Equation Manipulation Unit 1: Physics Toolbox Topics: Units Scientific Notation Conversion of Units Equation Manipulation Trigonometry Graphing Vectors

I. Scientific Notation For very small and very large numbers we use scientific notation Express decimals places as powers of ten Converting standard form to scientific notation: 25 students = 2.5 x 10 1 2. 2008 years = 2.008 x 10 3 3. 100,000,000,000 stars = 1.0 x 10 11

I. Scientific Notation B. Converting scientific notation to standard form: 1. 9.6 x 10 -5 mL = 0.000096 2. 8.3 x 10 -2 s = 0.083 3. 5.6 x 10 0 m = 5.6

II. Significant Figures They are the valid digits in a measurement Rules for Sig. Figs.: Nonzero digits are always significant 2. All final zeros after the decimal point are significant 3. Zeros between two other significant digits are always significant 4. Zeros used solely as placeholders are not significant Ex) 235 => 0.45 => 3 sig. fig. 2 sig. fig. 2.3400 => 5 sig. fig. 2008 => 0.203 => 4 sig. fig. 3 sig. fig. 300,000 => 0.0030500 => 1 sig. fig. 5 sig. fig.

II. Significant Figures Examples: How many significant figures does each of these measurements have? 3.1 m 3.0001 kg 1.20 x 10 -4 km 0.007060 cm 2 5 3 4

II. Significant Figures For this course, just round answers to 3 sig. figs. IMPORTANT: When using your calculator to perform scientific calculations ALWAYS use the EE (EXP) button. Example: 3 x 103 would be entered as: 3 EE 3 Round the answers to three significant figures and in scientific notation: 9.16 x 10 5 1.98 3.44 x 10 - 7 0.569 6.71 x 10 - 24

Common SI (Système International d'Unités) III. Units Base Units – these units are used along with various laws to define additional units for other important physical quantities Common SI (Système International d'Unités) Base Units

1 kg = 2.20 lb = approximate mass of a textbook III. Units Remember: We will only be using the metric system in this course, so here are some useful ways of relating the English System (US’s measurement) to the rest of the world 1 kg = 2.20 lb = approximate mass of a textbook (175 lb person = 79 kg) 1 m = 3.28 ft = approximate height of a doorknob (a basketball hoop (10ft) = 3.05 m) 8

III. Units Derived Units: Combination of base units Examples velocity/speed: m/s acceleration: m/s2 force: kg * m/s2 = newton (N)

III. Units centi- micro- kilo- k 103 c 10-2 milli- m 10-3 µ 10-6 The SI system is a decimal system, meaning prefixes are used to change SI units by a power of 10 Common prefixes we will use are: Prefix Symbol Notation centi- micro- kilo- k 103 c 10-2 milli- m 10-3 µ 10-6

III. Units Find the common prefixes on the Reference Tables (front page) Convert the following: 1) 25 centimeters to meters: 10 - 2 m 1 cm 25 cm x = 0.25 m 2) 0.25 g to milligrams (mg): 1 mg 10 - 3 g 0.25 g x = 250 mg

III . Units 3) 3.56 x 10-4 kg to grams: 10 3 g 1 kg 3.56 x 10 - 4 kg x 4) 500,000 micrometers to m: 10 - 6 m 1 μ m 500,000 μm x = 0.5 m 12

III . Units 5) 5.7 x 105 m to gigameters (Gm): 5.7 x 10 5 m x = 5.7 x 10 - 4 Gm 6) 4.5 x 10-7 megagrams to milligrams: 10 6 g x 1 Mg 1 mg x 10-3 g 4.5 x 10 - 7 Mg x = 450 mg 13

IV. Conversion of Other Units Any quantity can be measured in several different units, therefore it is important to know how to convert from one unit to another. Conversion scales are used to convert one unit to another 1. How many miles are in 28,465 ft ? 1 mi 5280 ft 28,465 ft x = 5.39 mi 2. How many gallons are in 5 L? 1 gal 3.79 L 5 L x = 1.32 gal

IV. Conversion of Other Units 3. How many meters in 1075 nanometers? 10 - 9 m 1 nm 1,075 nm x = 1.075 x 10 - 6 m 4. How many thingamajigs are in 5.5 x 10 4 whatchamacallits? 320 t 1 w 5.5 x 10 4 w x = 1.76 x 10 7 t 5. How many miles/hour is 40 m/s? 40 m x s 1 mile x 1609 m 3600 s 1 hr = 89 mi hr 15

V. Equation Manipulation In physics we will be using a lot of equations. To solve for specific variables in the equations we want to first manipulate the variables around to solve for what we are looking for. In the following examples solve the given variable: Work on the rest with your partner on the whiteboard.

