"force displacement" ... or not

[JohnDenker]

In section 10.5 on page 236 it says:

we need to look at the force displacement

the force displacement is zero

I don't think there is any such thing as a force displacement. The force is a vector, and the displacement is another vector. You can't displace a force.

Suggested rewording:

we need to look at both the force and the displacement

or

we need to look at the dot product, force • displacement

[ericmazur]

I explain that the term is shorthand for “displacement of the point of application of the force”. That is an absolutely CRUCIAL point.

It’s all there.

[ericmazur]

Response from @JohnDenker:

As an author, you are free to define terms however you want, ... but I have to ask, why would you want this?

1) There is an important principle of mathematical physics here: A vector has magnitude and direction, but it does not have a position. It is a common misconception to think that a force vector is attached to a given point. This is profoundly wrong. The force is a vector, and the displacement is another vector. Using the term "force displacement" inflames this misconception.

2) "Force displacement" is jargon ... highly non-standard jargon. I did some googling and was unable to find anybody else who uses such a term.

3) We agree that there is a "CRUCIAL" point to be made about the point of application, but it seems to me that hiding this point behind a layer of jargon is not helpful. Indeed it is a step in the wrong direction, because the /point of application/ is important whether it gets displaced or not. Specifically: The scalar F•x is work, whereas the bivector F∧x is torque.

So I reckon "force displacement" mashes together /three/ things that ought to be kept separate: force, point of application, and displacement.

4) Students would be better off if they could spend less time memorizing jargon and more time reasoning about the physics.

I renew my suggestion: AFAICT every mention of "force displacement" could be replaced by simple "displacement" with no loss in clarity, and indeed a net gain in clarity. (If you're worried about conflict with some other displacement, you could refer to the /relevant/ displacement ... but that should vanishingly rare.)

Summary: -- force is force -- point of application is point of application -- displacement is displacement

These are three important ideas, but they are not the same idea!

[ericmazur]

"every mention of "force displacement" could be replaced by simple "displacement" with no loss in clarity, and indeed a net gain in clarity."

A TREMENDOUS loss in clarity! Students end up taking the displacement of the center of mass of the object and getting the wrong answer! Plain and simple. For single rigid objects there is no difference between the displacement of any point on the object, but for a deformable object (or a system comprising more than one rigid object) work IS NOT equal to the product of the force and the displacement of the center of mass. It is equal to the scalar product of the force and the displacement of the point of application of that force. Since that is such a mouthful, I define "force displacement" as shorthand for "displacement of the point of application of the force" (definition on p. 204; the crucial point is at the bottom of page 203).

[JohnDenker]

The point remains, the force is not what's being displaced. To my ears, "force displacement" sounds like either the displacement of the force (which is silly) or force times displacement (which is something else entirely).

There's got to be a more descriptive term. Possibilities include:

• The i th displacement ... corresponding to the i th force.
• The application displacement. It is after all the point of application that is being displaced.
• The local displacement ... local to the place where the force is applied.
• The relevant displacement.
• ?????

A force is a rather narrowly defined thing, in accordance with the vector space axioms. As such, it cannot possibly have a location, as discussed in item #129. By the same token, it cannot possibly have a displacement. Force is an important concept and line of action is an important concept, but they are not the same concept.

I reckon that every introductory textbook in the last 400 years has gotten this wrong. There are lots of misconceptions about this.

[JohnDenker]

There are, as usual, about 400 different misconceptions to worry about. When fending off one misconception, one must be careful not to exacerbate others.

Without overlooking other misconceptions, let me call attention to the idea of "bound vector". This is not mentioned in the book. However, the concept is floating around in the literature. Students may have been exposed to it already, and will likely encounter it eventually. I mention this because as soon as you mention "force displacement" -- no matter how carefully defined -- some students will get the impression that the force itself is being displaced. In fact the force and the displacement are two different things. The displacement does not "belong" to the force nor vice versa; they both "belong" to some higher-order concept. I guarantee you that some people are confused about this, including students as well as some professors at Big Name universities. This confusion is exacerbated by so-called "extended" free-body diagrams where the arrow that supposedly represents a force is bound to a particular location. See e.g. figure 12.11 in section 12.3 on page 287, as mentioned in item #129.

Suggestion: It should be emphasized that force is a vector; as such it has direction and magnitude ... but not location. The force vector and the displacement vector are both important ideas, but they are not the same idea. The displacement does not "belong" to the force, nor vice versa.

The discussion in the book is "mostly" careful about this, but there are occasional lapses that could be cleaned up. For example:

• Exercise 9.1 on page 204 speaks of "displaced forces".
• On page 205 it speaks of a force and "its" force displacement.

See also next comment.

[JohnDenker]

Here's yet another thing to consider:

In section 9.1 on page 204 we find a concise definition of "force displacement". The same ten-word definition can also be found in the chapter summary on page 225.

That definition is consistent with a proper understanding of the physics, but it does not guarantee it. The problem is that students have been trained for years on end to rote-memorize definitions like that. All too often, they latch onto a legalistic, pharisaical interpretation. All too often, they emphasize literal words rather than concepts.

As an example of the sort of thing I'm talking about: There are lots of displacement vectors in the world. In particular, even though Δr is a displacement vector, r itself is also a displacement vector; it is the displacement of the point in question relative to the origin.

I'm making one-and-a-half different arguments here:

• The main point is that pithy, legalistic definitions are not good. They invite rote regurgitation rather than reasoning.
• The minor point is that this particular definition is imperfect. However, returning to the main point, the optimal solution is not to come up with a more-intricate legalistic definition ... but rather to get rid of the definition entirely. If you get rid of the jargon term, you can get rid of the definition. Rely less on jargon and rely more on conceptual understanding.

In this case, the central concept is _work_. The notion of so-called "force displacement" exists only as a stepping-stone in support of the concept of work.

Work is quite a tricky concept. In general, it depends on the _path_ from point A to point B. This leads immediately to a simple, constructive suggestion: Rather than relying on the non-traditional jargon term "force displacement", instead rely on the concept of path. In simple cases, Δr tells us what we need to know about the path.

(In non-simple cases, the concept of Δr does not survive, but the concept of path does.)