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Re: [Phys-L] fundamental notion of force --> using an arrow to represent something more than a vector

Some comments:

I think of an interaction as something that involves two forces, for instance, the gravitational interaction between my desk and the earth involves the force that the earth is exerting on my desk and the force that my desk is exerting on the earth.

I think of a vector as a mathematical entity and a force as a physical entity.

Sometimes a force can be represented by a vector. In such cases the object whose motion is under study can be represented by a point even if it is a block.

Sometimes a force can be represented by a vector plus a line of action.

Sometimes you need both a vector and a point of application to represent a force. The line of action doesn't cut it if you are interested in internal stresses and strains not too close to the point of application.

A force on some macroscopic object is typically an effective concentrated load representing the effect of a distributed load. Sometimes you need to use the distributed load, for instance, when you are interested in stress and strain near the region of application of the load.

Associated with a force are an infinite set of torques corresponding to an infinite set of points about which one might wish to calculate the torque.

I usually think of the lever arm as the scalar distance between the point about which one is calculating the torque, and, the line of action of the force (measured along a line that is perpendicular to the line of action of the force). The position vector of the point of application of the force (or some other point on the line of action of the force) relative to the point about which one is calculating the torque is "the position vector of the point of application of the force (or some other point on the line of action of the force)."

Perhaps replacing the word "force" with "concentrated load" would imply that it has a point of application.

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of John
Denker
Sent: Friday, October 23, 2015 5:08 PM
To: Forum for Physics Educators
Subject: [Phys-L] fundamental notion of force --> using an arrow to
represent something more than a vector

Hi Folks --

Suppose you have a cart supported on fully-castering wheels.
It is initially at rest.
a) You attach a string to the front-left corner and pull.
The cart moves forward and rotates CW.
b) You attach a string to the front-right corner and pull.
The cart moves forward and rotates CCW.

To diagram this, the traditional approach is to draw the cart and draw an
arrow at the appropriate corner. So far so good.

However it is *not* a good idea to call that arrow a force.
A force is a vector, and by definition a force has magnitude and direction but
*not* location ... whereas the arrow we just drew has a partially-significant
location.

The arrow is representing a vector and something more. It is representing a
force and something more. In expert hands such a representation is
convenient, but in less-skilled hands it tends produce profound
misconceptions about the concepts of "force" and "vector".

AFAICT every introductory textbook in the last 400 years has made a mess of
this issue. After much struggle, and with help from the folks in this forum,
I've come up with something that seems like a significant improvement. The
idea is to carefully distinguish between a /dynamic interaction/ and a force.
-- In simple cases, we have: interaction = (force, line of action)
-- If we pick a datum, we can write: interaction = (force, torque)
i.e. χ = (F, r∧F)
where χ is the interaction, F is the force, and r is
the lever arm.

The convenient (but tricky) thing is that we can use an arrow to represent
the entire interaction χ, not just the force F.
The length and orientation of the arrow represent the force vector, while the
location of the arrow represents the line of action.

This is a trap for the unwary, because you can’t necessarily tell by looking at
an arrow whether it is meant to represent a vector or something more. You
can’t necessarily tell by looking whether its location is significant.

Suggestions:
-- When in doubt, draw the lever arms and forces explicitly.
Use one arrow for each lever arm, and one arrow for each
force. Don't try to use one arrow to represent two vectors.
-- Make sure students have a firm grasp of the explicit
representation before moving on to the tricky representation.
-- If/when you introduce interaction arrows, be sure you
call them that. The interaction is not a vector; it's
something more than that. Don't confuse interaction with
force; the force is a vector, but the interaction is more
than that.
-- When you draw the interaction vector, label it χ not F.
-- Teach the students how to reconstruct the explicit lever
arms, starting from the implicit representation i.e.
starting from just the interaction arrows.

All this is discussed in more detail, with diagrams, at
https://www.av8n.com/physics/force-intro.htm#sec-interactions
and
https://www.av8n.com/physics/force-intro.htm#sec-more-interactions
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