🚀 go-pugleaf

On Thu, 4 May 1995, Mike Macallister wrote about physics problems
requiring analysis, synthesis and insight:

> Anyhoo, I am not terribly creative at making up my own such problems like
> this.  I also have not found any good sources as yet.  If you know of any
> such source, or have pet problems like this you wouldn't mind sharing, could
> you pass it along to me?

Mike,

This a very good suggestion, and such problems should be shared with the
entire group. About a month ago I solicited ideas for such problems, ones
which would focus on concepts as opposed to *mere* number-crunching. No
one took the bait, and none were submitted.

I have a few of my own, which I can post soon (soon enough for your final
exams? :-), once I reformat them for ASCII. You'll have to imagine any
diagrams. Till then, here's one I just used in an optics exam.

Dick and Jane are lab partners. They are doing the standard spectrometer
experiment, getting a spectrum from a prism. They know the formula for
index of refraction at minimum deviation: n = sin[(A+d)/2)]/sin(A/2) where
A is the angle between the prism faces and d is the minimum deviation
angle. They are given a prism with two 45 degree angles and one 90 degree
angle, with all three faces polished. Jane wants to use two faces which
make a 45 degree angle. Dick says that they'd get a greater deviation and
greater dispersion by using the faces making 90 degrees, and therefore get
better resolution of the spectral lines. Who is right, and why?

The answer is somewhere below:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Answers:

(1) Larger angles do give greater deviation. Test 20, 45, 60, etc. Now
try 90 degrees.

(2) Larger angles give greater dispersion for the flint glass usually
used in these experiments (answer based on student experience and astute
observation).

(3) Greater dispersion and deviation do not result in greater resolution.
The Rayleigh criterion rules here, and the resolution is due to the
optics of the spectrometer, not on the prism.

(4) For a refractive index of 1.6 or 1.7 (typical, based again on student
experience with this experiment) the light wouldn't emerge from the
second face, it would have total internal reflection. Therefore useless
for the experiment (Jane is right, so long as she gave the right reason).

Note that this answer involved (1) lab experience, (2) understanding of
minimum deviation, (3) understanding of the Rayleigh criterion of
resolution, (5) some math, (5) synthesis of these, (6) ability to
brainstorm a problem, and (7) ability to express physical ideas in words.
This particular question is a bit *much* for elementary physics courses,
but does illustrate the kind of problem we seek: one which uses concepts,
insight, understanding, experience, *and* the ability to handle the math.
It's not merely 'plug and solve'.

I'm not sure the classic 'Fermi' problem is appropriate for exams, but
certainly is for homework. (I'm speaking of the "How many piano tuners
are there in New York City" type of problem.)

Another one from optics:

Someone once looked at the Van Gogh painting "Starry Night" (available in
a nice GIF on the internet) and some other of his paintings, and wondered
why he painted things in the sky (sun, moon, stars) as fuzzy, indistinct
swirls and blobs, while foreground objects, houses, trees, etc, were
painted with far more detail. He speculated that Van Gogh was nearsighted,
and did not wear glasses, therefore he'd never seen anything in the sky
clearly. But he could walk up to nearby objects and see their details, and
therefore painted them that way. Discuss.

And still another 'arty' one:

Look at some paintings by El Greco. Note how he painted people tall and
skinny. Some have suggested this is because he had uncorrected
astigmatism. Explain why this explanation is wrong. Use physics in your
discussion.

Are you ready for another? This is from Meyer-Arendt, _Introduction to
Classical and Modern Optics_ 3rd ed. Prentice-Hall, 1972.

The neo-impressionist painter Georges Seurat was a member of the
pointillist school His paintings consisted of an enormous number of
closely spaced small dots (approximately 1/10 inch) of pure pigment. The
illusion of additive color mixing was produced only in the eye of the
observer. How far from such a painting should one stand in order to
achieve the desired blending of Color?

Who was it here, Fred Bucheit, I think, who said we don't often enough
cross disciplinary boundaries in our teaching? Such boundary-crossing
questions are 'out there' if you only look for them.

The answer to these are interesting. See below, but don't peek till
you've thought about them for several days.
















The Van Gogh problem came from a textbook I've forgotten, and from an
astronomy professor who quoted it straight-faced. I'm not sure this is a
good one to pose to physics students, for there's no physics to be
brought to bear on this. If Van was nearsighted all his life, and never
wore glasses (questionable assumptions crying out for library research),
and had never seen anyone else's paintings, drawings or photographs (Were
there photographs of these in Van Gogh's time? Back to the library) then
the conjecture might carry some weight. Most likely he painted them that
way because he liked the effect. Did he *always* paint that way (back to
the library).

Even if El Greco had a severe eye defect which elongated objects
vertically, his canvas would appear to him elongated as well, and *if* he
painted things 'as he saw them' the painting would look normal to everyone
else. Besides, astigmatism in the eye doesn't cause that much size
disparity, but mainly causes a resolution disparity. Again, he probably
simply liked the startling effect.

Seurat: The human eye has a pupil diameter of about D = 2mm in bright light.
With a wavelength of 550 nm (middle of visible spectrum) the smallest
resolvable angle is 1.22(wavelength)/D = 1.153 minute of arc. (1 minute
of arc is taken as a rough rule of thumb.) The eye's focal length is
about 20mm (in normal room light) so the resolution on the retina is 6700
nm. This is roughly twice the mean spacing between the receptors on the
retina. The human eye should therefore be able to resolve two points, an
inch apart, at a distance of some 2980 inches (about 1 in 3000, or 1 inch
at 100 yard). Answer: 1 part in 3000, 3000x(1/10 inch) = 300 inches = 25
feet = 8.33 yard.

This kind of homework problem encourages library research.

I found this in Janson, H. W. _History of Art_ Prentice-Hall, 1965: "In
Seurat's later works, such as _Side Show_, the flecks (of paint) become
tiny dots of brilliant color that were supposed to merge in the
beholder's eye and produce intermediary tints more luminous than those
obtainable from pigments mixed on the palette. This procedure was
variously known as Neo-Impressionism, Pointillism, or Divisionism (the
term preferred by Seurat). The actual result, however, did not conform to
the theory. Looking at _Side Show_ from a comfortable distance (...7' to
10' for the original), we find that the mixing of colors remains
incomplete; the dots do not disappear, but remain as clearly visible as
the tesserae of a mosaic... Seurat himself must have liked this
unexpected effect..."

Do we really want to go interdisciplinary?

A colleague at another school told of his experience teaching a "Physics
for Poets" type course. He expected the non-science students in the
course to know hardly any science, or even to have any initial interest
in it. But found, to his dismay, that they had no interest in liberal
arts content either. They simply had no interest in *anything* academic.
How many of your students have even heard of El Greco or Seurat, or can
remember seeing any specific pictures by them? How many will, on their
own initiative, use the library to seek out more information? Shouldn't
we be encouraging such habits, and such breadth of interests? How many
will use mathematics (as appropriate) in an 'essay' question?

        -- Donald