1. Physics Lesson 2 Math - Language of Physics Eleanor Roosevelt High School Chin-Sung Lin

2. Why Math?

3.  Ideas can be expressed in a concise way  Ideas are easier to verify or to disapprove by experiment Why Math?  Methods of math and experimentation led to the enormous success of science

4. “How can it be that mathematics, being a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” ~ Albert Einstein ~ Why Math?

5. Math – Language of Physics  SI Units  Scientific Notation  Significant Figures  Precision & Accuracy  Graphing  Order of Magnitude  Scalar & Vectors

6. Math – Language of Physics  Mathematic Analysis  Trigonometry  Equation Solving

7. SI Units

8. What is SI Units?  Système Internationale d’Unités  Seven fundamental units  Dozens of derived units have been created

9. SI Units What is UI Units? QuantityBase Unit Timesecond (s) Lengthmeter (m) Masskilogram (kg) TemperatureKelvin (K) Amount of a substancemole (mol) Electric currentAmpere (A) Luminous intensitycandela (cd)

10. SI Units Prefixes Used with SI Units PrefixSymbolScientific Notation teraT 10 12 gigaG10 9 megaM 10 6 kilok 10 3 decid 10 -1 centic 10 -2 millim 10 -3 micro  10 -6 nanon 10 -9 picop 10 -12

11. Scientific Notation

12. What is Scientific Notation?  Shorthand for very large / small numbers  In the form of a x 10 n, where n is an integer and 1 ≤ |a| < 10

13. Scientific Notation Scientific Notation & Standard Notation  1.55 x 10 6 = 1.550000 x 10 6 = 1,550,000 for a positive exponent, move the decimal point 6 place to the right  2.5 x 10 -4 = 0.00025 for a negative exponent, move the decimal point 4 place to the left

14. Scientific Notation Calculating with Scientific Notation  (2.5 x 10 3 )(4.0 x 10 5 ) = (2.5 x 4.0)(10 3 x 10 5 ) Rearrange factors = 10.0 x 10 3+5 Multiply = 10.0 x 10 8 Add exponents = 1.0 x 10 x 10 8 Write 10.0 as 1.0 x 10 = 1.0 x 10 9 Add exponents

15. Scientific Notation Using Calculator to Calculate Scientific Notation

16. Significant Figures

17. What is Significant Figures?  The result of any measurement is an approximation  Include all known digits and one reliably estimated digit

18. Significant Figures Zeros & Significant Figures  Each nonzero digit is significant  A zero may be significant depending on its location A zero between two significant digits is significant All final zeros of a number that appear to the right of the decimal point and to the right of a nonzero digit are also significant Zeros that simply act as placeholders in a number are not significant

19. Significant Figures Addition / Subtraction & Significant Figures  the number of digits to the right of the decimal in sum or difference should not exceed the least number of digits to the right of the decimal in the terms  (1.11 x 10 -4 kg) + (2.22222 x 10 -4 kg) = 3.33222 x 10 -4 kg = 3.33 kg

20. Significant Figures Multiplication / Division & Significant Figures  to round the results to the number of significant digits that is equal to the least number of significant digits among the quantities involved  (1.1 x 10 -4 kg) * (2.222 x 10 3 m/s 2 ) = 2.4442 x 10 -1 kg m/s 2 = 2.4x 10 -1 kg m/s 2

21. Precision & Accuracy

22. Precision  The degree of exactness of a measurement  Precision depends on the tools and methods  The significant digits show its precision

23. Precision & Accuracy Accuracy  The degree of agreement of a measurement with an accepted value obtained through computations or other competent measurement  Accuracy describes how well two descriptions of a quantity agree with each other

24. Precision & Accuracy Comparison of Precision & Accuracy

25. Precision & Accuracy Comparison of Precision & Accuracy

26. Precision & Accuracy Comparison of Precision & Accuracy

27. Precision & Accuracy Comparison of Precision & Accuracy

28. Precision & Accuracy Percent Error |accepted – measured| accepted Percent Error =x 100%

29. Graphing

30. Scatter-Plot  Showing Measured Data

31. Graphing Line Graph  Showing Trend

32. Graphing Bar Graph  Compare Nonnumeric Categories

33. Graphing Circle Graph  Showing Percentage

34. Order of Magnitude

35. Definition  Describe the size of a measurement rather than its actual value  The order of magnitude of a measurement is the power of 10 closest to its value

36. Order of Magnitude Example  The order of magnitude of 1024 m (1.024 x 10 3 m) is 10 3  The order of magnitude of 9600 m (9.6 x 10 3 m) is 10 4

