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Using Math Symmetry Operations to Solve a Problem in Elementary
PhysicsSubmitted to The Physics Teacher as:
“Applying Symmetry and Invariance to a Problem with Two Parallel Current
Carrying Wires”Sandy Rosas and Marc Frodyma
San Jose City College
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Symmetry Operations: A Quantity of Interest is Left Invariant by the
Operation
Operations leaving the roots of equations invariant:
1: Linear Equations:
Ax + B = 0
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Apply a Translation and Magnification:
Make the Substitution for x:
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Substitute x’ = 0 Into the Transformation Equation
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2: Quadratic Equations:
or
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Translation Eliminates Linear Term:
With This Substitution, Equation Becomes:
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Solution to Transformed Equation:
Undo the Translation:
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Scipione del Ferro (1465 – 1526)
Niccolo Tartaglia (1499 - 1557)
Girolamo Cardano (1501 - 1576) .
A good secondary source:Stillwell, J. (1994). Mathematics and Its History, New York, NY: Springer-Verlag,p54.
3. Cubic Equations:
Who Gets the Credit?
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Cubic Equation:
Creative Step: Do a Translation to Eliminate Quadratic Term:
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New Equation:
Coefficients p, q Combinations Of A, B, and C
Good Algebra Exercise for Students!
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Results for p and q:
and
Check My Algebra!
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Change of Variables on Left Side!
So we have:
Algebra!
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Recall x’ = u + v
We Claim:
and
Why! Hint: x’ is Arbitrary!
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Eliminate v Between Equations for p and qGet a Quadratic in u Cubed
Solution:
More Good Algebra for Students!
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But We Had:
and
Top Equation Symmetric With Respect to Interchanging u and v
So u and v Have Same Solutions!
with
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Solution for v:
Use “+” for u and “-” for v To Satisfy
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But Recall:
So We Have:
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Finally, Undo the Translation:
We Could Also Examine The Quartic But Alas, Not the Quintic or Higher
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http://www.storyofmathematics.com/images2/diophantus.jpg
Diophantus, 2nd Century AD
Hellenistic Mathematician
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The Most Famous Diophantine Equation!
The Pythagorean Theorem
Find Integer Solutions to:
How Do We Find Them?
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Example Problem Requiring Pythagorean
Triples for its Solution
A right triangle with integer sides has its perimeter numerically equal to its area. What is the largest possible value of its perimeter?
Problem 11 Math Contest 2015 Round One www.AMATYC.org
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Is There A Formula That Generates
All of the Pythagorean Triples?
YES!
And It Is Very Clever!
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Here is the Formula:
With m, n Arbitrary IntegersGregory Melblom, Private Communication
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Previous Formula Generates All Primitive Triples (no common factors other than one)
But It Misses Some Non-Primitive Triples,For Example: (9, 12, 15)
So We Add A Common Factor!
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Add a Common Factor of k:
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A Right Triangle With Perimeter = Area
Set A = P and Cancel Common Factors
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With Common Factors Canceled In the Equation A = P We have:
k = 1, n = 1, and m = 3 gives x = 8, y = 6, and z = 10
k = 1, n = 2, and m = 3 gives x = 5, y = 12, and z = 13
k = 2, n = 1, and m = 2 gives x = 6, y = 8, and z = 10
The Only Possibilities!
Largest Perimeter is 5 + 12 + 13 = 30
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Consider the Following Equations
What is the Ratio n/m?
Problem 20 Math Contest 2015 Round One www.AMATYC.org
Another Great Problem!
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Use the Definition of Logs:
Original Equation Becomes:
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Divide Both Sides By Smallest Term:
Rewrite As:
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Make a Substitution:
And We Have:
Equation for the Golden Section!
