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

[Moses Fayngold <moshfarlan@yahoo.com>]

 On Thursday, September 15, 2016 9:01 PM, Philip Keller <pkeller@holmdelschools.org> wrote:
 
> I have a question about this thread.  I am trying to puttogether some
> thoughts about force-as-vector, bound vector, etc.  


 
A synonym for "bound vector" may be "localvector". Either term identifies the same case, opposite to a free vector (https://en.wikipedia.org/wiki/Euclideanvector).


 
> A force is a vector.  It has magnitude anddirection.  I am not ready to
> ask if it has a "point of application".  (I believe that thepoint of
> application exists.  I just don't know that it is a property of the force.)


 
   A bound (local) vector hasan application point. A more general term would be a "location" or "positionpoint". The term "applicationpoint" usually relates to a force vector, which is applied to acertain point in a body or to a point-particle. We need to place a test charge q at a point r to measure the electric field vector at this point. Electricfield E and electric force F=qE, albeit intimately connected, aredifferent vectors due to factor q.  But electric field exists of its own at agiven point regardless of whether a test charge is there or not.

  In some cases, a force can also be "applied" to a vacuum point. The resultant gravity force on a doughnut is"applied" to its center of mass where there is nothing of doughnut'sstuff. In such cases, I would prefer the term "location" as moregeneral. BTW, the mathematical formalism for determining the resultant force implies that each participant vector hasits location at the corresponding element of mass. 

     
> So I guess my question is: can a force have ANY other property or
> information assigned to it other than magnitude and direction?  


 
  This is a tricky question, with the answers"Yes" or "No", depending on situation. Its formulation is alsotricky – it implicitly assumes that "property" is the same thing as "information".But they are different concepts, therefore I will consider them separately.

  1).  Focusing on the "information" aspect, my answer would be: generally, Yes, if the information is about location. But even then,there are cases when the answer may be "No". 

  Either of these answersholds for both – a vector and a scalar. Consider a scalar field, say, localtemperature distribution T(r) in a medium. Temperature T assuch has nothing to do with location r,but local temperature T(r)depends on r (answer "Yes"to Philip's question if extended to scalars). Mathematically, we have acorrespondence between set of values Tand set of points r. One is afunction of the other, hence the notation T(r). Here the information about r isvital for determining T(r), so that we could assign to each r its respective T(r) or vice versa if weconsider the inverse function (answer "Yes"). But the answer may be"No" in a special case T(r)=const. Then T is uniquely determined for all considered domain of space, and wedo not need any info about r.

  Similar situation is withthe vector field. Vector as such hasonly magnitude and direction, but in a vector field we have a correspondence E(r)(answer "Yes"). But the answer may be "No" in a specialcase E(r)=const (e.g., electric field in a semi-space on one side ofuniformly charged plane).  

  The comparison of vectorsand scalars reasonably explains the origin of assertions denying any attachmentof a vector to a specific location. In contrast to a scalar function T(r),void of any spatial visualization of scalar T,a vector E (as well as a bi-vector A^Betc.) can be visualized as a geometrical object in Euclidean space. This naturally invitesfor its parallel transport without any change in shape or orientation, so itmay be tempting to deny any possibility of its attachment to a speciallocation. This does make sense when allowed by corresponding physical reality,e.g. when E(r)=const (electric field within a parallel plate capacitor) or B(r)=const(magnetic field within a long solenoid with direct current). Then all relevant points,no matter how far separated, are mapped onto the same vector, so we can saythat a vector has no location (a free vector, the answer No to Philip'squestion). This is even more evident for an eigenvector of momentum operator inQM, as mentioned in my previous comment on this topic. 

  But in general case of an arbitrary vectorfield we do assign specific location to each relevant vector (answer "Yes"to Philip's question). This does not contradict the fact that magnitude and directionare the only properties of a vector as such. We just need additionalinformation about location to determine numericalvalues of these properties at this location. So we have some additional information assigned to thevector but not constituting its innate properties.   


 
2) Focusing on the "property" aspect. 


 
> Is the thing that associates a
> force with the object a property of the force?

  This question takes us back to the"property" aspect. And, as in the above examples, even when anexternal influence affects some properties of an object, it does not warrantincluding that influence into the list of properties. So regarding the "properties", the general answermust be "No". Even though location of a test charge determines theforce vector at this location in a given field , it is still not the defining propertyof a vector. Even though my location on Earth largely affects my well-being, myfeelings, etc., and therefore must be added to the list of factors determiningmy state, it is not the intrinsic property of my body or my personality.


Moses Fayngold,NJIT 

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