Subject:
Sans work, part 4 (finis), Long & Wordy
Date:
Thu, 20 Nov 1997 01:42:37 -0500
From:
Bob Sciamanda <sciamanda@worldnet.att.net>
To:
PHYS-L <phys-l@mailer.uwf.edu>
BCC:
sciamanda@worldnet.att.net
(This is the 2nd posting; the first seems to have got lost)
I develop here the general, three dimensional counterparts of the one
dimensional properties of the potential function U(r) discussed in part
3. I have tried to make things transparent enough for an introductory
course, but each teacher will have to judge for herself just how far
time and reality permit her to go in each class (some of the language
is perforce loose and does violence to complete rigor).
We have seen that in one dimension, any honest to goodness F(x) is
necessarily conservative. In three dimensions, the necessary and
sufficient condition for F(r) to be conservative is that the magnitude
of F(r) cannot vary in value as one moves infinitesimally in a direction
perpendicular to the direction of the vector F(r). More precisely, if
the direction of F(r) at a space point is defined to be the x direction,
then at that point dF(x,y,z)/dy = 0 , and dF(x,y,z)/dz = 0 (partial
derivatives). The drawing below will show that if this condition is not
satisfied, you can do the line integral around a closed infinitesimal
path and get a non-zero result:
F(x,y+dy,z)
-------------------->
---------->
F(x,y,z)
If F(r) behaves as shown (F points in the x direction, y is up the
page), the line integral counterclockwise around the square (dx by dy)
would yield F(x,y,z)*dx - F(x,y+dy,z)*dx . This is zero only if
F(x,y+dy,z) = F(x,y,z), or equivalently, the partial derivative
dF(x,y,z)/dy = 0. In the same way, dF(x,y,z)/dz =0 is required for a
conservative force.
In mathematical language, this requirement is worded: Curl F(r) =0.
This makes clear the physical requirement on a conservative force,
without going into the full, general machinery which generates the
(vector) Curl F(r) . The helpful paddle wheel picture of the curl might
be introduced here; ie., if F(r) in the above drawing is taken to
represent the water velocity in a stream, a paddle wheel introduced into
the stream with its axle pointing out of the page will not rotate if
Curl F(r) = 0. Not surprisingly, a vector field with zero curl is
often called an irrotational field. (Advanced students would hear about
Stokes' theorem at this point.)
To get the general, three dimensional, relation which recovers F(r)
given U(r), observe that a general vector displacement dr from some
space point r will encounter a change dU(r) in the scalar U(r). From
U(r) = - int{F(r) (dot) dr} we have dU(r) = -F(r) (dot) dr, for any
vector "step" dr. Now if dr is chosen to be completely in the x
direction, we have dU(r) = -F_x * dx. Similarly for steps in the y, z
directions the changes in U(r) are given by dU(r) = -F_y * dy and dU(r)
= -F_z * dz, respectively.
These statements are equivalent to F_x = -dU(x,y,z)/dx ; F_y =
-dU(x,y,z)/dy ; F_z = -dU(x,y,z)/dz (partial derivatives).
Conventional notation combines these three component statements into the
single vector statement F(r) = -Grad{U(r)}, "F of r equals minus the
gradient of U of r". An equivalent statement is that at any space point
r the component of the conservative force F(r) in any chosen direction
is equal to minus the space rate of change of its potential function
U(r) in that direction, or symbolically
F_s = -dU(r)/ds for any direction s.
The F(r) <==> U(r) mathematical machinery which we have developed has
analogous applications in several other fields of physics and
engineering. For example, if the scalar U(r) gives the temperature at
each space point r, there is a heat conduction equation, analogous to
F(r) = - Grad{U(r)}, which says that thermal energy will "flow" in the
direction -Grad{U(r)}, ie.; from positions of higher temperature to
positions of lower temperature. Students can more easily visualize this
thermal situation and can then be led to the "picture" of a particle
subjected to the force F(r) accelerating toward lower values of the
potential function U(r).
Addendum:
I have avoided using the word "work", not to champion any
crusade to
abolish use of that word, but simply to avoid the senseless semantic
arguments which inevitably occur when ever one asks, or answers the
question "Does such and such a force do work in such and such a
situation?" The physics can be unambiguously expressed, and such
arguments avoided, by simply avoiding use of that multi-valued word.
Thank you if you have read this far. I hope that there is some help
here for
someone.