Re: [Phys-l] Motion in 1D, vectors and vector components

["Jeffrey Schnick" <JSchnick@Anselm.Edu>]

> -----Original Message-----
> From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
> bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
> Sent: Tuesday, August 14, 2007 8:21 AM
> To: Forum for Physics Educators
> Subject: Re: [Phys-l] Motion in 1D, vectors and vector components
/SNIP/
> One common cause of confusion in kinematics problems in general,
> affecting students from high school to grad school inclusive, has
> to do with drawing physical vectors versus drawing basis vectors.
>
> In one dimension, suppose the basis vector points to the right,
> and suppose the physical vector points to the left.  The physical
> vector can be drawn to scale (including magnitude as well as
> direction) as an arrow pointing to the left.  The physical vector
> can be represented numerically as a negative number times the basis
> vector.
>
> This does *NOT* mean that you have a negative amount of the leftward
> arrow!  That would be double-counting the minus sign (and squaring
> the magnitude as well).  The leftward arrow is complete unto itself.
>
> Constructive suggestion:  Always label your vectors so as to make
> it clear what is a basis vector and what is not.
>
>
>
>
>
>     True story:  This was one of my first experiences in graduate
>     school:  The very first homework assignment involved a vector
>     pointing to the left.  Every first-year student diagrammed it
>     as an arrow pointing to the left, and quantified it as a negative
>     number.  The grader (a third-year grad student) marked every
>     student wrong.
>
>     We got out our torches and pitchforks and marched to the grader's
>     office:
>        http://mag.awn.com/issue9.02/9.02images/wolff09_VanHelsing-
> villager.jpg
I think it would be beneficial to physics learners in general if we
could clear up this confusion once and for all.  Consider the following
scenario: A block has a bunch of forces, all lying in one plane, acting
on it, including a force exerted on the block by a rigid strut (of
negligible mass) which lies in the plane of the forces and is
pin-connected (frictionless pins perpendicular to the plane of the
forces) on one end to the block and on the other end to something else.
Other than the pins, nothing is exerting a force on the strut.  The
orientation of the strut is specified in the problem statement.  The
question is, what is the force exerted on the block (plus the pin on the
block, which is considered to be part of the block) by the strut?
Enough information about the other forces is provided to make the
problem solvable.  Each student that solves the problem draws a free
body diagram using a point of view in which the strut extends up and to
the right at the specified angle above the horizontal.  The students
draw an arrow, with the tail on the pin of the block, extending to the
right and upward away from the block at the specified angle above the
horizontal.  They label the arrow F_s (without any vector notation such
as a little arrow over it).  Upon solving the problem they arrive at F_s
= -15 N.  As it turns out, the strut is indeed in compression and it is
indeed exerting a force of magnitude 15 N on the block.  The teacher
wants to clear up the issue raised above.  Before addressing it class,
the teacher wants to get an idea of what the students were thinking when
they solved the problem.  The teacher calls in four students one by one
and asks each one what the arrow labeled F_s in the diagram represents
and what F_s represents.

/Student 1/ explains that the arrow is a physical vector representing
the force being exerted on the block by the strut.  F_s represents the
magnitude of that vector.  The negative value for the magnitude means
the same thing as a vector in the opposite direction with a magnitude of
|F_s|.

/Student 2/ explains that the arrow is her/his way of defining a
one-dimensional coordinate system.  Any vector in that one dimensional
coordinate system can be represented by a number with units.  F_s is a
vector in that one dimensional coordinate system.  A positive value for
F_s means that F_s is in the same direction as the direction in which
the arrow is pointing.  A negative value means that the one dimensional
vector F_s is in the direction opposite that in which the arrow is
pointing.

/Student 3/ explains that the arrow represents one axis of a
three-dimensional coordinate system.  F_s represents a number-with-units
that is the component of the strut force, along that axis.  A positive
value for F_s means that the strut force component vector along that
axis (which happens to be the entire vector) is pointing in the same
direction as the positive direction for the coordinate axis itself.  A
negative value means that it is pointing in the opposite direction.

/Student 4/ explains that the arrow is a basis vector a.k.a. a unit
vector.  The strut force is F_s times the basis vector.  A negative
value for F_s means that the strut force is in the direction opposite
that of the basis vector.

Further discussion with the students reveals that the other arrows on
the free body diagram represented physical vectors labeled with their
magnitudes (which are all positive).

Now it is time for the teacher to prepare a mini lecture for the entire
class.  If you were the teacher, besides the obvious statement about the
magnitude of a vector being positive by definition, what would you say
to the students?

If you choose to designate the strut-force-related arrow as a basis
vector, how would you indicate on the diagram that it is a basis vector
rather than a physical vector?  Label it "basis vector"?  (Does one then
also need to label each physical vector "physical vector"?)  Or would
you use two parallel lines ==== to depicted the shaft of a basis arrow
vector rather than the single line ---- that you use to depict the shaft
of a physical vector?  Where would you put the "F_s"?