Re: Work-Energy or Work-Kinetic Energy??

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Thu, 6 Nov 1997, Timothy J. Folkerts wrote:

> I just got into a discussio with a collegue on the interpretation of 
> the work-energy theorem, and we had quite different views on this basic 
> topic.  Is it W(net) = Delta(E), or W(net) = Delta(KE)?
>
> Consider a system consisting of a single mass being lifted:
> View 1)  You do work, so energy of the system increases.  This shows up 
> as potential energy.
> View 2)  You do work, but gravity does equal negative work.  The total 
> work done is zero, so KE stays the same.
> ...

I think I will refrain (this time) from getting involved in the 
debate about what "the" work-energy theorem says.  Most long term 
members of this list have heard me express my opinion many times 
in the past and have either accepted, rejected, or ignored it.

I would like, however, to weigh in on the slippery concept of 
potential energy due to the work done "on" a system by "a" 
conservative force.  Potential energy is a property *of* a system 
and, must properly be considered as an *internal* energy that 
arises from *pairs of internal* forces, NOT *single external* 
forces.

All forces are "half interactions" and we should really speak of 
the potential energy due to conservative "interactions."  In the 
cases of the gravitational and electrostatic interactions between 
point particles, the interaction generally does work on *both* 
particles and it is the sum of those works that depends only on 
the initial and final separations of the particles.  We designate 
the interaction "conservative" as a result of this fact, and 
assign a potential energy to the interaction.  That energy belongs 
to the system consisting of *both* particles and can not sensibly 
be allocated between them.

The spring is a trickier beast.  The elementary approach is to 
model it as an object with one degree of freedom (length) and no 
mass (so that it *must* exert equal and opposite forces on 
whatever is attached to its ends.)  We then further model its 
behavior with the specification of a simple force law that depends 
only on its length and not at all, for instance, on how much it 
may have been stretched in the past, what the temperature is, etc.  
It is only within this very restrictive model that we can define 
the usual spring potential energy function.#  We (should) do so by 
noting that the total work done on *two* particle-like objects 
connected to the ends depends only on the initial and final 
lengths of the spring.  This potential energy belongs to the 
spring; it cannot be sensibly allocated between the two particles 
attached to its ends.

In intro physics we often get away with talking about the 
gravitational potential energy "of" an object because the other 
object is so massive that it barely moves in response to the 
interaction force and any work done by the interaction is done 
primarily on the single object of interest.  Technically, however, 
we need to include both masses in any system to which we assign 
gravitational potential energy.  Nevertheless, I'm not sure there 
is enough pedagogical advantage to justify the extra effort that 
is required to explain all of this.  (Two of the best intro texts 
on mechanics--one by Bruce Sherwood and the other by Randy Knight--
do so, however.)

We often do a similar thing with springs that have one end rigidly 
attached to an "immovable" object when we (carelessly) assign the 
elastic potential energy to the object attached to the free end.  
In this case, it is simple and sensible to remedy the situation by 
carefully defining the system to include the spring.

Perhaps my main point is this: Potential energy only makes sense 
in the context of an extended, nonrigid system.  If we want to 
treat potential energy properly, we must first recognize that 
internal forces *do* perform work on such a system and that this 
work, like all other work, contributes to changes in the system's 
total kinetic energy.

John

# Real springs (like all real objects) have bulk kinetic energy, 
vibrational and rotational kinetic energies, and extremely 
complicated potential energy functions that depend on the detailed 
configuration of the spring.
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 A. John Mallinckrodt      http://www.intranet.csupomona.edu/~ajm
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