[Phys-L] Re: notation for initial velocity components

[jsd at AV8N.COM (John Denker)]

   BTW I think this is a nifty discussion.  This is the sort of discussion
   that could only take place on Phys-L.  To a layperson, these might seem
   like fine points, but really it is important to sort them out.





Rauber, Joel wrote:

> OTOH, the subscript, particularly in a FBD, usually refers to the object
> causing the force, not the object receiving the force.
>
> E.g.  F_s for spring force acting on an object, or F_T for the tension
> in a rope acting on an object, etc.

Yeah, that's a good point.

Actually, this is just the tip of an iceberg.  To see the more general
problem, consider the example of Newton's cradle.  (I represent the balls
as squares partly because I'm lazy, and partly to make a point;  see below.)

       _______ _______ _______ _______ _______
      |       |       |       |       |       |
      |       |       |       |       |       |
      |   A   |   B   |   C   |   D   |   E   |
      |_______|_______|_______|_______|_______|

      +-------+-------+-------+-------+-------+-----
      0       1       2       3       4       5   x -->

I label the balls with letters A through E, and I label points on the x-axis
using numbers in the usual way.

Ball A has just swung in from left to right.  Balls B,C,D, and E were initially
at rest.  At this moment, ball A is coming to rest and ball E is about to fly
away.

It would be nice to express the force as a force-field, as a function of x.
The problem is, you cannot think of "the" force at point 1, because there
are two forces, an AB force and a BA force.  In my experience, it is a
nightmare trying to keep track of which force is which.  This is a nightmare
not just for naive students, but even for professionals.  Maybe there is a
way to do it, but I don't know how.  I also don't much care, because there
is a good way of making the whole problem go away, namely by considering
momentum instead of force.  It is very simple and very powerful to just
talk about the momentum flowing across the AB boundary ... which is the
same as the momentum-flow at point x=1.  You don't think in terms of a
force field, but rather a momentum-flow field.  It just works.

By the way, the figure shown above can be re-interpreted as a fluid
dynamics picture.  The same line of reasoning leads to the same conclusion:
you are much better off thinking about the momentum in region A, rather
than worrying about forces at the A/B boundary.  But in an elementary class,
you want to express it in terms of Newton's cradle, because you'll scare
'em if you start talking about fluid dynamics.  Halloween is upon us, but
there's a limit to how much you want to scare the students :-)



As a related point, a lot of students think the fact that F_AB and F_BA
are "equal and opposite" is the embodiment of the third law ... but it
is not!  This can be understood as follows:  The third law, when correctly
understood, is isomorphic to conservation of momentum.  The F_AB/F_BA
relationship is not.  This can be seen by considering a non-conserved
quantity, such as the number of photons.  We could have a radio station
inside ball B endlessly emitting photons, i.e. the grossest possible
non-conservation of photons.  But it would still be true that the net
number of photons crossing right-to-left across the AB boundary would
be equal-and-opposite to the net number crossing left-to-right.  It
takes a lot more than F_AB = -F_BA to express the third law.


Bottom line:  There are many situations where thinking about momentum-flow
is a huuuge help.  I'm not advocating using momentum-flow to the exclusion
of force, but I am quite sure that force should not be used to the
exclusion of momentum-flow.