High School Mathematics-Physics SMILE Meeting
07 December 2004
Notes Prepared by Porter Johnson

Leticia Rodriguez [Peck Elementary School]           Magnetic Fishing Pole
Leticia tied one end of a string to the end of a wooden stick, and attached a small magnet to the other end.  Holding the stick, she showed how this fishing pole could be used to test materials for magnetization.  Some materials (aluminum keys, US quarters, plastic spoons, etc) are not attracted to the magnet, whereas others (steel keys, old Canadian quarters, chewing gum wrappers, etc) are attracted to it.  Students made two lists:  Magnetic Objects and Nonmagnetic Objects.  This was a very good introduction to scientific observation and "experiencing science".  Thanks, Leticia!

Bill Blunk [Joliet Central, retired and getting "mellow"]           Tactile Magnifier: Cellophane
Bill reminded us of these observations  made at the last MP SMILE meeting [mp112304.html] , concerning the apparent "bumpiness" of a glass surface when rubbed:

Mode of rubbing \ ® \  Surface: window glass   telescope lens 
   (a) finger on glass
   (b)  finger through cellophane
    --  cellophane on glass
smooth
-
bumpy
smooth
-
smooth

The manufacturer of the cleaner had claimed that bumpiness felt through the cellophane is caused by surface imperfections in the material.  Although these results seemed to confirm that point, we looked for other explanations:  imperfections in the cellophane itself, oil on our fingers, waxy yellow buildup, ...  Bill then told us that his description of the experiment had not been complete.  Actually he had very carefully wiped the telescope lens beforehand.  He posed an explanation involving surface dust on the lens.  To verify this point, he smacked two blackboard chalk erasers together, thereby scattering some chalk dust on the glass lens surface.  It felt bumpy after this -- just like ordinary glass. Aha!

Why, then, does cellophane enable us to feel the "bumpiness" of surface dust? Perhaps the dust serves as a sort of "tent pole" to raise the cellophane around it, creating a larger bump for our sense of touch to detect.  Research instruments such as the Scanning  Electron Microscope (SEM) and Scanning Tunneling Microscope (STM) [ See http://www.mos.org/sln/sem/ and ] are used to create images of sub-microscopic surface irregularities --- even down to the atomic level! 

 The role of the non-uniform response of nerve endings in the fingers and elsewhere to tactile sensations was also discussed.  It is a fact that  you cannot tickle yourselfBeing tickled  requires being surprised by another person.  In other words, it's more psychological than physical.  Then, how can ever we trust our sense of touch? And yet, we must!

Very thought provoking, Bill!

F J Schaal [Lane Tech, mathematics]           Spheres to Cubes 
Fred reminded us that a cube of side b0 has 6 square faces, a total surface area S0 = 6 b02, and total volume V0 = b03.  Let us enlarge the cube uniformly until its volume is doubled: V1 = 2V0 = b13. The length of a side is therefore equal to b1=  21/3  b0 ~ 1.27  b0.  Correspondingly, let us double the surface area S2 = 2 S0 = 6 b22 .  We obtain b2=  21/2  b0  ~ 1.41  b0 and V2 = b23 = 2.82 V0.  These results are the same as those obtained by Fred at the last MP SMILE meeting [
mp112304.html] for a solid sphere. Bill Shanks pointed out that, if  an inverted hollow cone filled halfway to the maximum height (with snow or ice cream -- pick your favorite), the volume of edible material is only 1/8 of that when it is filled to the top.  What a rip-off! Thanks for the ideas, Fred and Bill!

Ann Brandon  and Debby Lojkutz [Joliet West HS, physics]           Non-scrambled Eggs 
Ann and Debbie  held opposite ends of a fitted bed sheet so it was open and mostly spread out in a vertical plane.  From two meters away, Fred S, Benson U, and visiting student Nicole each threw a raw egg at the sheet.  None of the eggs were broken in the process.  Why not?

The answer lies in the Impulse-Momentum Theorem, which is a direct consequence of Newton's Second Law:

F = DP / Dt
... or ...
I = F Dt = m Dv
For a given mass (m) and stopping speed (Dv), a small average force results when the stopping time is large (soft landing: egg survives intact). For a small stopping time the average force must be large (hard landing: egg breaks).  In the notation of Conceptual Physics by Paul Hewitt :
F D t = F Dt = m Dv
Arlyn van Ek described a variation of this experiment, in which students throw a water balloon (a balloon filled with water) at a bed sheet.  According to Arlyn, at least one person in every class manages to break the balloon.  Now, why does that happen?

Porter Johnson described an Egg Crush video demonstration, in which an egg is placed with its long axis vertical into a crushing apparatus with heavy, strong rubber padding on the top and bottom against the egg.  The egg was easily able to stand a steady load of 10 - 20 - 30 - 40 -50 kilograms.  For visual impact, that egg was then dropped into a frying pan from a height of 30 cm --- and its shell broke into piecesPorter mentioned the Diamond Anvil [http://scienceworld.wolfram.com/chemistry/DiamondAnvilCell.html] as a tool for achieving high pressures (up to 106 atmospheres), to study the properties of materials such as solid Helium at room temperature. John Scavo called attention to the production of industrial diamonds that are of the same quality as the best natural diamonds. It is believed that natural diamonds were created over eons of time under conditions of high pressure and high temperature, deep within the earth.

Thanks, Ann and Debbie.

