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Re: [Phys-L] defining energy



The work-energy theorem (that change in 0.5mv^2 is equal to Fd) is
typically derived from F = ma. The work-energy theorem looks like it has
something to do with energy, but at base, being derived from F = ma =
dp/dt, it really deals with momentum, not energy. Note for example that the
change in 0.5mv_x^2 is equal to the integral of Fdx, and similarly for y
and z: there are three separately valid relationships, whose sum is the
work-energy theorem. Also, with m the total mass of an arbitrarily complex
translating, vibrating, rotating, deforming system, and a the acceleration
of the center of mass of that system, and F the net force, the sum of
various forces that may have different displacements at their points of
application, the work-energy theorem is still valid. It has sometimes been
called the pseudowork-energy equation because the integral of the net force
through the displacement of the center of mass has been called pseudowork
and is not in general equal to work, and it is equal to the quantity
0.5m_sys*v_cm^2, which is called the translational kinetic energy.

For a very simple example, consider stretching a spring with equal and
opposite forces applied at the ends. The net force is zero, the
displacement of the center of mass is zero, the pseudowork is zero, and the
change in the translational kinetic energy is zero. There is however real
positive work done by both forces, both of which act through displacements
in the direction of the force, and there is an increase of internal energy
of the system (the spring).

Nevertheless, a sensible way to introduce energy is through the work-energy
theorem, derived from F = ma for a point particle or any system whose
internal energy does not change (including rotational kinetic energy). The
next step might well be identifying potential energy of pairs of
interacting objects (e.g. the falling rock and the Earth). For coherence
and consistency, one should not ascribe potential energy to the rock but
rather to the interacting pair. This comes down to being careful about the
choice of system:

System = rock: K_f - K_i = +mgh

System = rock+Earth: K_f - K_i + (U_f - U_i) = 0, where (U_f - U_i) = -mgh

If one is not careful about the choice of system issue, it's easy to
double-count, to say that the Earth does an amount of work +mgh AND to say
that there is also a decrease in U = -mgh.

(The change in the kinetic energy of the Earth is negligible, as can be
seen either through the fact that it gains the same magnitude of momentum
as the rock but very little 0.5mv^2 =0.5 p^2/m because m_Earth is huge, or
by seeing that the Earth gains a speed v = p/m that is tiny, so it moves a
very short distance, so the work done on the Earth as system is very tiny,
so the Earth's final kinetic energy is very tiny.)

After that, it seems reasonable to me (and not inconsistent with the lovely
Feynman parable) to say that the energy equation associated with energy
conservation is "change of energy of a system is equal to the net inputs to
that system" and that we've identified one example of an input, Fd, and one
example of a change in energy, change of translational kinetic energy, and
that starting in the early 1800s scientists (chemists, physicists, and
biologists) slowly came to identify many additional kinds of inputs to a
system and many additional forms of energy in a system, and that this is
NOT something that can be "derived" from Newton's laws but instead
represents a long accretion of experimental evidence.

Bruce