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Units and Dimensions




Without thinking about it more deeply, I have assumed the the torque/work
case implies that there is a dimensional difference between "parallel
metres" and "perpendicular metres", i.e. between the result of a dot product
and of a cross product. Can quantities have the same dimensions and yet be
completely different physically?

Mark

Yes. The present example is a good one. For a related example which
will give you some reason to think about such things I recommend
reading the first chapter in "Spacetime Physics" by Edwin F. Taylor
and Jonathan Archibald Wheeler. It is called "The Parable of the
Surveyors". The fact that the dimensions of torque and energy are
the same is an artifact only of the conventional manner in which
these quantities were separately introduced into physics.

Leigh


Well, I'm not convinced. I know the Parable of the Surveyors: the story line
first deals with using different units for the different axes, along with
coordinate systems rotated wrt each other. There are no dimensional issues
here. Then Taylor & Wheeler introduce the idea of Spacetime. Since we are
doing Relativity and now understand that Space and Time are wondrously fused
into Spacetime, we can swallow (maybe uncomfortably) the business of
measuring distances in seconds. The point is that here we have a theoretical
basis for saying that what we thought were two different physical quantities
are aspects of the same one.

Now nobody is saying that torque and work are really the same: it's just a
banal matter of radial vs circumferential distances. The other similar case
that I know of, i.e. angular momentum vs energy x time, as in dimensions of
the Planck constant, has the same origin. Otherwise, there seems to be a
clear relationship between dimensionality and physical quantity. So why
don't we just legislate away this inconsistency, as John Mallinckrodt
suggests, so that I can sleep peacefully?

Mark.

Mark Sylvester
UWCAd, Duino, Trieste, Italy.