High School Mathematics-Physics SMILE Meeting
25 February 2003
Notes Prepared by Porter Johnson

Don Kanner [Lane Tech HS, Physics]     The Search for the Missing Jug Equation
began by describing an assignment for his class to find web-based information on the collapse of the Tacoma Narrows Bridge in November 1940.  One student discovered that the frequency of vibration of the bridge was 30 Hz, so that the torsional wave might more appropriately be described as a "hummer", rather than a Galloping Gertie: http://www.ketchum.org/tacomacollapse.html.  Also, at about the same time that Professor Faquharson was running down the nodal line on one side, Leonard Coatsworth, a Tacoma newspaper reporter was running along the edge on the other side --- and hanging on for dear life.  The wind-induced oscillation was described in one place in terms of "negative damping", and an "aerodynamically assisted self-excitation", which they insisted that it was not a "resonance", as such: http://www.ketchum.org/wind.html.  For additional information see the NASA Report From Bridges and Rockets: Lessons for Software Engineering by C Michael Holloway: http://shemesh.larc.nasa.gov/fm/papers/Holloway-Bridges-Rockets.pdf.

Don then took out an empty 5 liter wine jug, blew across the lip of the jug, and measured the frequency f as D#: 75 Hz on his music note audio detector.  He computed the wavelength for this note using the velocity of sound (c    »  350 meters/second) by  l  = c / f = 350 / 75 = 4.68 m. Don asked how this could possibly be correct, in light of the fact that the lowest resonant frequency for a long, thin tube of length L, enclosed at one end, corresponds to a quarter wavelength; i.e. l / 4 = L.  There are higher harmonics in vibrations of, say, an organ pipe, but not here!  Don then opened a half-liter soft drink bottle, drank from it until the fluid level was down to the major diameter of the bottle, and blew across the bottle, obtaining the frequency  A: 440Hz, corresponding to a wavelength  l  = c / f = 350 / 440 = 0.80 m.  Again Don was puzzled, in light of the fact that the height of the air inside the bottle was only 7 cm = 0.07 m.  He continued to experiment with an Erlenmeyer flask (straight sides) and a Florence Flask (curved sides), finding that the frequencies were different, under similar conditions.  How come?  Larry Alofs [Kenwood HS, Physics] wisely suggested that the devices in question were called Helmholtz Resonators, which would provide a key for determining the Jug Equation, or the equation for the vibrational frequency of the jug in terms of its size and shape.

For additional information, see the Helmholtz Resonance website of Professor Joe Wolfe of the Musical Acoustics Group, University of New South Wales, Sydney Australia: http://www.phys.unsw.edu.au/~jw/Helmholtz.html.  The following information has been extracted from that source:

"A Helmholtz resonator or Helmholtz oscillator is a container of gas (usually air) with an open hole (or neck or port). A volume of air in and near the open hole vibrates because of the 'springiness' of the air inside. A common example is an empty bottle: the air inside vibrates when you blow across the top, as shown in the diagram at left. (It's a fun experiment, because of the surprisingly low and loud sound that results.)

Some small whistles are Helmholtz oscillators. The air in the body of a guitar acts almost like a Helmholtz resonator. An ocarina is a slightly more complicated example. Loudspeaker enclosures often use the Helmholtz resonance of the enclosure to boost the low frequency response.

The vibration here is due to the 'springiness' of air: when you compress it, its pressure increases and it tends to expand back to its original volume. Consider a 'lump' of air at the neck of the bottle. The air jet can force this lump of air a little way down the neck, thereby compressing the air inside. That pressure now drives the 'lump' of air out but, when it gets to its original position, its momentum takes it on outside the body a small distance. This rarifies the air inside the body, which then sucks the 'lump' of air back in. It can thus vibrate like a mass on a spring. The jet of air from your lips is capable of deflecting alternately into the bottle and outside, and that provides the power to keep the oscillation going."

