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Using Math Symmetry Operations to Solve a Problem in Elementary Physics

This article discusses how symmetry and invariance can be used to solve a problem involving two parallel current carrying wires in elementary physics. It explores the application of symmetry operations and the use of mathematical transformations to simplify and solve equations. The article also delves into the history of symmetry in mathematics and includes examples and exercises for students to practice. The text language is English.

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Using Math Symmetry Operations to Solve a Problem in Elementary Physics

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  1. Using Math Symmetry Operations to Solve a Problem in Elementary Physics Submitted 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

  2. 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

  3. Apply a Translation and Magnification: Make the Substitution for x:

  4. Substitute x’ = 0 Into the Transformation Equation

  5. 2: Quadratic Equations: or

  6. Translation Eliminates Linear Term: With This Substitution, Equation Becomes:

  7. Solution to Transformed Equation: Undo the Translation:

  8. 3. Cubic Equations: Who Gets the Credit? Scipionedel Ferro (1465 – 1526) NiccoloTartaglia (1499 - 1557) GirolamoCardano (1501 - 1576) . A good secondary source: Stillwell, J. (1994). Mathematics and Its History, New York, NY: Springer-Verlag,p54.

  9. Cubic Equation: Creative Step: Do a Translation to Eliminate Quadratic Term:

  10. New Equation: Coefficients p, q Combinations Of A, B, and C Good Algebra Exercise for Students!

  11. Results for p and q: and Check My Algebra!

  12. Change of Variables on Left Side! So we have: Algebra!

  13. Recall x’ = u + v We Claim: and Why! Hint: x’ is Arbitrary!

  14. Eliminate v Between Equations for p and q Get a Quadratic in u Cubed More Good Algebra for Students! Solution:

  15. But We Had: with and Top Equation Symmetric With Respect to Interchanging u and v So u and v Have Same Solutions!

  16. Solution for v: Use “+” for u and “-” for v To Satisfy

  17. But Recall: So We Have:

  18. Finally, Undo the Translation: We Could Also Examine The Quartic But Alas, Not the Quinticor Higher

  19. Diophantus, 2nd Century AD Hellenistic Mathematician http://www.storyofmathematics.com/images2/diophantus.jpg

  20. The Most Famous Diophantine Equation! The Pythagorean Theorem Find Integer Solutions to: How Do We Find Them?

  21. 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

  22. Is There A Formula That Generates All of the Pythagorean Triples? YES! And It Is Very Clever!

  23. Here is the Formula: With m, n Arbitrary Integers Gregory Melblom, Private Communication

  24. 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!

  25. Add a Common Factor of k:

  26. A Right Triangle With Perimeter = Area Set A = P and Cancel Common Factors

  27. 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

  28. Another Great Problem! Consider the Following Equations What is the Ratio n/m? Problem 20 Math Contest 2015 Round One www.AMATYC.org

  29. Use the Definition of Logs: Original Equation Becomes:

  30. Divide Both Sides By Smallest Term: Rewrite As:

  31. Make a Substitution: And We Have: Equation for the Golden Section!

  32. And the Answer to the Problem is:

  33. An Example of Symmetry Transformations In Physics Reduction to the Equivalent One-Body Problem Consider a Binary Star System

  34. "Orbit5" by User:Zhatt - Own work. Licensed under Public Domain via Commons - https://commons.wikimedia.org/wiki/File:Orbit5.gif#/media/File:Orbit5.gif

  35. Two Stars Orbit in Ellipses About Common Center of Mass Quantity of Interest: Separation R(t) Two-Body Problem

  36. Apply Transformation:“Mass” m Orbits Fixed Force Center. Separation R(t) and Force Left Invariant

  37. Magnetic Field, North and South Poles https://www.google.com/search?q=magnetic+dipole&tbm=isch&tbo=u&source=univ&sa=X&ved=0CFoQsARqFQoTCPiOwointscCFcgwiAodG84M6A

  38. Magnetic Field of a Wire No North and South Poles!

  39. Magnitude of the B Field Field Strength Proportional to Ratio of Current to Distance To Keep Field Invariant, Change Current and Distance by Same Factor

  40. Original Version of Problem

  41. Symmetry Transformation: Move Wire 2 Into Symmetric Position

  42. 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!

  43. More General Problem, Point P Out of the Plane of the Wires Find the Unknown Current I2

  44. Find the Base Angles With Laws of Cosines and Sines

  45. Move I1 Towards Point P To Make BNet Horizontal But No Change in Magnitude Base Angles Change by 5 Degrees Why?

  46. New Base Angles The New Distance a of I1

  47. 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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