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

I think I am missing the point.

A force is a vector. But a rope is not. And a "lever-arm" is a vector. But a beam is not.

When a rope pulls on a beam, we define the force vector with magnitude and direction. But the force vector is not a physical entity. It is a mathematical abstraction. And it does not have location. When we are working with those abstractions, we find it convenient to move them around, such as when adding them tip-to-tail. We can move the force but we are not moving the rope. We are moving them in an abstract vector space, not the real world. [Just this week, I have been teaching first year students how to take the forces on a free-body diagram and and add them tip-to-tail.]

Similarly, lever-arm is a vector. We can define it as the displacement vector from the chosen pivot axis to the point of application of the _rope_ (not "the force", the rope). Moving the real-world rope will change the lever arm vector. Moving the abstract (vector-space) force will not.

Then, we can define torque = r x F = |r| times |F| times sine of angle between them. Go ahead and move F or r wherever you want...torque won't change.

On 10/23/2015 5:08 PM, John Denker wrote:
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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