Don Kanner [Lane Tech HS, Physics] The
the Missing Jug Equation
Don 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
Acoustics Group, University of New South Wales, Sydney Australia:
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.)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:
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."
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,
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
Approximation of Functions
Walter 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:
Bill Shanks [Joliet Central,
Your Eyes Out
Bill 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:
We are dazzled by your brilliance and great ideas, Bill!
Michelle Gattuso [Sandburg HS, Orland Park,
Physics] Kinetic and Potential Energy /
Michelle 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:
You dropped the ball, but didn't drop the ball, Michelle! Great job!
Monica Seelman [St James School]
Tension with Cheerios
Monica 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