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Re: [Phys-l] units, dimensions, scaling

It is interesting that you should bring up scaling in this thread. I
have been thinking about its application to processes being discussed in
the currently active [Phys-l] T dS versus dQ thread. Suppose we use as
a container for the gas discussed in that thread, a cylinder whose
height is roughly the same as its diameter D. Consider two cylinders,
one with characteristic length D and the other with characteristic
length 2D. Do a sudden compression which in each case, under thermally
insulated conditions would take the gas from the an initial state that
is the same (in terms of intensive quantities such as T, P, density,
specific entropy, etc.) for the two cylinders to a final state (after
the system comes to equilibrium) that is different from the initial
state but the same for the two cylinders. I'm assuming that we can do
our best to thermaly insulate the containers but can't get it perfect.
In both cases we wait the same amount of time before taking our
measurements on the final state. Assuming the rate at which the energy
leaks out through the walls is, for the same state of the gas,
proportional to the surface area of the cylinder, that rate scales with
D^2. But the volume of the gas scales with D^3 so the rate at which the
intensive variables are changing goes like 1/D. Hence, with similar
materials the larger we make the cylinder the closer we approach a
thermally insulated system. Perhaps by carrying out the same experiment
at several different scales we can get a handle on how much energy leaks
out through the walls and arrive at an experimental result for what the
final state would be if the system were perfectly thermally isolated.

-----Original Message-----
From: [mailto:phys-l-] On Behalf Of John Denker
Sent: Tuesday, January 19, 2010 7:43 AM
To: Forum for Physics Educators
Subject: [Phys-l] units, dimensions, scaling

On 01/18/2010 02:16 PM, Jeff Loats wrote:

On the first day of class I do a brief example to illustrate unit
(snore) and I usually spice it up ....

On a completely serious note, here are some points
that you might want to think about in connection with
units. There is a fairly natural segue from units
to dimensions, and from dimensions to scaling.

1) Units are *not* the same as dimensions. The
existence of dimensionless units such as "degrees
of arc" should suffice to prove this point.

Units and dimensions are not entirely the same,
but they're not entirely unrelated, either. If
you know the units you know the dimensions (but
not conversely).

A lot of students have misconceptions about this.

2) Sometimes dimensional analysis is presented as
if it were a law of nature ... which it is not.

Dimensional analysis should be considered a
heuristic for guessing the scaling behavior.
Like all heuristics,
-- In skilled hands, it is a way of getting
the right answer quickly.
-- In unskilled hands, it is a way of getting
the wrong answer quickly.


I mention this because scaling laws are tremendously
important. Given any discussion of units, I would
be unable to resist the temptation to segue from
units to dimensions, and then from dimensions to

Scaling laws have been central to physics since
Day One of the modern era (1638) and remain so even
now. IMHO, they are grievously underemphasized in
the typical curriculum. They are IMHO more useful
*and* easier and more age-appropriate than most of
the stuff that is in the curriculum.

For more on all this, see
and especially

Forum for Physics Educators