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On Sat, 15 Nov 1997, Bob Sciamanda wrote:
The use of potential energy functions will usually be restricted to
. . .
As a very useful example consider the
dumbbell system of two masses interacting through a central force
F(|R|); R is the vector locating mass 1 from mass 2 and F(|R|) depends
only on its magnitude. This could be the model for a variety of
physical systems, from a hydrogen molecule to a binary star system. We
are interested in the time behavior of the vector R, ie.; the motion of
M1 relative to M2, assuming the dumbbell system is isolated from other
forces.
R2 R1
M2<-----------------------CM----------------------------------------------------->M1
------------------------------------------------------------------------------->
R
The difficulty is that if we define M1 as our system, the force on it
depends not only on its location, but also on the location of M2, which
will not stand still for us. However:
. . .
Using Eq#2, m*R'' = F(|R|) * R / |R| , where m = (M1*M2)/(M1+M2), the
"reduced" mass. (Eq #4)
Eq (4) describes the behavior of the vector R, which locates M1 from
(the moving) M2, and says that its behavior is the same as that of a
particle of mass m under the influence of the central force F(|R|) of a
FIXED source. Obviously, the MET can use a potential energy function to
describe the behavior of R in Eq #4.
Bob,
All of the above is true, of course; it follows the exposition found in
any intermediate level textbook on mechanics. But I don't agree with the
conclusion in your last sentence because I don't see what it has to do
with the MET (or, as it is more commonly known, the "pseudowork-energy
theorem," PET.) The MET/PET talks *only* about the effects of *external*
forces on the motion of the *CM* of a system. Of course, the extremely
restricted case of an isolated system of two point particles interacting
only via a separation-dependent attractive or repulsive force is
isomorphic to the equally restricted case of a single particle subject
only to a radius-dependent central force. And since the latter case *is*
restricted enough to be usefully described by your extended MET (EMET),
one can use the results from the EMET analysis to describe the former
case. But when the MET/PET *itself* is applied to the two body system what
one finds is no "pseudowork" and no change in CMKE--i.e., 0 = 0, hardly an
illuminating result.
. . .
John
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A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
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