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*From*: "Jeffrey Schnick" <JSchnick@Anselm.Edu>*Date*: Tue, 19 Jan 2010 09:26:25 -0500

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: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-

bounces@carnot.physics.buffalo.edu] 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 unitconversion

(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.

http://www.av8n.com/physics/dimensionless-units.htm

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.

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

http://www.av8n.com/physics/dimensional-analysis.htm

and especially

http://www.av8n.com/physics/scaling.htm

_______________________________________________

Forum for Physics Educators

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**References**:**[Phys-l] Fun/cool unit conversion example?***From:*Jeff Loats <jeff.loats@gmail.com>

**[Phys-l] units, dimensions, scaling***From:*John Denker <jsd@av8n.com>

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