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

Why Math?

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

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 ~

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

Math – Language of Physics Mathematic Analysis Trigonometry Equation Solving

SI Units

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

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)

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

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

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

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

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

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

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

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

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

SI Units SI Unit Conversion 1 MHz = __________________ Hz 106

SI Units SI Unit Conversion 1 kg = __________________ g 103

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

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

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

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

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

Scientific Notation

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

Scientific Notation Scientific Notation & Standard Notation 1.55 x 106 = 1.550000 x 106 = 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

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 103+5 Multiply = 10.0 x 108 Add exponents = 1.0 x 10 x 108 Write 10.0 as 1.0 x 10 = 1.0 x 109 Add exponents

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: 2.5 2ND EE 3 x 4.0 2ND EE 5  ENTER Calculate in scientific notation

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 = 0.00125 = 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

Significant Figures

Significant Figures Significant Figures and Measurement

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

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

Significant Figures 4 4 5 6 4 1.47 x 105 Significant Figures Exercises No. of significant figures of 125400 = No. of significant figures of 1001 = No. of significant figures of 12300. = No. of significant figures of 1254.00 = No. of significant figures of 0.001250 = (1.234 x 105) + (2.4 x 104) = 4 4 5 6 4 1.47 x 105

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 x 10-4 kg

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) = 2.4442 x 10-1 kg m/s2 = 2.4x 10-1 kg m/s2

Precision & Accuracy

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

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

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?

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

Precision & Accuracy Which ruler will provide more precision?

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 ± 0.0005

Precision & Accuracy Comparison of Precision & Accuracy

Precision & Accuracy Comparison of Precision & Accuracy

Precision & Accuracy Comparison of Precision & Accuracy

Precision & Accuracy Comparison of Precision & Accuracy

Graphing

Graphing Scatter-Plot Showing Measured Data

Graphing Line Graph Showing Trend Helicopter Motion

Graphing Bar Graph Compare Nonnumeric Categories

Graphing Circle Graph Showing Percentage

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

Order of Magnitude

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

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

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 13000 = Order of magnitude of 8200 = Order of magnitude of 12345 = 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

Scalars & Vectors

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

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

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

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

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

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

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

Mathematical Analysis

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

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

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

Mathematical Analysis Inverse Relationship y = a / x y x

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

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

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

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

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

Equation Solving

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

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

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

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

Equation Solving Solve Simple Quadratic Equations 20 = 5 / x2

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

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

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

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

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

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

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

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

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

Trigonometry

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

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

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

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

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

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

The End