In addition, the Eric W Weisstein / Wolfram Research website on the Helmholtz Resonator ( http://scienceworld.wolfram.com/physics/HelmholtzResonator.html) contains the following formula for the angular frequency w= 2 p f of resonance:
w = c [ A / (V ´ L) ]1/2
Here the frequency is expressed in terms of c---  the velocity of sound, A --- the cross-sectional area of the column, L --- the height of the vertical column, and V --- the volume of air inside the container.

Don admitted that he played the jug [and other improvised instruments] in the Windy City Jammers Group, which plays fine music upon request.  Although you played well, you should not quit your day job, Don.  Interesting and Thought-Provoking!

Roy Coleman [Morgan Park HS, Physics]      Replacement of Mercury Thermometers
Roy explained that he had taken all the mercury thermometers in the physics laboratory to central headquarters, where he gave them to somebody in a hazardous waste disposal costume, in exchange for new thermometers that did not contain mercury.  When he read the documentary information, he learned that these thermometers contained toluene, kerosene, aniline compounds, and other Volatile Organic Compounds, which posed a potentially significant carcinogenic and hazardous risk.  Roy expected to receive thermometers containing a colored alcohol-water mixture, but because of some cross-up in the order, he had received the wrong thermometers. 

Roy also called attention to the next two meetings of the Chicago Section of the American Association of Physics Teachers http://orion.neiu.edu/~pjdolan/CSAAPT.html:

Very interesting, as always, Roy!

Walter McDonald [CPS Substitute -- VA Hospital Technician]      Approximation of Functions
first passed around an article by Jim Ritter [Health Reporter] in the Metro Section of the Chicago Tribune of Friday 13 December 2002, which described the use of magnetic fields to guide the course of a catheter with a magnetic tip, which is used in connection surgery on the brain.  The catheter is fed through the femoral artery close to the surface in the groin area, and then magnetically guided through the large blood vessels to the region of interest.  Its course is tracked with X-rays that are taken every 3 seconds during surgery.  This catheter can reach 85-95 % of the brain , in contrast to conventional catheters that can only reach 60 % of the brain.  The treatment shows promise in treating blood clots, strokes, aneurisms, brain tumors, epilepsy, Parkinson's disease, and other brain disorders, with surgery that avoids the need for drilling holes in the skull.  It also shows promise in opening clogged arteries in the heart.

Walter next showed us how to do Taylor Series polynomial expansions of trigonometric functions on his HP 48GX Programmable Calculator.  The idea is to truncate the expansion of, say, sin x in powers of x after a few terms, and to plot the results on the calculator.  The expansion:

sin x = x - x3 / (1 ´ 2 ´ 3) + x5 / (1 ´ 2 ´ 3 ´ 4 ´ 5) - ...
requires about 10 terms for accurate representation of the graph of sin x vs x over one period -p < x < p. He obtained similar results for the series involving cos x:
cos x = 1 - x2 / (1 ´ 2) + x4 / (1 ´ 2 ´ 3 ´ 4) - ...
He found that the series for the functions [ tan x,  sec x, tan-1x, sin-1x, cos1x ] converged only at small x, and that the functions cot x and csc x did not possess series expansions.  Porter Johnson suggested expanding the functions (x cot x and x csc x)  in the latter cases, and then dividing by x at the end.  This feature of  automated Taylor series expansions is not present on the TI 83 or TI 86 Programmable Calculators, but does occur on the HP 48.

Fascinating, Walter!

Bill Shanks [Joliet Central, Retired]      Knock Your Eyes Out
began by putting five colored sheets of construction paper onto the white board in the classroom:

First he shined a powerful spotlight on the red sheet, and we looked steadily at that sheet. After 30 seconds or so, we began to see a greenish halo around the sheet, and then it seemed as though the color in the central region of the red sheet became less intense. We shifted our gaze to somewhere else on the white board, and began to see cyan, the color complementary to red.  We repeated this procedure for the other sheets, and saw the following after-image colors:
Color  Red  Green Yellow Orange Blue
After-image   Cyan   Magenta Deep blue Blue Yellow-orange
Why is this happening to us?  In the original theory of color vision developed by Hermann Helmholtz, there were 3 color receptors in the eye, corresponding to the primary colors (red, green, blue).  When you look at, say, an intense red field of view, the red color receptors become fatigued --- unable to restore the chemicals necessary to "see red" in the retinal area.  As a consequence, the interior red field becomes less intense.  When we look at the white board, our red receptors are temporarily out of commission, so we see the complementary color cyan, a mixture of blue and green.  According to the modern theories of color perception, these three types of color receptors accept a range of colors (wavelengths), but are most sensitive at (red, green, blue), respectively.  For details see Light Science:  Physics and the Visual Arts by Thomas D Rossing and Christopher J Chiaverina [Springer 1999] ISBN 0-387-98829-0.

