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

2. Why Math?

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

4. Why Math?
“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 ~

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. SI Units 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? Quantity Base Unit Time second (s) Length
meter (m)
Mass
kilogram (kg)
Temperature
Kelvin (K)
Amount of a substance
mole (mol)
Electric current
Ampere (A)
Luminous intensity
candela (cd)

10. SI Units Prefixes Used with SI Units tera T 1012 giga G 109 mega M 106
Symbol
Scientific Notation
tera T
1012
giga G
109
mega M
106
kilo k
103
deci d
10-1
centi c
10-2
milli m
10-3
micro 
10-6
nano n
10-9
pico p
10-12

11. kilo, 103 SI Units Prefixes Used with SI Units
What does the prefix symbol “k” mean?
kilo, 103

12. milli, 10-3 SI Units Prefixes Used with SI Units
What does the prefix symbol “m” mean?
milli, 10-3

13. nano, 10-9 SI Units Prefixes Used with SI Units
What does the prefix symbol “n” mean?
nano, 10-9

14. mega, 106 SI Units Prefixes Used with SI Units
What does the prefix symbol “M” mean?
mega, 106

15. giga, 109 SI Units Prefixes Used with SI Units
What does the prefix symbol “G” mean?
giga, 109

16. micro, 10-6 SI Units Prefixes Used with SI Units
What does the prefix symbol “m” mean?
micro, 10-6

17. tera, 1012 SI Units Prefixes Used with SI Units
What does the prefix symbol “T” mean?
tera, 1012

18. centi, 10-2 SI Units Prefixes Used with SI Units
What does the prefix symbol “c” mean?
centi, 10-2

19. SI Units
SI Unit Conversion
1 MHz = __________________ Hz
106

20. SI Units
SI Unit Conversion
1 kg = __________________ g
103

21. SI Units
SI Unit Conversion
1 G Bytes = __________________ Bytes
109

22. SI Units
SI Unit Conversion
2 mm = __________________ m
2 x 10-3

23. SI Units
SI Unit Conversion
5 ns = __________________ s
5 x 10-9

24. SI Units
SI Unit Conversion
4 cm = __________________ m
4 x 10-2

25. SI Units
SI Unit Conversion
3 mm = __________________ m
3 x 10-6

26. Scientific Notation

27. Scientific Notation 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

28. Scientific Notation Scientific Notation & Standard Notation
1.55 x 106 = x 106 = 1,550,000
for a positive exponent, move the decimal point 6 place to the right
2.5 x 10-4 =
for a negative exponent, move the decimal point 4 place to the left

29. Scientific Notation Calculating with Scientific Notation
(2.5 x 103)(4.0 x 105) = (2.5 x 4.0)(103 x 105)
Rearrange factors
= 10.0 x Multiply
= 10.0 x Add exponents
= 1.0 x 10 x 108 Write 10.0 as 1.0 x 10
= 1.0 x Add exponents

30. Scientific Notation Using Calculator to Calculate Scientific Notation
(2.5 x 103)(4.0 x 105) =
Step 1: MODE  4 (SCI)  CLEAR Change to scientific notation mode
Step 2: ND EE 3 x ND EE 5  ENTER Calculate in scientific notation

31. Scientific Notation 1.25 x 105 1.25 x 10-3 1.9 x 10-1 2.4 x 1018
Scientific Notation Exercises
125,000 = =
1.9 x 102 mA = A
2.4 x 1021 g = kg
(3.2 x 105) (2.5x104) =
1.25 x 105
1.25 x 10-3
1.9 x 10-1
2.4 x 1018
8.0 x 109

32. Significant Figures

33. Significant Figures
Significant Figures and Measurement

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

35. 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

36. Significant Figures 4 4 5 6 4 1.47 x 105 Significant Figures Exercises
No. of significant figures of =
No. of significant figures of 1001 =
No. of significant figures of =
No. of significant figures of =
No. of significant figures of =
(1.234 x 105) + (2.4 x 104) = 4 4 5 6 4
1.47 x 105

37. 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) + ( x 10-4 kg)
= x 10-4 kg
= 3.33 x 10-4 kg

38. 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 103 m/s2)
= x 10-1 kg m/s2
= 2.4x 10-1 kg m/s2

39. Precision & Accuracy

40. Precision & Accuracy Accuracy
The accuracy is a measure of the degree of closeness of a measured or calculated value to its actual or accepted value
Accuracy describes how well two descriptions of a quantity agree with each other

41. Precision & Accuracy Accuracy & Percent Error
Accuracy is often reported quantitatively by using percent error
|accepted – measured|
accepted
Percent Error =
x 100%

42. Precision & Accuracy Percent Error Two experiments:
(A) measured value = 125, accepted value = 100
(B) measured value = 100, accepted value = 75
Which one has higher percent error?

