Re: Newton's 2nd Law Lab

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Concerning Joel Rauber's comments:
> My comment/questioned (hopefully designed to get some response) is this:  It
> seems to me that the situation can be regarded as intrinsically one
> dimensional.  After all, if I try to solve the problem using lagrangian
> mechanics I need only *one* generalized coordinate and can therefore regard
> the background manifold as one dimensional.
>   ...
> In what sense is it legitimate for me to say that the problem is therefore a
> one-dimension problem.  This isn't unrelated to saying the problem is
> intrinsically two dimensional; rather than three, since we know the motion
> is confined to a plane.

Of course it is true that the Lagrangian formulation of this problem
involves only 1 generalized coordinate, so the underlying configuration
space for the problem is 1-dimensional (and the corresponding phase space
is 2 dimensional) even though the motion of the masses in ordinary
spatial 3-space is in two different directions.  I don't think there ever
was any controversy related to the idea that the motion could be
described (and the problem solved) in terms of a single dynamical
variable.  Rather, the contested question related to whether or not the
motion of the composite system demonstrated N2 when contributions to the
net external force acting on the composite system such as the cart's
weight, the track's normal force and, *especially* the pulley's force are
ignored.  There are also internal forces (e.g. tape tensions) which are
also ignored which could have been used to show N2 for the separate moving
masses if they were explicitly included, but since the tape tension is not
found or measured (in the described lab), N2 cannot be shown for them
either.  N2 is formulated in *ordinary spatial 3-space* (or lower
dimensional subspaces of this when the motions of the system's constituent
parts are confined to a lower dimensional plane or line).  N2 is not
formulated as such for the generalized configuration (or phase space) of
Lagrangian (or Hamiltonian) mechanics.

There is, however, an analog of N2 in Lagrangian mechanics, it's the
Euler-Lagrange equations of motion.  But these equations do not actually
explicitly describe N2 *unless* the generalized coordinates appearing
in those equations happen to be the components of position vectors in
ordinary spatial 3-space for the individual constituent particles or are
the components of the center of mass vector of a composite assembly of
particles, *and* there are no forces of constraint acting on the system
which are explicitly suppressed by using the constraints to eliminate
the redundant generalized coordinates (which are redundant by virtue of
those constraints).  When the generalized coordinates of the Lagrangian
are all components of position vectors (in ordinary space) *and* there
are no suppressed coordinates due to constraints, then the E-L equations
*do* explicitly represent N2 for a dissipationless system.  Unfortunately,
in the case of the half-Atwood problem (as described by Rick) the tape
tension and the pulley forces are intentionally suppressed so that when
the single E/L equation is solved, the motion is described without
reference to some of the crucial force contributions needed to show N2 for
either the individual masses or for the composite system.

Anyway, since the half-Atwood setup was introduced as a lab exercise for
intro students, it doesn't make much difference to those students what a
Lagrangian formulation of the problem does or does not show, since they
(presumably) would have not been exposed to such a treatment of the
problem anyway, nor would they be in a position to understand it.

David Bowman
dbowman@georgetowncollege.edu