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



Bob,

You have a serious word wrap problem.

Again, I rest my case regarding the confusion generated by corrupting
the pristine definition of work.

OK Bob, I'll bite. What is the "pristine" definition of work and by
whom was it proclaimed to be so? Which of the meanings of "pristine"
is intended here?

Leigh

Chill, old man! I knew "pristine" would draw fire as soon as I typed
it!

Then you should have erased it. If you won't tell me what you meant,
please tell me what you meant to mean!

Seriously, please don't drag me into a semantic argument, especially since
I and the list have chewed this cabbage at length in the very recent past.
In fact a post of mine laboriously spelled out the conventionaly accepted
definition of work and the work-energy theorem.

OK then. What is the conventionally accepted definition of work?
Whose convention are you following? Please; I really have no idea
what your definition of work might be.

However . . .
To summarize, by the WE theorem I refer to the usual (dot product)
integration
of F=ma over the trajectory of the CM of a system of one or more
particles.
(F= the vector sum of all external forces acting on the particle
system.)

Does this mean that a couple applied to the system does no work?
I'm sure you don't mean that, but that is a logical consequence
of your definition, if I read it correctly. I don't like that
very much.

I have six textbooks which are devoted to classical mechanics on
my shelf here at home: French, Davis, Synge & Griffith, Goldstein,
Fowles & Cassiday, and an excellent book, "New Foundations for
Classical Mechanics" by David Hestenes. Of these only the last has
an index entry for anything like the work energy theorem. Hestenes'
book gives two versions, one which includes the internal kinetic
energies of the particles and one which does not. He calls the
first version (which allows for nonconservative internal forces)
the general work energy theorem. A version which constrains the
internal forces to be conservative is also derived. Thus the only
text I have that even mentions the theorem does not make it unique,
and neither agrees with either your version or mine.

I have one elementary textbook on my shelf here at home, Fishbane,
Gasiorowicz and Thornton. (I'm teaching from this text now. I did
not select it and I would not advise others to do so.) It has
several index entries for the work energy theorem, one of which
(called "the work-energy theorem") deals with a single "object"
(which I take to be a particle) and equates the change in kinetic
energy of the particle with the work done on it by all external
forces. I have no problem with that result, but I judge it to be
so trivial as to be unworthy of being singled out for its own
designation. F, G & T go on to define "the work-energy theorem for
rotations". Unsurprisingly this one allows couples to do work
(perhaps because that's the only way they can make it today).

It seems neither of these is the same animal as your work-energy
theorem either.

Nothing more than F=ma and mathematical logic is required to show a
numerical
equality between this "work integral" and the change in the CM kinetic
energy
of the particle system. If energy conservation were abandoned this WE
theorem
would still be valid. It is simply a re-statement of F=ma; it applies
to all
forces, frictional included; it says nothing about universal energy
conservation;
it knows about no other kinds of energy than kinetic- and this only as a
convenient numerical quantity as expressed in the theorem.
It implies nothing about a transfer of energy from the agents of F to
the particle
system. The term dw in the first law of thermodynamics is a different
animal,
both conceptually and numerically except in some carefully contrived
circumstances
(I suggest that dw in the FLT not be called work, call it adiabatic (or
non-thermal?)
energy transfer.)

Well, that sure clears up a lot, doesn't it? Are you aware that
conventionally the terms W & Q in the first law of thermodynamics are
called, respectively, heat and work?

The WE theorem knows nothing about the energy concept implicit in the
FLT.

It seems to me that the work energy theorem isn't even unique. My versions
are the first law of thermodynamics applied to adiabatic and alternatively
to adiabatic reversible processes. The latter theorem (and need I prove it
to you?) is the work energy theorem as I use it in classical mechanics.

When my version is applied to a system of particles it is equivalent to
Hestenes' second version. It also applies to systems with other mechanical
degrees of freedom, but not to systems with internal nonconservative forces.
That would be equivalent to Hestenes' "general work energy theorem" for a
system of particles. Mine's more general, that's all.

My contention is that recent tendencies to confound the WE theorem and
the FLT,
involving different uses of the word "work", are leading to needless
confusion and shedding
no light; examples abound on this list (like making the WET a part of
thermodynamics)!
(all of this has been endlessly discussed quite recently on this
list.)

It bothers me a lot to see some kinds of work, such as that done by a
couple acting on a rigid body, called something other than work. I
agree that there is potential for confusion here, but reserving the
term "work" to the pristine (meaning 1: primitive) case of the integral
of the net external force over the displacement of the center of mass
of a system of particles is unlikely to reduce that confusion.

I conclude that there is no unique work energy theorem, and I believe
that is the answer sought by the original questioner.

Finally, let me illustrate the application of these versions in a
simple example. Consider a particle of mass m to which is affixed
a massless Hooke spring of spring constant k. Starting at time t=0
I pull on the spring with a force F=Ct, where C is a constant. The
system is initally at rest.

After a time T:

(a) How much elastic potential energy will be stored in the system?

(b) What will the kinetic energy of the system be?

(c) How much work will I have done on the system?

Answers:

(a) F(T) = CT = k x where x is the spring's elongation.

1 2 1 2 2
PE = --- k x = ----- C T
2 2 k

(b) 1 2
KE = ----- p where p = m v is the momentum of the system.
2 m

Since the spring is massless, the force exerted on one end is
transmitted undiminished to the other end, where it accelerates m:

/ T / T 1 2
p(T) = | F dt = C | t dt = --- C T
/ 0 / 0 2

1 2 4
KE = ----- C T
8 m

(c) Here I will apply the first law of thermodynamics. Since the
initial kinetic and potential energies were both zero,

1 2 2 1 2 4
W = PE + KE = ----- C T + ----- C T
2 k 8 m

The definition of work done *on the particle* is the definition I
understand Bob wants to call exclusively the work, and this would
indeed be equal to KE, the change in the particle's kinetic energy.
Thus his version of the work energy theorem would be interpreted
with, I think, more difficulty than mine. Of course that is caused
solely by the introduction of the spring into the system.

Leigh