On Tue, 19 Aug 2003 23:34:07 -0400, wrote:

Good day Richard,

You have picked an example that simply has different representations
for power. I do not believe there has been any dispute about whether
conversions between different units of power are valid; they are.

The general question is: if two things can be simplified to the same
set of units are they the same thing.
At least two counter examples have been offerred to demonstrate that
just because two things have the same units, they are not the same.

Torque is not work; though they both have N-m as their units.

If you take a solid axle, fix it at one end and twist at the other,
Torque is the plastic deformation in the form of that twist being
distributed along the length of the axle as shearing stress. That
twist allows for some rotation at the end where the rotational force
is applied and that is obviously work. I know, I've calibrated 100's
of Torque wrenches (mostly micrometer click wrenches) from 15 pound-in
to a 600 pound-ft and broke a bench doing it.

Modulus of elasticity is not stress; though they are both expressed
as Pascals (after simplification).

This has a close association with your observation above, so I will
continue with the same model. But first, the definitions that you
seem to accept, but tied into this discussion. From "University
Physics," Sears and Zemansky, containing a chapter called "Elasticity"
whose second section is titled "Stress" (the first section is titled
"Introduction").

"Stress is a force per unit area."

"Strain. ...refers to the relative change in dimensions ... subjected
to stress." As this is distance over distance, strain has no
dimension (the units cancel as has been pointed out by others).

"Elastic modulus. The ratio of a stress to the corresponding strain
is called an elastic modulus. ... Since a strain is a pure number,
the units of Young's modulus are the same as those of stress, namely,
force per unit area. Tabulated values are usually in lb/in² or
dynes/cm²."

Returning to that same axle. We score a line along its length from
free end to fixed end with a scribe that travels a path parallel to
the axis. We apply some force, hold it, and scribe a second line. We
go to the middle of its length and scribe two lines around the
circumference of the axle (a short distance apart). These last two
lines describe opposite shears due to torsion. The stress varies as a
function of depth into the axle (greater at the periphery, less in the
interior). We then examine the enclosed area which describes a twist
per length (area for the applied force - stress). The axial lines are
parallel to the compression and the circumferential lines are parallel
to the tension. If this axle were made of wood, it would fail under
compression when its elasticity was pushed beyond its limit. In
comparison, it would also exhibit a larger rotational displacement
with the same force applied to an iron axle.

The last observation is simply reduced, or normalized as described
above in the definition of modulus and what you obtain for the two
materials is either a constant force with different rotational
displacements (and different scribed areas); or the same rotational
displacement (constant scribed areas) with different applied forces.
You still have torsion, you still have stress and strain, and you
still have rotational displacement - the only difference is in the
material's characteristic which is described by the modulus.


This seems sufficient to prove that two things with the same units
are not necessarily the same.

It proves you have two different materials which is the point of a
modulus in any discipline.


It leaves open the question as to how does one know whether two
things with the same units are the same (or not); a much more
challenging problem, I suspect.

...Keith

Hi Keith,

The only question is what is different, the why follows from fairly
obvious implications of being material based. Your problem is in the
definition of the application of the terms, not their expressions.
You might want to consult a slim volume called
"Elements of Strength of Materials," Timoshenko.

You will note that this bears no relation to ohms being different,
because as you observed with the horsepower example, it is simply
flipping through translations until you hit the units you want.

73's
Richard Clark, KB7QHC