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Suppose the P-V diagram for a solid is plotted, with P along the x
axis. Most graph readers would probably say that it is a "weak"
relation--V changes very little when P is changed. But show the same
relation with V along the x axis and most graph readers would probably
say it is a "strong" relation-- P changes a lot when V is changed.
Is my prediction correct? Perhaps someone will answer this question in
a large auditorium, for example by showing two plots and asking which
relation is strong and which relation is weak. How many will say "they
are equally weak," or something equivalent?
Ludwik
================================================
On Jan 12, 2012, at 11:39 PM, John Denker wrote:
On 01/12/2012 08:42 PM, Jeff Bigler wrote:
Part of the reason I don't personally need a mnemonic for which
axis is
which is because pretty much every plot I've ever worked with over
several decades depicts y as a function of x and not the other way
around.
However, personal experience isn't the sole determining factor.
I find this list hugely valuable because I can learn from other
folks who have had different experiences ... or who had similar
experiences but noticed things that I didn't.
*) A world-line plotted on a conventional spacetime diagram displays
the
independent variable vertically and the dependent variable
horizontally.
*) On a conventional PV diagram, the independent variable is
horizontal
_if_ you think volume is the independent variable. Partly in jest I
suggest that physicists like to hold the volume constant and let the
pressure do what it will ... whereas chemists and biologists like to
hold the pressure constant and let the volume do what it will, in
which case it is conventional to put the independent variable on the
vertical axis.
*) In many cases, x is a function of y _and vice versa_. Certainly
this
is true for any linear relationship. It's also true for the P versus
V relationship in an ideal gas. Et cetera.
*) In some cases, neither the horizontal variable nor the vertical
variable is a function of the other. The _power curve_ example
previously mentioned is an important real-world example.
If you tell students that y "must" be a function of x, you are
making my job harder. I have had students tell me that my power
curve diagram is "impossible" or "illegal" because y is not a
function of x.
*) If you want a more introductory pedagogical example that is
not a function of either axis-variable, have students plot sin(θ)
versus cos(θ), for all θ from -1.1 π to +0.75 π.
Then have them plot θ sin(θ) versus θ cos(θ) for all θ from 0 to
10. These examples (and dozens more) are rather widely known
in math classes at the high school level. Sometimes even middle
school.
*) On a magnetic hysteresis loop diagram, the horizontal axis is
"considered" the independent variable, and the vertical axis is
"considered" the dependent variable, but neither is a _function_
of the other.
*) On a root-locus plot, as used in electrical engineering and
elsewhere, it is verrry common to find that neither Re(z) nor
Im(z) is a function of the other.
Conventions may be broken, but it is generally unwise to do so
capriciously or out of ignorance.
I don't think the examples itemized above are based on caprice or
ignorance.
Sometimes the physics demands that both x and y are functions of
each other. Sometimes the physics demands the neither x nor y is
a function of the other. Sometimes convention calls for plotting
the independent variable vertically.
At some point, the teacher needs to ask what the real goal should
be. Is the goal for students to do well in school ... or is the
goal for students to do well in the rest of their life? It may be
that every graph in the textbook plots dependent-y as a *function*
of independent-x ... but the real world plays by very different
rules.
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Forum for Physics Educators
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=
======================================================================
Ludwik Kowalski, whose profile is at:
http://pages.csam.montclair.edu/~kowalski/my_profile.html
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Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l