Re: [Phys-L] treating force as a vector ... consistently

[Richard Tarara <rtarara@saintmarys.edu>]

I'd suggest that the 'confusion' here is from using a strict 
mathematical definition of vectors that I doubt almost anyone who has 
taught the introductory level of physics even knows much less pays any 
attention to.  We treat the vector representations of the two forces 
being considered (book on earth and table on book) as distinct.  It may 
disturb the mathematician's sensibilities, but this is not the only 
place that physics fudges with the math without damage to the 
physics--at least at this level of instruction.  I know for sure that no 
student I ever taught has objected to having vectors of the same 
magnitude and direction shown applied at different locations and 
considering how little 'vector' math seems to exist in many math 
curricula, it is very unlikely to be a problem for virtually any 
student.  In other words John, your objections would introduce more 
confusion into standard physics instruction.  The momentum approach 
might well work, but I never tried it and won't have the opportunity to 
do so.

rwt

On 8/20/2016 2:55 PM, John Denker wrote:
> On 08/20/2016 11:11 AM, Philip Keller wrote:
> > But somewhere in
> > our definition of forces, we have to build in (at least implicitly) the
> > idea that a force is something that is applied to an object and that a
> > given force can only affect the object it is applied to
> We agree that's where the problem lies.
>
> We have a physical intuition and a mathematical formalism ... which are
> telling us different things!  That's nasty!  I suggest renaming one of
> them, to reduce the amount of confusion;  see below.
>
> > I'd say that two force vectors can be equal and yet distinct.
> Let's talk about that.  I recommend the idea of /equivalence relation/.
> There are lots of different equivalence relations.  For example:
>    A red square is equivalent to a red triangle with respect to color but not shape.
>    A red square is equivalent to a blue square with respect to shape but not color.
>
> An equivalence relation is by definition reflexive, symmetric, and transitive.
>     http://mathworld.wolfram.com/EquivalenceRelation.html
>     http://www.math.umaine.edu/~farlow/sec33.pdf
>
> Clearly two objects can be equivalent in one sense and inequivalent
> in another.
>
> HOWEVER things get sticky when we talk about vectors, because the
> definition of vector comes with its own equivalence relation!  Two
> vectors are the same if they have the same magnitude and direction.
>
> The representation of a vector in terms of an arrow drawn on paper
> is not an entirely faithful representation, because the arrow has a
> location as well as magnitude and direction.  This adds to the confusion.
>     https://www.av8n.com/physics/vector-intro.htm
>
> We can define a notion of /dynamical interaction/ that comprises a
> force and a point of application ... but then the dynamical interaction
> cannot be represented by a vector.
>
> > I'd say that two force vectors can be equal and yet distinct.
> It might be better to say that two dynamical interactions can have
> the same force vector but distinct points of application.
>
> Don't foist onto the vector concept a burden it cannot handle.
>
> Again:  A lot of this confusion evaporates instantly if we express
> the physics in terms of momentum conservation and momentum transfer.
>
> ===============
>
> The third law expressed in terms of equal-and-opposite force
> vectors is not a complete statement of what we know about the
> physics.  We need equal-and-opposite /torque/ as well.
>
>    You could "mostly" express the physics in terms of force vectors
>    and points of application, but it is safer and easier to use
>    force vectors and torque bivectors.
>
> The third law expressed in terms of conservation of momentum is
> not a complete statement of what we know about the physics.  We
> need to conserve /angular momentum/ as well.
>
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--
Richard Tarara
Professor Emeritus
Saint Mary's College

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