Re: Ohne die Arbeit, part 3 (Long & wordy)

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Sat, 15 Nov 1997, Bob Sciamanda wrote:

> The use of potential energy functions will usually be restricted to
> problems involving a single particle or a single rigid body,  because
> the conservative force must be a function of the CM position only. 
> However, one can sometimes rework the model description so that the
> mathematics is the same as if the above were true, even though it was
> not true in the original model. 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:
> > From the definition of the CM,   M1*R1 + M2*R2 = 0			(Eq #1)
>
>
> By construction,  R = R1 - R2 = R1*(1 + M1/M2)  , after using Eq
> #1.         	(Eq #2)
>
> Since the CM is an inertial origin,  M1*R1'' = F(|R|) * R / |R|	; ('' =
> 2nd time derivative)  (Eq#3)
>
> 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.

On the other hand one *can* obtain the same results *directly* from other
work-energy theorems (WETs) which *do* take into account the works done by
internal forces, casting them more properly into the form of potential
energies of *interaction*.  For instance, one such WET says

                 w_ext = delta(K_int + phi)

where w_ext is the work done by all external forces (using the motions of
the points of application of the external forces wrt to the system CM),
K_int is the sum of all kinetic energies of system components (again
calculated wrt the system CM), and phi is the sum of all internal
potential energies of interaction.  One remarkable thing about this WET is
that the CM need not even be an inertial frame.  This WET, like *all* good
WETs including the MET/PET, can be applied without exception to *all*
mechanical systems. The isolated, two body system would merely be a
particularly simple case in which w_ext = 0 and there is only one
potential energy of interaction. 

John
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