High School Mathematics-Physics SMILE Meeting
09 December 2003
Notes Prepared by Porter Johnson

Don Kanner (Lane Tech HS Physics Teacher)      Helicopter Whirligigs, General Relativity, and Forces
At our last meeting John Scavo was wondering how the number of blades on a propeller related to airflow and Richard Goberville had propeller whirligigs for all of us to take and experiment with. Don had collected all of the whirligigs from the other Lane Tech teachers at SMILE and modified them in an attempt to answer John's question. Here is the data table:

Helicopter Whirligig Data

Number of Blades Mass / grams   Result
4 blades  11.3 hovers
2 blades 11.3 hits ceiling  
2 blades 5.8 hits ceiling
2 blades 4.0 hits ceiling
2 half-blades   9.5 halfway up
1 blade (balanced) 9.5 hovers
It was shown that while a four blade whirligig would only hover, the two blade reached the ceiling. With the speculation that the mass of the whirligig was the factor, a second two blade was prepared by adding mass to match with the four blade. Surprisingly, it also reached the ceiling. Thus, we have good reason to believe that two blades move more air than four blades. However, rotational speed was not controlled. Christopher Jarr, a student at Lane Tech, suggested that the whirligigs might be slipped over the shaft of a drill bit on an electric drill to produce a constant speed for each. Lane Tech Earth science teachers pointed out that hand-held anemometers were available in most of the science supply house catalogs.

Having just used Karlene Joseph's and Dan Caldwell's paper plate and marble centripetal force demonstration in his classroom, Don showed us how to illustrate a celestial object being pulled into a black hole (sparing no expense -- ha!) using a marble whirling around inside a the top of a 1 liter plastic pop bottle held vertically with its mouth pointed toward the floor. When one stops rotating the bottle, the marble continues to whirl around the inside until it falls out the mouth. In Einstein's General Theory of Relativity, gravity corresponds to a distortion (intrinsic curvature) of the space around a mass. We are thus led to following  the question: Assuming that the mouth of the bottle is analogous to a black hole, what portion of the area near the mouth of the bottle best fits Einstein's description of space-time distortion?

Now who says that you can't teach about black holes in high school? Don referred to the classic film Frames of Reference by Hume and Ivey. The following description is adapted from information on  the website of the Department of Physics and Astronomy of the University of Victoria (BC, Canada), [http://www.phys.uvic.ca]:

FRAMES OF REFERENCE (Educational Services, Inc., 1960) 25 min, snd, b.w.
Professors Patterson Hume and Donald Ivey of the University of Toronto demonstrate the behavior of a body under the force of gravity as viewed from different frames of reference and the behavior of a frictionless puck on a rotating table in the laboratory. Two excerpts from this film are also available which present the above material in a condensed form:
1. "Excerpt 1", 7 min. QA839 F7. Shows gravitational effects.
2. "Excerpt 2", 5 1/2 min. QA839 F72. Shows rotational effects.

PJ Comment:  The idea of "tunneling into the center of force" is very old.  Isaac Newton criticized the Descartes model of the solar system, pointing out that this would be precisely the outcome of that mode.  The Bohr Atomic Model was roundly criticized during the period 1915-1925, because electrons in atomic orbits would be expected to "wind down" into the center of the nucleus, in a time of about 10-9 seconds, because of the total power radiated by a free point charge q experiencing an acceleration of magnitude a. According to the Larmor formula

P = (2 k q2 a2 ) / (3 c3)
Note that the velocity of light c ~ 3 x 108 m/sec, and k ~ 9 x 109 Nt m2/Coul2 is the constant appearing in Coulomb's Law. In addition, it is a consequence of Quantum Mechanics that magnetic monopoles, if they happen to exist, will -- in effect -- gobble up both negatively and positively charged particles that come directly at them (zero angular momentum). This idea of orbiting into the center of attraction is a timeless, recurrent theme in physics!  For additional information see the History of Mathematics Website at St Andrews University [ http://www-groups.dcs.st-and.ac.uk/~history/] and especially the entry for Sir Isaac Newton, as well as Theories of Gravitation.

Finally, Don had a modified version of Gary Guzdziol's vacuum disk [mp111803.html]. By reducing the internal plastic disk's diameter, it was used as a washer for a nut on the end of an eyebolt to which a heavy cord was attached. Without the worry of string breaking, Don appeared to lift a stool with the device but left us wondering which force really lifted the stool.

