High School Mathematics-Physics SMILE Meeting
23 October 2001
Notes Prepared by Porter Johnson

Don Kanner (Lane Tech HS, Physics)  Inertia
Don
handed out selected portions of the authorized English translation of landmark book Philosophiae Naturalis Principia Mathematica by Sir Isaac Newton, which included explanations of the following definitions and laws: [For the Latin text see http://www.maths.tcd.ie/pub/HistMath/People/Newton/Principia/Bk1Sect1/].

Definitions
1. The quantity of matter is the measure of the same, arising from its density and bulk conjointly.
2. The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.
3. The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.
4. An impressed force is an action exerted upon a body, in order to change its state, either or rest, or of uniform motion in a right line.
5. A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.
Laws
1. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
2. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
3. To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Curiously, Newton distinguishes an innate force of matter (inertia?)  in #3 from an impressed (applied) force in #4.

Porter Johnson commented that the question of whether Newton actually discovered his laws by himself has been hotly debated over the years. Consider this excerpt from a Newton Biography:  http://www.ing.iac.es/PR/int_info/intisaac.html

Isaac Newton was born at Wolsthorpe, Lincolnshire on 25 December 1642. Born into a farming family and first educated at Grantham, Isaac Newton was sent to Trinity College, Cambridge, where as an undergraduate, he came under the influence of Cartesian philosophy. When confined to his home at Woolsthorpe by the plague between 1665 and 1666 Newton carried through work in the analysis of the physical world which has profoundly influenced the whole of modern science.

On returning to Cambridge, Newton became a Fellow of Trinity College, and was then appointed to the Lucasian Chair of mathematics in succession to Isaac Barrow. In the 1670s lectures, demonstrations and theoretical investigations in optics occupied Newton.  In 1672 he constructed the reflecting telescope today named after him, but in the early years of the 1680s correspondence with Robert Hooke re-awakened his interest in dynamics. After Edmond Halley's visit to Cambridge to encourage him in this work, Newton laid the foundations of classical mechanics in the composition of his fundamental work Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which was presented to the Royal Society in 1686, and its subsequent publication being paid for by his close friend Edmund Halley.

Consider also this excerpt from the Biography of Robert Hooke:  http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hooke.html

In 1672 Hooke attempted to prove that the Earth moves in an ellipse round the Sun and six years later proposed the inverse square law of gravitation to explain planetary motions. Hooke wrote to Newton in 1679 asking for his opinion:-

... of compounding the celestiall motions of the planetts of a direct motion by the tangent (inertial motion) and an attractive motion towards the centrall body ... my supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall...

Hooke seemed unable to give a mathematical proof of his conjectures. However he claimed priority over the inverse square law and this led to a bitter dispute with Newton who, as a consequence, removed all references to Hooke from the Principia.

For balance, look at the corresponding Newton Biography on the St Andrews website:  http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html.

Consider also the excerpt from this source:  http://physics.ucsc.edu/~michael/

Michael Nauenberg, University of California, Santa Cruz, who organized the session, presented a paper entitled, "Newton's Early Computational Method for Dynamics." He began by observing that despite considerable historical research, very little is known about how Newton developed the mathematical theory of orbital dynamics which culminated in the Principia. A letter from Newton to Hooke, written on Dec. 13, 1679, reveals that Newton had made considerable more progress in understanding central force motion than had been previously realized. In particular a careful analysis of the original diagram which appears in this letter reveals that by then Newton understood by the fundamental symmetries of orbital motion for central forces. Moreover, the text of the letter indicates that he had developed a computational method to evaluate orbital motion for arbitrary central forces. Nauenberg went on to show that the early mathematical method Newton used to solve orbital motion for general central forces in his letter to Hooke was based on the calculus of curvature which he developed in the late 1660's. In correspondence with Newton in late 1679, Hooke suggested an alternative physical approach to which Newton gave a mathematical formulation without acknowledging Hooke (later in 1686 he wrote to Halley emphatically denying that Hooke had made any important contributions). This approach led Newton immediately to the discovery of the physical basis of Kepler's area law, which remained hidden in his earlier curvature method. The new approach is described in Proposition I, Theorem I of the Principia, and constitutes the cornerstone for the geometric methods in the book.

Bill Colson (Morgan Park HS, Mathematics)  Kitty Ball
Bill
obtained the hard plastic ball from Pet Smart for around \$4.  It lights up when it is shaken, or when a force is applied to it.  It does not bounce well, but certainly lights up when accelerated.  Very nice, Bill!

