefulness, and elegance of physics.

[JohnDenker]

On page 500, Laplace's law is defined in terms of surface tension. This may leave the impression that it applies only to surface tension, when a liquid is bounded by a free surface.

However, the same concept applies to volumes enclosed in solid boundaries. This point is made in example 18.11 on page 499, in connection with blood pressure.

Another minor suggestion:

This point should have been made (but wasn't) in connection with aircraft cabin pressure. See #72.

This is the poster child for why we emphasize the principled approach to physics. Students have a choice:

• Laplace's law was derived in connection with the free surface of a liquid, and defined that way on page 500. Alas, if you remember it that way, you won't be able to apply it to more general situations.
• In contrast, the principle of virtual work can be applied over and over again, including liquid drops, soap bubble films, blood vessels, airliner cabins, etc. etc. etc. etc.