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



I'm cross-posting this from the chemistry list to the physics list,
because it ties in so nicely with the current discussion of vectors
and components. My apologies to those who receive two copies.....







On 08/14/2007 03:13 AM, M. Farooq wrote:

Let us suppose we have a diatomic molecule, with Re as equilibrium
distance between them. Now most of authors including Herzberg's
"Spectra of Diatomic Molecules" write that we give a *displacement* R
to the molecule, and thus write the Hooke's law as:

Force = - k (R - Re) = negative ---------(1)

OK.

This expression appears true for stretching because R > Re and tells
us that stretching displacement was force are in opposite directions.
But if we, compress the diatomic molecule, and while using the same
expression and knowing that R<Re, Hooke's law will finally give a
positive sign:

Force = -k(R - Re) = positive -----------(2)

OK.

implying that force and displacement are in the same direction.

No. The displacement is in the negative direction while
the force is in the positive direction. There is nothing
"same" about it.

=========================

I don't know /exactly/ why there is confusion in this case, but
I can guess.

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