When Fred L first posted the boater, anchor, and boat problem
(http://lists.nau.edu/cgi-bin/wa?A2=ind0210&L=phys-l&F=&S=&P=10799),
he wrote (correctly IMO), "The expected response is that the work
should equal
the total KE aquired by the boat, boater, and anchor." Nevertheless,
in previous posts I have shown why that "expected response" is not
the correct answer if one uses the most conventional definition of
"work."
It turns out, however, that there *is* a less well-recognized
work-like quantity that *is* equal to the total KE acquired by the
boat, boater, and anchor. It is what I call the "system-specific
external work" (SSEW) and it differs from the conventional definition
of work--what I like to call the "frame-specific external work"
(FSEW) to avoid ambiguity--in a subtle but critically important way.
First let's review:
To calculate the FSEW (conventional work) done by a system, we first
identify every force that it exerts on an external system. Then, for
each such force we determine the path of its point of application
relative to some (too often only implicitly specified) inertial
reference frame. Next, for each identified force and for each
associated path, we perform the integral of the force dotted into the
infinitesimal elements of that path. The FSEW done by the system is
obtained by adding the results of all of these integrals.
It can be shown(1) that, as long as all external interactions are via
contact forces, the FSEW done by a system is equal to the opposite of
the sum of the changes in the system's internal AND bulk
translational kinetic energies (where internal energy here includes
rotational kinetic energy.) That is
FSEW = -(dU_system + dK_system)
To calculate the SSEW done by a system, we follow the same procedure
*except* that the paths of the points of application are determined
relative to the center of mass of the system. Notice that, since the
center of mass may accelerate, the integrals for the SSEW are
performed in a reference frame that is *not*, in general, inertial.
Under the same contact force only assumption, it can be shown that
the SSEW done by a system is equal to the opposite of the change in
the system's internal energy. That is
SSEW done by a system = -dU_system
(Note that the SSEW is directly connected to the change in the
internal energy of a system and is, therefore, invariant with respect
to changes of reference frame. This is in marked contrast to "work"
as it is conventionally defined and is one of the reasons that the
SSEW is such a useful quantity.)
The SSEW is not actually all that obscure. For instance, consider a
thermodynamic analysis of an gas in a container that is accelerating
while ALSO being allowed to expand. I think most of us would simply
ignore the force that is causing the bulk acceleration and calculate
the work done by the gas as the integral of PdV. In other words, we
would simply calculate the work done in the noninertial frame of the
gas itself. This is the SSEW.
It seems to me that people often use SSEW and FSEW (along with other
useful work-like quantities) interchangeably without realizing that
they are *different* quantities connected to *different* energy
changes. The unwitting use of different definitions for work
commonly explains the confusion that arises in work-energy problems
like this one.