Re: units and dimesions

[palmer@sfu.ca (Leigh Palmer)]

> Could you clarify the difference between units and dimensions for me?

The units in which a physical quantity is expressed are specific to
some system. For example in SI the unit of energy is the joule (J),
where 1 J = 1 kg m^2 s^(-2). The dimensions of a physical quantity
are the kinds of units in which the quantity may be expressed. The
dimensions of energy are M L^2 T^(-2). Dimensions are usually not
abbreviated, and units are, as in this case, where the quantity is
so common that it deserves its own unit. It would be unacceptable
to express a torque in joules, for example, even though torque and
energy have the same 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?

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