Re: Choosing Coordinate Systems

[Nathaniel Davis <haphaestus@EARTHLINK.NET>]

The question of choosing a co-ordinate system for force diagrams can
sometimes be a sticky problem.  First, it has been my experience, one should
reinforce that the choice of ANY co-ordinate system is correct; but, as you
have noted, the choice of one over another sometimes makes it easier, either
conceptually for the neophyte student, or mathematically (i.e., one might
have to invoke fewer trig. functions).

You might first introduce that, historically, the choice of a co-ordinate
systems has been hotly debated.  The debate started with the
philosophical/religious question of, "What is the center of the Universe?"
According to the Aristotlean system, Earth is the logical choice.
Copernicus introduced the Sun as the center of the known universe.  (After
this, all heaven broke loose!  Copernicus was shunned, Giodarno Bruno was
burned at the stake and Galileo was put under house arrest.)  Hundreds of
years later (of long debate) mathematicians started to play with the idea of
dimensions and VECTORS (including unit vectors i, j, and k representing the
x, y, and z axis, respectively).  Although not widely credited for the
"discovery", it was Gibbs who was the first to engender the modern notion of
VECTORS.  Come to find out, it doesn't matter what co-ordinate system we use
(largely because everything is relative) just as long as the magnitude and
direction of the vector remain constant.  For more on the history of vectors
and their independence of co-ordinate systems, see "Mechanical Universe:
'Vectors'"

In my classes I give a SHORT lecture on co-ordinate systems and vectors and
then start with example problems.  The choice of co-ordinate systems depends
on the problem -- if the cardinal directions (N,S,E,W) are invoked, axis
should be co-incident with the cardinal directions (N-S, E-W).  "Up" and
"Down" are, naturally, arbitrary and subjective (This is why half of physics
textbooks define 'down' as positive; the other half 'up' as positive!)  For
box-on-an-incline problem let me suggest that one axis ALWAYS be defined
along the direction of the Normal Force for the surface in question:  in so
doing, you automatically define the perpendicular axis ALONG THE DIRECTION
OF MOTION IF THE BOX WERE TO OVERCOME THE FORCE OF FRICTION AND SLIDE.
Also, students seems to "get" better the idea of forces acting parallel to
the surface of the incline, and forces perpendicular.  (This also means one
only has to decompose the weight vector acting 'down', rather than the
Normal Force.  Students 'like' the notion that the Normal Force is always of
equal magnitude but opposite in direction to the perpendicular component of
the weight with respect to the surface.  BE CAREFUL HERE NOT TO SAY, "THE
NORMAL FORCE IS OPPOSITE BUT EQUAL TO THE PERPENDICULAR COMPONENT OF THE
WEIGHT," as this invokes verbally Newton's third although the Normal Force
and the Weight are not an action-reaction pair (one is a long-range force,
the other a contact force!)

Hope this helps...


--
Nathaniel Wayne Davis
Physics Teacher
Mountain View High School
haphaestus@earthlink.net


> When you doing force problems/examples for your students, how do you =
> explain why you choose the coordinate system you do?  I told them I p=
> ick the coordinate system that will make it easy.
>
> Actaully a more appropriate way to see this is the orientation of the=
> coordinate axis.  We are only using x-y coordinates at this point.
>
> TIna