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And the Answer to the Problem is:
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An Example of Symmetry Transformations In Physics
Reduction to the Equivalent One-Body Problem
Consider a Binary Star System
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"Orbit5" by User:Zhatt - Own work. Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Orbit5.gif#/media/File:Orbit5.gif
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Two Stars Orbit in Ellipses About Common Center of MassQuantity of Interest: Separation R(t) Two-Body Problem
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Apply Transformation:“Mass” m Orbits Fixed Force Center. Separation R(t) and Force Left Invariant
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Magnetic Field, North and South Poles
https://www.google.com/search?q=magnetic+dipole&tbm=isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A
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Magnetic Field of a WireNo North and South Poles!
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Magnitude of the B Field
Field Strength Proportional to Ratio of Current to Distance
To Keep Field Invariant, ChangeCurrent and Distance by Same Factor
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Original Version of Problem
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Symmetry Transformation: Move Wire 2 Into Symmetric Position
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Questions
Why is I2’ Up? (I2 was Down)
Claim: I2’ = I1 = 10A. Why?
Distance of I2’ Reduced By Factor of Three. What was Original I2?
Original Problem Solved!
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More General Problem, Point POut of the Plane of the Wires
Find the Unknown Current I2
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Find the Base Angles With Laws of Cosines and Sines
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Move I1 Towards Point P To Make BNet HorizontalBut No Change in Magnitude
Base Angles Change by 5 Degrees Why?
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New Base Angles
The New Distance a of I1
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Original Distance of I1 was 4
New Distance of I1’ is 3.05
So To Keep BNet Invariant:
Recall I1 was 10A
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By Symmetry:
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Apply Correction Factor 1.25 To I1’ to Make B1 = B2
Then Apply Rotation With Point P as AxisP is Back in the Wire Plane
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Net Field At P is Now Zero
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Complete Symmetry Restored!
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References with Slide Numbers1. Find information about The Physics Teacher at: https://www.aapt.org/Publications/2. Photo Courtesy of Jamie Alonzo, Dean of Math/Science, SJCC.6. Stillwell, J. (1989). Mathematics and its History, New York, NY: Springer-Verlag, New York, p51.9. Stillwell, p54.20. Image Retrieved from: http://www.storyofmathematics.com/images2/diophantus.jpg22. A limited number of past exam questions are available to the public at:http://www.amatyc.org/?page=SMLPastQuestions23. Melblom, G., EVC, Private Communication. Material on Pythagorean formula used with permission.29. See Reference for #22.33. See a definition of the Golden Ratio in: Huntley, H.E (1970). The Divine Proportion,New York, NY: Dover, pp24-27.36. "Orbit5" by User:Zhatt - Own work. Licensed under Public Domain via Commons – https://commons.wikimedia.org/wiki/File:Orbit5.gif#/media/File:Orbit5.gif37. Image Retrieved from: https://www.google.com/search?q=binary+orbit&ie=utf-8&oe=utf-8 38. Image Retrieved from: https://www.google.com/search?q=elliptical+orbit&tbm=isch&tbo=u&source=univ&sa=X&ved=0CEkQsARqFQoTCLn6n43YtccCFcqViAodHpkE6Q39. Image Retrived from: https://www.google.com/search?q=magnetic+dipole&tbm=isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A40. Image Retrieved from: https://www.google.com/search?q=magnetic+field+of+a+wire&tbm=isch&tbo=u&source=univ&sa=X&ved=0CC0QsARqFQoTCKXtlYjatccCFUk5iAodtBUMYw
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References, Cont.42. See for example: Giambattista, A., Richardson, B. & Richardson, R. (2007).College Physics, 2nd ed. Boston, MA: McGraw-Hill, p728; Walker, J. Physics, 3rd ed. (2007). New Jersey, NJ: Pearson/Prentice Hall, p766; Serway,R. & Vuille, C. College Physics, 10th ed. (2015). Boston, MA: Cengage Learning, p695; Young, H. D., Freedman, R. A. & Ford, A. L. University Physics, 13th ed. (2014).Boston, MA: Pearson, p951.
44. Image Retrieved from: http://www.amazon.com/Brilliant-Blunders-Einstein-Scientists-Understanding/dp/1439192375/ref=sr_1_1?s=books&ie=UTF8&qid=1449596241&sr=1-1&keywords=Brilliant+Blunders
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