Monica Seelman (ST James School)  Casting out Nines
Monica
pointed out that you can check arithmetical operations by calculating the entries modulo base 9, and then checking the arithmetical operations modulo 9.  This check on arithmetic would work on any base, but it is especially convenient using modulo 9, since you get the number mod 9 by repeatedly summing the digits.  For example, 1285 ---> 1+2+8+5=16 ---> 1+6 = 7.  She did the following sample problems

 Addition: Original Problem    Modulo 9       362        --->          2       256        --->          4       147        --->          3  ----------                 -------       765        --->          9 Subtraction: Original Problem    Modulo 9      5273        --->          8   -   189        --->           0  ----------                 -------      5084        --->          8 Multiplication: Original Problem     Modulo 9       635     --->             5    ´   24     --->         ´  6  ----------                 -------   15240     --->             3

Fred Schaal mentioned that in hexadecimal notation, in which the counting sequence is

1  2  3  4  5  6  7  8  9 A  B  C  D  E  F  10 ...

he just turned the age of 3F, and next year would become age 40.  Dream on about hex code and remember what the Beatles said [http://www2.uol.com.br/cante/lyrics/Beatles_-_When_I_am_64.htm], Fred!

Fred Schaal (Lane Tech HS, Mathematics) 9 ´ 9 Magic Squares
Fred handed out a sheet containing the following empty lattice:

Fred then asked for a start number (we chose 11), as well as an add number (we chose 17).  Next, he put 11 into the middle square on the top row.  The idea is to implement "toroidal topology" with periodic boundary conditions, and to add 17 sequentially to the 11, and the total placed one square above and to the right of the previous element.  The first few numbers are shown below:

 11 147 130 113 96 79 62 45 28

At this point we do not put 164 (147 + 17) into the location already occupied by 11; instead we put the 164 under the 147, and continue until we hit the next snag:

 11 198 147 181 130 164 113 300 96 283 266 79 62 249 45 232 28 215

once again, we proceed by putting the next number, 317, under the 300, and continue the procedure. to get

 793 980 1167 1354 11 198 385 572 759 963 1150 1337 147 181 368 555 742 776 1133 1320 130 164 351 538 725 912 946 1303 113 300 334 521 708 895 929 1116 96 283 317 504 691 878 1065 1099 1286 266 453 487 674 861 1048 1082 1269 79 436 470 657 844 1031 1218 1252 62 249 606 640 817 1014 1201 1235 45 232 419 623 810 997 1184 1371 28 215 402 589

The sum of every row, every column, every diagonal, and every set of numbers symmetrically placed about the diagonal is 6219, which is 9 multiplied by the central element, 691.  Why?

Porter Johnson suggested subtracting the start number 11 from each element, and then dividing each element by the add number 17, to obtain the following array:

 46 57 68 79 0 11 22 33 44 56 67 78 8 10 21 32 43 45 66 77 7 9 20 31 42 53 55 76 6 17 19 30 41 52 54 65 5 16 18 29 40 51 62 64 75 15 26 28 3 50 61 63 74 4 25 27 38 49 60 71 73 3 14 35 37 48 59 70 72 2 13 24 36 47 58 69 80 1 12 23 34

The sums are equal to 9 multiplied by the central number 40, or 360.  This property remains valid if you make the following operations:

• Reflect the matrix about the middle horizontal row (elements in bold).
• Reflect the matrix about the middle vertical row.
• Reflect the matrix about either diagonal.

Very interesting, Fred!

Ann Brandon (Joliet West HS, Physics) Pressure
Ann  began by showing a heavy rubber insulating pad obtained used from the local electrical power company for electrical line maintenance. Then she had cut it into a circular disk of diameter about 10 in, sheet, she had poked a hole in the middle, passed a piece of strong fishing line cord through the hole, and tied it to a heavy washer.  She placed the disk on a smooth flat object, and when she pulled up on the cord, the object was lifted, thanks to air pressure.  Since the air pressure P is about 15 lb/in2, and the cross-sectional area  A of a circle of diameter d of about 10 in is  A = p d2/4 » 80 in2, the total force available because of  air pressure  F = P A is about 1200 lb.

As an additional application of air pressure, she showed a pair of dent pullers, available at local hardware stores for about \$1.  Dent pullers work better, and they cost less than the Magdeberg Hemispheres available at science supply houses http://www.sciencekit.com/store/catalog/product.jsp?product_id=8879857.

Ann next showed the Bed of Nails Demo, showing the effects of a uniform force distributed over multiple points, and then only at one point. This apparatus, shown below, is available from the following Educational Supply house:

Tonawanda Products Inc.
653 Erie Ave
N Tonawanda, NY 14120
Phone: 716-743-2021
Fax: 716-743-2787

She blew up a balloon, and placed it under a platform held in place on a bed of nails.  Then, she placed weights on top of the platform, until the balloon burst.  She then repeated the experiment, using only one nail instead of the bed of nails.

Marilynn Stone (Lane Tech HS, Physics) Home Made System to Illustrate Circuits
Marilynn gave us the following diagram for her circuit from the last meeting,  09 October 2001, with directions: