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Sans Work, part 1 (Long and wordy)



The following is (part 1 of) a review for introductory mechanics
teachers of a useful corollary of Newton's laws which I will call the
Mechanical Energy Theorem (MET). A conscious motivation will be to
contrast the MET with the First Law of Thermodynamics (FLT). In the
spirit of introductory texts and lectures I may do violence to
completeness and strict rigor; I welcome correction and amplification by
others. I promise to NEVER use the word "work":

The MET can be applied to the translational motion of the center of mass
(CM) of a single particle, a rigid body, a gas, a liquid, or any system
of particles or objects. For example, it can be applied to a system
defined as the collection: a taxi cab in NY city, a horse in Wyoming, a
commuter train in California, a mosquito in the Everglades, an airplane
flying over Moscow and the planet Uranus. The system of interest is
simply defined by enumerating the particles and/or objects to be
included; the rest of the universe constitutes the "environment" for the
defined system. Mechanical interaction between the system and the
environment is modeled by Newtonian forces represented by vectors. At
every instant of time these forces can be summed vectorialy to obtain
the "net external force" on the system. (Even though these forces act
on different particles and objects, perhaps widely separated, the vector
sum is performed in the usual manner, just as if one were combining the
forces acting on a single particle.) Internal forces (interactions
between system members) need not be included; by Newton's third law they
would contribute zero to the force sum. Also, at every instant of time
the position of the CM of our system can be calculated and its
trajectory in space can be defined.

The MET states that the line integral of the net external force over the
space trajectory of the system CM yields a scalar quantity (with sign)
which is numerically equal to the change in the quantity .5 M V^2,
calculated at the beginning and end of the trajectory (M is the total
mass of the system and V is the speed of the CM). Mathematically: the
integral of Fnet (dot) dRcm from point 1 to point 2 equals (.5MV^2)2
minus (.5MV^2)1. This theorem may be applied over any part, or all, of
the CM trajectory; of course, the calculations are to be performed in an
inertial frame. Within Newtonian Mechanics, there is no exception to
this theorem. It is conventional to refer to the quantity .5MV^2 as the
CM kinetic energy of the system.

If one already knows about energy conservation and the transfer of
energy between objects (who doesn't?), it may be tempting to leap to the
conclusion that the MET implies a transfer of energy from the agent(s)
of Fnet to the system; ie., that these agents are the sources (or sinks)
of the system kinetic energy change. This may or may not be so; in
either case the MET says nothing on the subject. For example, when an
ice skater pushes off from a wall and thereby gains speed, the MET
states that the line integral of the force of the wall on the skater is
numerically equal to the skater's kinetic energy increase. However, the
wall has not given up any energy to the person; the wall has not lost
any energy; the source of the skater's kinetic energy increase is body
metabolism acting through body muscle forces. The wall provides a
"leverage fulcrum" against which these forces can operate; in so doing
the wall force takes measure (through its line integral) of the kinetic
energy change. In sum, the MET uses the line integral of the wall force
as a measurement of the skater's kinetic energy change. It says nothing
about the source of this energy. It is the FLT which performs this
function.

The FLT does something the MET does not; it identifies and classifies
the environmental interactions responsible for the system energy
changes. More to come . . .

-Bob
--

Bob Sciamanda sciamanda@edinboro.edu
Dept of Physics sciamanda@worldnet.att.net
Edinboro Univ of PA http://www.edinboro.edu/~sciamanda/home.html
Edinboro, PA (814)838-7185

"To me it seems as if many of those who are discussing this question of
the conservation of [energy] are plunging into the fog of mysticism."
-Written in 1858 by William Barton Rogers, founder of MIT.