High School Mathematics-Physics SMILE Meeting
24 February 2004
Notes Prepared by Porter Johnson

Arlyn Van Ek [Illiana Christian HS, Physics]           The Gravity Probe B Mission
Arlyn set up his laptop computer and played a DVD he had obtained about the joint NASA - Stanford University project called Gravity Probe B,  which is scheduled to be launched soon on a polar orbit about the earth. We saw beautiful graphical images describing this exciting project. In addition,  he passed around an article appearing in the November 1 2003 issue of Science News .  Here is an excerpt from that article (p 280):

"A satellite designed to test one of the more twisted predictions of Albert Einstein's general theory of relativity is finally at its launch site after 40 years of preparation. The probe will look for evidence of a gravitational effect known as frame dragging. Just as a dipper drags honey along as it twirls in a honey jar, any spinning body in space, including Earth, ought to drag some space-time along with it. That was Einstein's prediction, anyway. The effect has never been convincingly observed. ... To create gyroscopes sensitive enough to register such minute rotations, the GP-B team has crafted niobium-coated, solid quartz spheres the size of ping-pong balls.  Nowhere do these silvery orbs deviate by more than 40 atoms from perfect sphericity. In each gyroscope, one of these balls will spin at 10,000 revolutions per minute while floating weightless within a chamber."
In addition see the Stanford University website on Gravity  Probe B: [http://einstein.stanford.edu/, and, particularly, http://einstein.stanford.edu/RESOURCES/education-index.html]. 

In Einstein's theory of General Relativity, gravity is not merely a "force" of usual variety (i.e., electric, magnetic, etc). Rather, it is treated as a distortion of the fabric of space and time (metric tensor), resulting in an intrinsically curved space-time, as described by Riemann geometry, rather than Euclidean geometry. The theoretical situation is briefly summarized on the ST Andrews [UK] Mathematics History websites, on webpages Non-Euclidean Geometry [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html#67],  as well as General Relativityhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/General_relativity.html#23 --- from which the following has been excerpted:

" ... In 1907, two years after proposing the special theory of relativity, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life, namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence: ... 'we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.' ..."

Unfortunately, Einstein's theory has been notoriously difficult to test in full detail, because its effects are extremely small in the "weak" gravitational field of the earth.

A real mind-opener, Arlyn!

Carl Martikean [Crete Monee HS, Physics]           Galileo's Study of Motion (handout)
Carl pointed out that, in Galileo's time, the subject of algebra was very new to Europeans -- even the well-educated elite.  The scientists of his and Isaac Newton's time were more comfortable with geometrical arguments, since the study of Euclidean geometry was part of their education.  Thus, Galileo's analysis of Accelerated Motion [in effect, blocks sliding down inclined planes] was given in the language of geometry. Carl discussed Galileo's proof of the following proposition:

Theorem II, Proposition II
"The spaces described by a body falling down from rest with a uniformly accelerated motion are to each other as the squares of the time intervals employed in traversing these distances."
[Today we are more comfortable in describing such motions using algebra: d = vo t +1/2 a t2, with vo = 0 for starting from rest.]   Using spark timer data to describe uniform acceleration, Carl outlined the steps in analyzing the information in the style of Galileo.  We were struck with the difficulty we encountered in following Galileo's approach.  Carl also mentioned that it took only 3-4 generations for investigators to develop calculus, once they had learned about algebra.  Still, the scientists of the late renaissance considered themselves first and foremost as geometers.  When Sir Isaac Newton said that there was no royal path to learning geometry, he used the word "geometry" to mean what we call "physics" today. For more details on Galileo's life and discoveries, see the ST Andrews [UK] History of Mathematics website [http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html], and especially their biographical information on  Galileo Galilei [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Galileo.html].

Thanks for sharing this new insight, Carl.