37. Scalars & Vectors

38. Comparison of Scalars & Vectors ScalarsVectors Magnitude Direction Physical Quantities

39. Scalars & Vectors Comparison of Scalars & Vectors ScalarsVectors Magnitude Direction Physical Quantities 3 m/s North

40. Scalars & Vectors Comparison of Scalars & Vectors ScalarsVectors Magnitude Direction Physical Quantities 3 m/s 60 o

41. Scalars & Vectors Examples of Scalars & Vectors ScalarsVectors distance speed Mass Displacement Velocity Force Physical Quantities

42. Scalars & Vectors Vector Representation  An arrow is used to represent the magnitude and direction of a vector quantity  Magnitude: the length of the arrow  Direction: the direction of the arrow Head Tail Magnitude Direction

43. Mathematical Analysis

44. Line of Best Fit  To analyze a graph, draw a line of best fit which passes through or near the graphed data  Describe data and to predict where new data will appear Line of Best Fit

45. Mathematical Analysis Linear Relationship  y = mx + b where b is the y-intercept and m is the slope x y b Slope = m

46. Mathematical Analysis Quadratic Relationship  y = ax 2 + bx + c where c is the y-intercept x y c

47. Mathematical Analysis Inverse Relationship  y = a / x x y

48. Mathematical Analysis Inverse Square Law Relationship  y = a / x 2 x y

49. Equation Solving

50. Solve Linear Equations  3x + 7 = 8x – 3

51. Equation Solving Solve Linear Equations  3x + 7 = 8x – 3 7 + 3 = 8x – 3x 10 = 5x 5x = 10 x = 2

52. Equation Solving Solve Simple Rational Equations  3 / 8 = 9 / x

53. Equation Solving Solve Simple Rational Equations  3 / 8 = 9 / x x = 8 (9 / 3) x = 24

54. Equation Solving Solve Simple Quadratic Equations  20 = 5 / x 2

55. Equation Solving Solve Simple Quadratic Equations  20 = 5 / x 2 x 2 = 5 / 20 = 1 / 4 x = 1 / 2 (most of the time, only pick the positive value)

56. Equation Solving Solve Rational Equations  1/R = 1/20 + 1/30

57. Equation Solving Solve Rational Equations  1/R = 1/20 + 1/30 1/R = 5/60 = 1/12 R = 12

58. Equation Solving Solve Radical Equations  3 = 2π√L/10

59. Equation Solving Solve Radical Equations  3 = 2π√L/10 3/(2π) = √L/10 (3/2π) 2 = L/10 9/(4π 2 ) = L/10 L = 45/2π 2

60. Equation Solving Solve Equations of Variables  d = ½ gt 2,solve for t

61. Equation Solving Solve Equations of Variables  v f 2 = v i 2 + 2gd,solve for d

62. Equation Solving Solve Equations of Variables  F e = kq 1 q 2 /d 2,solve for d

63. Equation Solving Solve Equations of Variables  L = L 0 √ 1 – v 2 / c 2,solve for v

64. Trigonometry

65. Trigonometric Ratios  In ΔABC, BC is the leg opposite  A, and AC is the leg adjacent to  A. The hypotenuse is AB sin A = a / c cos A = b / c tan A = a / b A a b c B C

66. Trigonometry Trigonometric Ratios of Special Right Triangles  30-60-90 & 45-45-90 special right triangles 1 1 √2 B C 45 o 1 2 B C 30 o √3 AA

67. Trigonometry Calculation Using Trigonometric Ratio  Identify the known angle/side & the unknown side  Establish the trigonometric ratio between the known side and unknown side using the angle  Solve for the unknown sin A = BC / AB sin 30 o = x / 10 x = 10 sin 30 o = 5 A x 10 30 o B C

68. Trigonometry Calculation Using Trigonometric Ratio  Identify the known angle/side & the unknown side  Establish the trigonometric ratio between the known side and unknown side using the angle  Solve for the unknown cos A = AC / AB cos 30 o = x / 10 x = 10 cos 30 o = 8.66 A x 10 30 o B C

69. Trigonometry Calculation Using Trigonometric Ratio  Identify the known angle/side & the unknown side  Establish the trigonometric ratio between the known side and unknown side using the angle  Solve for the unknown sin A = BC / AB sin 30 o = 10 / x x = 10 / sin 30 o = 20 A 10 x 30 o B C

70. Trigonometry Calculation of angles  Identify the known side & the unknown angle  Establish the trigonometric ratio between the known sides and unknown angle  Solve for the unknown using inverse trigonometric function tan θ = BC / AC tan θ = 10 / 15 = 2 / 3 x = tan -1 (2 / 3) = 33.7 o A 10 15 θ B C

71. The End