We are dazzled by your brilliance and great ideas, Bill!

Michelle Gattuso [Sandburg HS, Orland Park, Physics]      Kinetic and Potential Energy / handout
showed us a laboratory experiment that involved attaching special tape to a ball of mass m. The tape passed through a spark timer, and when the ball was released from rest, a record of its motion was made.  She used the Nakamura Electronic Spark Timer, which is listed at item P1-180 for $112 in a recent Arbor Scientific Catalog; see their website, http://www.arborsci.com. The timer operates at two settings, 60 Hz and 10 Hz.  According to Arlyn van Ek, there seemed to be considerably less friction than with the older timers containing carbon paper. When the ball is released from rest at an initial height H, its velocity v at height h should satisfy the condition of conservation of mechanical energy:

Etotal = P E + K E
m g H + 0 = m g h + 1/2 m v2
Since both sides of the second equation are independently measurable, energy conservation can be tested. [A variation of the experiment is to determine g, the acceleration due to gravity, with this apparatus.]  Here is a summary of her Procedure

  1. Attach the timer tape to the object.  Place two lab stools on top of each other and then place them on top of the lab table.  Measure the height and record this value in your data table.
  2. We know the time between sparks to be 1/60 second (it sparks 60 times every second).  By measuring the distance on the tape between spots produced by the sparks, we can calculate the average speed vavg = Dx/ Dt, since the object travels a distance Dx in time Dt.  The average velocity between two points in  vavg= (v1 + v2)/2, where v1 and  v2 are the initial and final velocities, respectively, over the interval.  Since the initial velocity on the first time interval was zero, we can calculate the velocity at every spark location.
  3. Record the distance fallen (by reading the position on the tape) in your data table, including the units.  Now record the speed of the object after the object has fallen this distance.
  4. Using your object and the speeds you have calculated, find the Kinetic Energy K E = 1/2 mv2 at each height, and record it in the data table.
  5. Record the distance between the object in the ground, h, by subtracting the initial height H from the distance the object has fallen.
  6. Recall that the stored potential energy is P E = m g h.  Calculate this stored gravitational potential energy for each height h, and record that potential energy in the data table.
  7. Finally, draw an energy versus time graph, noting the potential energy, kinetic energy, and their sum--  the total energy -- at each data point.  Is the total energy conserved?

You dropped the ball, but didn't drop the ball, Michelle!  Great job!

Monica Seelman [St James School]      Surface Tension with Cheerios
has always enjoyed eating Cheerios™ cereal for breakfast, and was particularly fascinated by the fact that these pressed toroidal cereal pieces tend to clump while floating on milk. How come?  At Monica's invitation, in groups of 2, we put some milk into a bowl and began to add a few Cheerios, which floated on the surface.  Monica had expressed some concern that she had only been able to get 2% milk, versus her usual skim milk at breakfast, and wondered how it would work.  We found that it worked very well, and that it worked at least as well, and possibly better, with water.  The cereal pieces floated on the surface until they came close, and then seemed to stick together along their edges.  Presumably, the surface energy, which is proportion the surface perimeter between cereal and fluid, is reduced by having the cereal pieces to adhere. The same principles apply to adhesion of algae in a pond, clotting of blood, etc.

Very interesting --- even though you haven't been eating your Wheaties™, Monica!

We ran out of time before Carl Martikean (you conduit yourself), Ann Brandon, and Arlyn van Ek could make their presentations.  They will go first at our next meeting, 11 March 2003!

Notes prepared by Porter Johnson