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On 01/18/2010 02:16 PM, Jeff Loats wrote:
On the first day of class I do a brief example to illustrate unit conversion
(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
Example: Pendulum
Let’s do an example. Consider a simple pendulum of length l and mass m subject to a gravitational field of magnitude g. We want to know τ, the period of oscillation. We won’t be able to find the period exactly, but we can find a scaling law, as follows:
τ ∝ m^a * l^b * g^c (3)
The dimensions in this equation are:
[t] = [m]^a * [l]^b * [l]^c * [t]^−2c (4)
where a, b, and c are (for the moment) unknowns, to be determined by dimensional analysis. We have used the fact that g has dimensions of acceleration, i.e. dimensions of length per time squared. . . . etc.