Larry Alofs [Kenwood HS, Physics]           "Armstrong" Flashlight
Larry held up a transparent, plastic flashlight, called the "Eternal Flashlight".  When he shook it back-and-forth along its principal axis, it would generate enough energy to light up.  We could see a solenoid coil near its center, and a magnet that shuttled back-and-forth along a tunnel passing through the solenoid.  So the changing magnetic flux within the coil would generate an EMF producing a current which charged a capacitor storing energy to light the bulb.  Visit the Heartland America website http://www.HeartlandAmerica.com, search for product number 95784 --- about $15. Larry then held up an LED flashlight, purchased for about $10 on the Harbor Freight website: http://www.harborfreight.com.  The flashlight that he showed us had 3 AA batteries and 2 LEDs, The current drawn for this LED flashlight was very low, and it would operate continuously for essentially the shelf life of the batteries!

You really brightened up our day ... and our nights!  Thanks Larry!

Ann Brandon [Joliet West HS, Physics]           Figuring Physics:  Light Bulbs
Ann called our attention to the March 2004 issue of The Physics Teacher [http://scitation.aip.org/content/aapt/journal/tpt], an official publication of the AAPT: American Association of Physics Teachers:[ http://www.aapt.org]/. The monthly column Figuring Physics by Paul Hewitt contained questions about two light bulbs, A and B. They were in sockets connected in series across a DC source (battery).  Bulb A is definitely brighter than bulb B.  What happens when the positions of the two light bulbs are switched?  We took the following "straw poll":

A will be brighter        17 Votes
B will be brighter   1  Vote
No Idea   4 Votes
Ann then switched the bulbs. Behold!  Bulb A was still definitely brighter than bulb B, in agreement with Hewitt's answer page.  Why?

Ann then asked which bulb would burn brighter when they were each placed directly across the battery? Curiously enough, bulb B burned brighter than bulb A in that situation. Why? 

The result can be understood with the formulas relating the voltage V, current I, resistance R, and power P for a resistor:

P = I2 R = V2 /R
The lower wattage bulb thus corresponds to higher electrical resistance. Tres simple!

You really lit us up! Thanks, Ann.

Camille Gales [Coles Elementary School,  Math]           A question from "The Chicken from Minsk ... "
Camille passed out the following problem, which was taken from an out-of-print (?) book mentioned by Bill Colson at the High School Physics SMILE meeting of  08 September 1998  [ph090898.htm].  Here is the question:

" ... Two mathematicians, Igor and Pavel, meet on the street. 'How is your family', says Igor. 'As I recall, you have three sons, but I don't remember their ages.' 'That's easy', says Pavel.  'The product of their ages is 36.'  Igor still looks confused, so Pavel points across the street.  'See that building?  The sum of their ages is the same as the number of windows facing us.'  Igor thinks for a minute and says, 'I still don't know their ages'.  'I'm sorry, says Pavel, 'I forgot to tell you:  my oldest son has red hair'.  'Now I know their ages', he says.
Do you? ... "
Source:  The Chicken from Minsk and 99 Other Infuriatingly Challenging Brain Teasers from the Great Russian Tradition of Math and Science, by Yuri B. Chernyak and Robert M. Rose
Can anybody help Camille to solve this problem? [Note:  there is a hint in the write-up of our 08 September 1998 meeting].
Thanks, Camille.

Fred Farnell [Lane Tech, Physics]           The Dissection of the Twinkling Shoes
Fred held up his daughter's tennis shoes, the same ones that he had showed us at the 18 November 2003 SMILE meeting [mp111803.html].  The lights embedded in the shoes still flashed on and off when he struck them.. His daughter's shoes had outgrown their usefulness, and Fred was allowed to study / destroy  them to determine how they work. He polled the audience on hypotheses as to how they work. Here is the "official tally":

Mechanism Number of Votes
Switch plus battery 8
Capacitor plus battery 0
Piezoelectric device 5
Spring plus battery 4
Magnet plus coil 1
Then Fred used scissors and a pocket knife to dissect one of the shoes to find the mechanism. After some effort, he found -- near the center of a sole -- a hard, transparent, plastic box, about 3 cm (square) and 1 cm thick.  He cut the wires connecting the box to the various LED lights on the shoe, and removed the box.  We could see a small battery inside the box, along with other stuff. We ran out of time to study the box in detail.  Maybe next time?!  

You really got to the sole of the issue! Very nice, Fred.

There was not time enough for Walter McDonald to present his work on mathematics problems that involve "sailing".  He will be scheduled at the beginning of our next class, Tuesday 09 March 2004 See you there!

Notes prepared by Porter Johnson