Don also posed the challenge of how to get all the marbles to stay on a paper plate when the plate is rotated, as an extension of the lesson given by Karlene Joseph at the last SMILE meeting [mp111803.html]. Go for it, Don --- you're on a roll!

Leticia Rodriguez  [Peck Elementary School]       Chemical Tests
Leticia brought in a tray of materials which we were to categorize by their physical properties of form, color and texture.  She had put various types of food coloring close to these samples, so that SMILE participants (role-playing as third graders) could more easily distinguish them.  We were permitted to see, feel, hear, and smell them, after being assured that these particular materials passed relevant safety tests.  Here is a table summarizing our observations:

  Observations of Properties of Unknown Solids   
Color Description: (secret identity)
Red crystals, crunchy,
rocky, clean, clear
Yellow transparent ,solid,
dull, white, powder
Green solid, white, smooth,
dull, perfume scent
Blue crystal, solid, white,
cracking, no odor
baking soda
Orange powder, no odor corn starch
This lesson is adapted from one described on the Carolina Scientific Website [http://www.carolina.com/] on the STC Units Descriptions. Good stuff, Leticia!

Fred Schaal [Lane Tech HS, mathematics]        Unleashing Complex Numbers
extended the consideration of zeroes of quadratic functions; ax2 + bx + c = 0 , which he began at the last SMILE class.  He wrote down the quadratic formula

x± = [ -b ± Ö(b2 -4ac) ] /(2a)

and asked what happens in the case (a, b, c) = (1, 2, 3)?, In that case one obtains x = -1 ± Ö(-2).  This case, as well as many, many others, involves taking the square root of a negative number.  By adopting the notation  Ö(-1)  = i or  i2 = -1, he introduced complex numbers and wrote the answer as  x = -1 ± 2i. He then showed, using the algebra of complex numbers, that these two complex numbers satisfy the original quadratic formula:

(-1 ± 2i) 2 + 2 (-1 ± 2i) + 3 ?=? 0

1 - (± 4i ) -4 + 2 (± 2i) + 3 ?=? 0

0 = 0  

All right! So, complex numbers are not so complex, after all!  Thanks, Fred!

Porter Johnson mentioned that complex numbers were originally used merely to solve polynomial equations, after Gauss showed that every n-th order polynomial equation has n (possibly degenerate) complex roots.  Much later, a mechanical engineer named Fourier made explicit use of the Euler formula, eix = cos x + i sin x, to develop Fourier series for the specific purpose of  solving problems related to time-dependent heat flow in conductors.  The electrical engineers introduced the complex impedance of a circuit as a means of analysis of time-dependent circuit behavior. In addition, complex numbers play a special role in descriptions of electromagnetic waves through Maxwell's Equations of electromagnetism.  In 1925, the young physicists Schrödinger and Heisenberg independently developed Quantum Mechanics. For the first time, complex numbers played  a central and unavoidable role in that theory, and in virtually all subsequent theoretical developments in physics.  In effect, the central theoretical concept (wave function, probability amplitude, state of the system) cannot be measured directly, although its effects can be seen all over the universe!  For additional information see these St Andrews University History of Mathematics pages: Quadratic, cubic, and Quartic Equations and The Fundamental Theorem of Algebra.

Arlyn VanEk [Illiana Christian HS, physics]        Air Resistance of Swinging Block
Arlyn VanEk
set up a bifilar pendulum in front of us. He explained that he had done this in his classroom. He used a wooden block (with two eye screws) on its top edge for its bob, and suspended it from the ceiling using light, inelastic cord tied to the eyes screws. He then swung the bob back in an arc, keeping the cord taut to make angle to the vertical. Then he released the bob from rest, and measured the time for it to pass through a Pasco® (http://www.pasco.com) timing gate at the bottom of its swing. Knowing the width of the block, its speed could be calculated. From this kinetic energy could be calculated at the bottom of its swing, and its gravitational potential energy could be calculated by measuring the decrease in altitude from the beginning to the bottom of its swing.

Assuming air friction is nil and noting that the motion of the block should not depend upon its mass, when he did careful measurements, he found that energy was not being conserved! Thinking that this might be due to air friction, Arlyn made a second block with a streamlined shape, resembling the head of a doubled-bladed axe. Repeating the experiment with this bob, he found this time that it had more energy at the bottom of its swing than it had potential energy at the beginning! How could this be?!

