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

Why Math?

 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

“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?

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

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

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)

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

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 10 6 = x 10 6 = 1,550,000 for a positive exponent, move the decimal point 6 place to the right  2.5 x = for a negative exponent, move the decimal point 4 place to the left

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 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

Scientific Notation Using Calculator to Calculate Scientific Notation

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 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 kg) + ( x kg) = x kg = 3.33 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 kg) * (2.222 x 10 3 m/s 2 ) = x kg m/s 2 = 2.4x kg m/s 2

Precision & Accuracy

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

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

Precision & Accuracy Comparison of Precision & Accuracy

Precision & Accuracy Comparison of Precision & Accuracy

Precision & Accuracy Comparison of Precision & Accuracy

Precision & Accuracy Comparison of Precision & Accuracy

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

Graphing

Scatter-Plot  Showing Measured Data

Graphing Line Graph  Showing Trend

Graphing Bar Graph  Compare Nonnumeric Categories

Graphing Circle Graph  Showing Percentage

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 10 3 m) is 10 3  The order of magnitude of 9600 m (9.6 x 10 3 m) is 10 4

Scalars & Vectors

Comparison of Scalars & Vectors ScalarsVectors Magnitude Direction Physical Quantities

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

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

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

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

Mathematical Analysis

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

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

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

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

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

Equation Solving

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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  & special right triangles 1 1 √2 B C 45 o 1 2 B C 30 o √3 AA

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 o 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 30 o = x / 10 x = 10 cos 30 o = 8.66 A x o 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 30 o = 10 / x x = 10 / sin 30 o = 20 A 10 x 30 o 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) = 33.7 o A θ B C

The End