43. Precision & Accuracy Precision
Precision is a measure of how well a result can be determined (without reference to a theoretical or actual value)
It is the degree of consistency and agreement among independent measurements of the same quantity
Precision depends on the tools and methods

44. Precision & Accuracy
Which ruler will provide more precision?

45. Precision & Accuracy Precision & Significant Figures
In many cases engineers and scientists choose to use an implied precision via significant digits.
Significant figures carry with them an implied precision of ± ½ unit in the rightmost significant digit For example,
3280 ± 5
220. ± 0.5
6.47 ± 0.005
0.190 ±

46. Precision & Accuracy
Comparison of Precision & Accuracy

47. Precision & Accuracy
Comparison of Precision & Accuracy

48. Precision & Accuracy
Comparison of Precision & Accuracy

49. Precision & Accuracy
Comparison of Precision & Accuracy

50. Graphing

51. Graphing
Scatter-Plot
Showing Measured Data

52. Graphing
Line Graph
Showing Trend
Helicopter Motion

53. Graphing
Bar Graph
Compare Nonnumeric Categories

54. Graphing
Circle Graph
Showing Percentage

55. Graphing Steps of Graphing Scattered Plots
Graph the axes with proper markings
Graph the data points on the grid
Draw the line/curve of best fit
Calculate the slope of the line if asked

56. Order of Magnitude

57. Order of Magnitude 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

58. Order of Magnitude Example
The order of magnitude of 1024 m (1.024 x m) is 103
The order of magnitude of 9600 m (9.6 x 103 m) is 104

59. Order of Magnitude 4 or 104 4 or 104 4 or 104 5 or 105 6 or 106
Order of Magnitude Exercises
Order of magnitude of =
Order of magnitude of 8200 =
Order of magnitude of =
Order of magnitude of 2. x105 =
Order of magnitude of 7. x105 =
4 or 104
4 or 104
4 or 104
5 or 105
6 or 106

60. Scalars & Vectors

61. Scalars & Vectors Comparison of Scalars & Vectors Physical Quantities
Magnitude
Magnitude
Direction

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

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

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

65. 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
Magnitude
Head
Direction
Tail

66. Scalars & Vectors 26 Scalars & Vectors Exercises
If John walks 10 m to the right, 5 m to the left, 3 m to the right, and then 8 m to the left.
Total distance: ____________________
Total displacement: __________________ 26

67. Scalars & Vectors 23 5 Scalars & Vectors Exercises
If John walks 8 m to the east, 7 m to the north, 4 m to the west, and then 4 m to the south
Total distance: ____________________
Total displacement: __________________ 23 5

68. Mathematical Analysis

69. Mathematical Analysis
Line of Best Fit
To analyze a graph, draw a line/curve 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

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

71. Mathematical Analysis
Quadratic Relationship
y = ax2 + bx + c
where c is the y-intercept y c x

72. Mathematical Analysis
Inverse Relationship
y = a / x y x

73. Mathematical Analysis
Inverse Square Law Relationship
y = a / x2 y x

74. Mathematical Analysis
Mathematical Analysis Exercise
If F = ma describes the relationship between the force (F) and acceleration (a) with constant mass (m), how to represent the relationship between F and a? F a
Linear Relationship

75. Mathematical Analysis
Mathematical Analysis Exercise
If F = ma describes the relationship between the mass (m) and acceleration (a) with constant force (F), how to represent the relationship between m and a? m a
Inverse Relationship

76. Mathematical Analysis
Mathematical Analysis Exercise
If KE = ½ mv2 describes the relationship between the kinetic energy (KE) and velocity (v) with constant mass (m), sketch the graph representing this relationship? v KE
Quadratic
Relationship

77. Mathematical Analysis
Mathematical Analysis Exercise
If Fe = kq1q2/d2 describes the relationship between the electric force (Fe) and distance (d). , sketch the graph representing this relationship? d Fe
Inverse Square Law

78. Equation Solving

79. Equation Solving
Solve Linear Equations
3x + 7 = 8x – 3

80. Equation Solving Solve Linear Equations 3x + 7 = 8x – 3

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

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

83. Equation Solving
Solve Simple Quadratic Equations
20 = 5 / x2

84. Equation Solving Solve Simple Quadratic Equations 20 = 5 / x2
(most of the time, only pick the positive value)

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

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

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

88. Equation Solving Solve Radical Equations 3 = 2π√L/10 3/(2π) = √L/10

89. Equation Solving
Solve Equations of Variables
d = ½ gt2 , solve for t

90. Equation Solving Solve Equations of Variables
vf2 = vi2 + 2gd, solve for d

91. Equation Solving Solve Equations of Variables
Fe = kq1q2/d2 , solve for d

92. Equation Solving Solve Equations of Variables
L = L0 √ 1 – v2 / c2 , solve for v

93. Trigonometry

94. Trigonometry 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

95. Trigonometry Trigonometric Ratios of Special Right TrianglesB B √2 2 1 1
30o
45o A C A C 1 √3

96. 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 30o = x / 10
x = 10 sin 30o = 5 A x 10
30o B C

97. 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 30o = x / 10
x = 10 cos 30o = 8.66 A x 10
30o B C

98. 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 30o = 10 / x
x = 10 / sin 30o = 20 A 10 x
30o B C

99. 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) = 33.7o A 10 15 θ B C

100. The End