Ann Brandon and Larry Alofs remarked that the separation of kinetic energy of a moving body into rotational and translational motion can be made only corresponding to a translation of the center of mass, and rotation with respect to the center of mass.  One may thus apply the principle of conservation of energy using that principle.  Correspondingly, it is crucial to be certain that the center of mass of the pendulum moves through a circular arc, and that translational speeds are measured for the motion of that center mass.  The standard arrangement for a ballistic pendulum is to suspend  a wooden block by four strings, attached on the top side on locations symmetric with respect to the front, back, left, and right edges.  One thereby makes a bifilar pendulum, with the desired properties.   PJ also mentioned that an aerodynamic shape should more properly represent an aircraft wing --- sharp in the front and smooth in the back, rather than being front-back symmetric.  He pointed out that the winning athletes in the platform ski jumps in the 2002 Winter Olympics (in addition to being anorexic) held their skis- cross-pointed at the front, to reduce air resistance, rather than in the standard railroad track position,  to reduce air resistance.

As a sequel, Arlyn showed that two steel balls collide almost elastically when one rolls into the other at rest on a smooth table.  However, if the balls smash against one another in mid-flight and stay together, all the mechanical energy must be converted into heat.  How do we know this?  Arlyn took two solid steel balls [mass of about 500 grams each; about 5 cm in diameter], and smashed them together while a sheet of ordinary paper was held between them.   It was quite plain to see that a small hole had been burned through the paper with each encounter.  Furthermore, when the experiment was repeated in a darkened room, we could see flashes of light with each collision.  Also, the smell of burnt paper was unmistakable  Remarkable!

Finally, on a non-destructive note, Arlyn held up a Thumb Drive Flash Memory Stick with a capacity of 64 MB that can be inserted into the USB plug on his fairly new computer.  He is using this small memory stick  (normally used for a digital camera) to transfer data from the school computer to his computer at home -- the hard drive recognizes it as a formatted [ROM] disc, so that files can be moved to and froNeato!

Arlyn, you showed us how it really is!  Thanks!

Fred Farnell [Lane Tech  HS, physics]     Rocket Balloons
took a long, collapsed balloon, and inflated it by inserting a special straw and blowing. Then he released it into the air.  It zoomed around the room, making a "screaming" sound.  Just for amusement and edification, he sent off several more balloons, with similar effect --- except for the one that exploded during inflation.  This is an ideal party favor, which Fred had obtained from The Party Corner®, in Orland Park Shopping Center.  It was described on the package as follows:

Flying Screaming Rocket Balloon -- Watch 'em Fly; Hear 'em Scream
36" length with blow tubes ...(choking hazard)
These balloons can also be ordered from the Rocket Balloon website: http://www.rocketballoons.com/. How would you connect this to Newton's Third Law?

Referring to his presentation at a previous SMILE meeting [mp111803.html], Fred promised that he would bring his daughter's old tennis shoes to SMILE in the near future, since she is nearly ready to donate them to us for scientific study. And, it's about time for her to wear winter shoes!

Those rockets really took off!  Thanks for showing us,  Fred.

Bill Shanks [retired physics teacher & member, Joliet Junior Chorale]        Clothes Pins that Light Up
showed us the perfect "party gag" gift --- a pack of 50 plastic clothes pins, complete with (LED) bulbs, which light when you use the clothes pin to clamp the LED leads to make contact with electrodes on a small "dime shaped" Lithium cell, such as CL 2016, 2025, or 2032, which are rated at 2.8 Volts.
PJ Comment:
A red or green LED can be lighted with a single cell, since a photon of energy 2.8 eV corresponds to a wavelength

l =  c / n =  h c /(h n) =  hc / E =  1240 eV nm / 2.8 eV =  442 nm.

How could you use this in class?

Bill, you must be the life of the party! Very nice!

Karlene Joseph [Lane Tech HS, physics]        Crash and Burn Website
showed us some video images of collisions of automobiles in which "crash dummies", as well as stunt drivers were sitting in the automobiles. The frame-by-frame sequence of images is quite fascinating.  It was clear to all of us that, in fact, Newton's Laws fully explain the occurrences during the crash.  In particular, when we saw the impact and damage when the head of the unrestrained occupant hit the dashboard, the warning "wear your seatbelts" was justified in graphic detail.  These images were located on the website of The Center for Injury Control, School of Public Health,  Emory University [http://www.sph.emory.edu/] on the Motor Vehicle Crash Video page.

Those daredevils and dummies showed how Newton's laws determine the course of collisions!  Thanks, Karlene!

 Notes taken by Porter Johnson