Karlene Joseph [Lane Tech HS  physics]           The Physics of Hopper Poppers
Karlene showed us a flexible rubber spherical segment (popper) about 3 cm in width and 1 cm high, which she had obtained recently as a party favor.  She pressed on the top of the popper so as to turn it "inside out", thus elastically "priming" it into a state of higher potential energy.  She then placed it on the table. After a few seconds, the popper spontaneously and suddenly relaxed to its original shape, jumping several meters into the air.  Then she primed it again, and placed it on the table upside down.  This time when it "jumped", it achieved a height of less than one meter.  Why the difference? There was some talk in the group about "needing a good push" off the launch pad.  To illustrate the effect, Bill Blunk primed the popper and put it on the edge of a film canister --- which was just the right size!  The launch fizzled ... Why?  These "hopper poppers" may be obtained in bulk from either the American Science and Surplus [http://www.sciplus.com/] or Oriental Trading Company [http://www.orientaltrading.com].

Good launch for a serious discussion of impulse, Karlene!

Bill Shanks [Joliet Central, happily retired]          Various Topics 
Bill first held in his hand a hexagonal socket used with a 3/8" (8 mm) square drive to fit a 14 mm spark plug. He struck the hexagonal end smartly against the palm of his hand several times.   Each time we heard a short "pop" sound with a certain pitch. Bill then asked us what pitch of sound would occur when  he hit his palm with the other end.  Our survey consisted of votes in all three categories --- lower pitch, same pitch, higher pitch.  Then he did it ---and we heard a "pop" sound of obviously higher pitch. Bill then referred to the wine jug instrument (Helmholtz Resonator) presentation made at the 25 February 2003 MP SMILE meeting by Don Kanner [mp022503.html].

At the last MP SMILE meeting [mp112304.html] Bill measured the length of a little wooden cube (a give-away), and obtained 1.27 cm (corresponding to a half-inch).  He calculated the volume of the cube, obtaining (1.27 cm)3, or a little more than 2 cm3.  We earlier had guessed that the cube was 1 cm on a side, with a volume of 1 cm3How can the volume of the cube more than double when its sides change only by a "small amount"?  To explain this, Bill put x = 1.00  and Dx = 0.27 into the expansion formula for (x + Dx)3:

(x + Dx)3 = x3 + 3 x2 D x + 3 x (D)2 + (D)3
... or ...
(1.00 +0.27)3 = 1.00 + 3(0.27) + 3(0.27)2 + (0.27)3
... or ...
(1.27)3 = 1.00 + 0.81 + 0.2187 + 0.0020 = 2.0484

Neat! Thanks, Bill!

Monica Seelman [ST James Elementary School]           Testing for Divisibility by 7: Follow-up 
Monica  reminded us of  the test described at the last MP SMILE meeting [mp112304.html] using the number 2164 --- which is not divisible by 7.  We begin by taking the last digit (4), doubling it, writing both numbers down (84), subtracting that from the test number (2164), and dropping the trailing zero.  We repeat the procedure until we cannot continue.  Is the remaining number divisible by 7?  If so, then so is the original number.  If not, then the original number is not divisible by 7.  Here is the example:

                PARTIAL               FULL   
               2  1  6  4          2  1  6  4
             -       8  4         -      8  4  
               ----------          ----------
               2  0  8             2  0  8  0 
             - 1  6  8            -1  6  8  0
               -------             ----------
                  4                   4  0  0

   4 IS NOT DIVISIBLE BY 7   --  AND NEITHER IS 400
                         STOP
Monica pointed out that the following numbers occur on the subtraction line:
Last Digit:12345 6789
Subtraction Line:    21 42 63 84 105  126 147 168 189
All numbers on the subtraction line are divisible by 7 --- in fact they are divisible by 21.   Thus, we are always subtracting a multiple of 7, even after the zeros are included. 
Ain't Mathematics Wonderful?? Thanks, Monica.

Sally Hill [Clemente HS]           Physics Catapult Project  (handout)
The following has been extracted from the handout passed out by Sally:

  1. Goal: Create a device that will launch a ball at a target with proper distance and accuracy.
  2. Competition Rules
  3. Testing Procedure
Sally brought in a winning catapult, and used it to fire a tennis ball across the room.  Stretched fabric was used to provide the potential energy needed for launch.  We found that the catapult was more powerful when a large, strong rubber band was wrapped around the pivot point of the catapult -- the tennis ball went about 8 meters across the room..

Porter Johnson reminded us of this question posed by Larry Alofs at the Math-Phys SMILE meeting of 09 November 2004 mp110904.html:

"I'm thinking of a 5 digit number.  When I put a "1" after it, the result is 3 times as large as when I put a "1" in front of it.  What is the number?"
We may represent the original five-digit number as just "x", as well as  "D:abcde" in decimal form. According to the problem we need:
(expressed in decimal form):
D:abcde1 = 3 ´ (D:1abcde)
(expressed in terms of x):
10 * x + 1 = 3 * ( 105 + x)
... or ...     7 * x = 3 * 105 - 1
solve:            x = 299999 / 7 = 42857
checking:           3 ´ 142857 = 428571
... ok! ...
Note that the solution of the algebraic equation 7 x = 299999 comes out as an integer, even though we must divide by 7. Ever in search of a mathematical quandary, Fred Schaal asked whether the problem has a solution for an N-digit number for any value other than N = 5. In that more general case, one must solve the equation
7 * x  = 3 *  10N  - 1
A unique real solution x of this algebraic equation exists for any N, but x is an integer only for certain special choices:  N = 5, 11, 17, 23, ... .  For the case N = 11, the solution is x = 42857,142857 (for emphasis, commas have been inserted every 6 decimal places), which satisfies the check 
´  142857,142857  =  428571,428571
Can you find the solution for N = 17, 23, ... ?

Equations that must be satisfied for integers are called Diophantine equations.  Our equation for x is a Diophantine equation, and Fermat's Last Theorem involves surely the most famous Diophantine equation:

x N + y N = z N
This equation has solutions for integers (x, y, z) only for N = 2.

Notes prepared by Porter Johnson