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Re: Work/Energy Theorem



On Tue, 11 Mar 1997, Donald E. Simanek wrote:

If a stick
of dynamite at rest blows and stuff flies all about, are we as a group
comfortable saying that no work was done? ... Or a system of two particles
connected by a compressed spring is freed??? No work???

These are not well defined questions. You need to specify the system on
which the work is to be calculated, the type of work you want to
calculate, and, for frame-dependent works, the reference frame.

Is there *any* frame you could choose in which there would be *no* work
done in these two examples?

O.K., for those who are interested, here goes ...

There are at least the following seven useful definitions of work:

W_tot = work done on system by all forces
relative to an inertial frame

W_ext = work done on system by all external forces
relative to an inertial frame (the usual work in mechanics)

W_int = work done on system by all internal forces
relative to an inertial frame

W_ps = work done on system by all forces
calculated AS IF each acts at the system CM
relative to an inertial frame (the so-called "pseudowork")

w_tot = work done on system by all forces
relative to the system CM frame

w_ext = work done on system by all external forces
relative to the system CM frame (the usual pdV work in thermo)

w_int = work done on system by all internal forces
relative to the system CM frame (= W_int)

Now consider two rigid (or point) bodies of different mass connected to
the ends of a non-massless compressed spring in "gravity-free" space.
Assume that at the start of the interval of interest the three objects are
moving with the same velocity and assume that at the end of the interval
the two bodies are noving apart relative to each other and the spring is
vibrating, the table below shows the algebraic signs (+, 0, -, or "any")
of the seven works for various subsystems during this process. I'm not
willing to guarantee that I haven't made a mistake or two in the table,
but it clearly demonstrates that the "sign of the work" depends on the
choice of system, the choice of work definition, and, for some works, the
choice of reference frame.

/--------------------SYSTEM-------------------------\
WORK one body both bodies spring entire system

W_tot (= W_ps) (= W_ext) any (= W_int)

W_ext (= W_ps) any any 0

W_int 0 0 + +

W_ps any any any 0

w_tot 0 + + or 0 +

w_ext 0 + - 0

w_int 0 0 + +

----------------------------------------------------------------
A. John Mallinckrodt email: mallinckrodt@csupomona.edu
Professor of Physics voice: 909-869-4054
Cal Poly Pomona fax: 909-869-5090
Pomona, CA 91768 office: Building 8, Room 223
web: http://www.sci.csupomona.edu/~mallinckrodt/