V. Equation Manipulation Answers to 3 - 10

VI. Dimensional Analysis Dimensional analysis is used to check mathematical relations for the consistency of their dimensions Meaning, the dimension (i.e. – length) on the right side of the equation, has to match the dimension on the left side of the equation

VI. Dimensional Analysis Based on the given units for the variables on the right, determine what would be the unit(s) for the variable left of the equal sign. Reduce the units to find the final simplest unit for the variable.

VI. Dimensional Analysis Based on the given units for the variables on the right, determine what would be the unit(s) for the variable left of the equal sign. Reduce the units to find the final simplest unit for the variable. Try problems 1 and 2 from Sect. IV from the Work Packet

VII. Trigonometry hyp. opp. adj. In physics we use trig. to find the dimensions of buildings and objects and to find the x and y components of vectors (i.e. – velocity, force, displacement) hyp. opp. adj. NOTE: the choice of which side the triangle to label opposite and adjacent is made after the angle is identified

VII. Trigonometry Example: A tall building casts a shadow of 70.0 m long. The angle between the sun’s rays and the ground is 50.0 degrees. Determine the height of the building.

VII. Trigonometry Inverse Trigonometric Functions: When values of two sides of a right triangle are given, the angle can be calculated by using the inverse trig. functions

VII. Trigonometry Example: A lakefront drops off at a certain angle. At a distance 15.0 m from the shore, the water is 2.25 m. What is the angle the lakefront drops off at?

VIII. Pythagorean Theorem Square of the length of the hypotenuse of a right triangle is equal to the sum of the square of the length of the other two sides:

VIII. Pythagorean Theorem Example: A highway is to be built between two towns, one of which lies 35.0 km S and 72.0 km W of the other. What is the shortest length of the highway that can be built between the two towns, and at what angle would this highway be directed?

IX. Graphing/Mathematical Models To describe how nature works, we collect data and look for patterns in that data We often try to map these patterns to mathematical models in the form of equations. These models are most important in physics because they not only describe the specific experiments we performed, but give you insights beyond our data set to help us understand the universe.

X. Linear Functions Linear functions are the easiest ones to recognize – they are _________________ straight They all have the general form: y = mx + b m = slope, b = y-intercept

X. Linear Functions V I numerator Slope =∆y = V = R V = IR ∆x I Direct Relationship Exists when both variables are located in the ______________ Physics Equation Example numerator Slope =∆y = V = R ∆x I V = IR (R Constant) V I

X. Linear Functions Example: Plot the data on the graph and analyze the relationship. Also, calculate the slope of the line.

XI. Power Function (Nonlinear Relationships) There are a lot of functions that have this form, but the two we will be most concerned are: Direct Square (Top opening parabola) Exists if both variables are located in the ____________ and one is variable _________ numerator squared

XI. Power Functions (Nonlinear Relationships) General Equation: Physics Equation Example y = ax2 P P = RI2 (R Constant) I

XI. Power Function (Nonlinear Relationships) 2. Inverse/Inverse Square Exists if one variable is located in the ____________ and other is in the _________ numerator denominator (maybe squared)

XI. Power Functions (Nonlinear Relationships) General Equation: Physics Equation Example y = a or y = a x x2 Never reaches zero R R = ρL A (ρ L Constant) A

XI. Power Functions (Nonlinear Relationships) Example: Plot the points and analyze the relationship Direct square

I. Vector Introduction What is a scalar quantity? Measurement/value that has only has MAGNITUDE (SIZE) Examples: Time, Temperature, Length, speed, mass

I. Vector Introduction What is a vector quantity? Measurement/value that has magnitude AND direction Examples: Displacement, velocity, force, momentum

I. Vector Introduction What is a vector? (Despicable Me) Scale used to show magnitude and direction 10 m (Magnitude) 30o (Direction)

I. Vector Introduction What do negative and positive signs mean when dealing with vectors? They mean direction, NOT magnitude y Same magnitude, but opposite direction -10 m +10 m x

I. Vector Introduction How do I draw a vector? Cartesian Coordinate System: Uses X and Y axes 10 m (Magnitude) 90o 30o (Direction) 180o 0o 270o

I. Vector Introduction How do I draw a vector? Navigational Coordinate System: Uses North, South, West, East 10 m (Magnitude) N North of East (Direction) W E S

II. Scaling Vectors

II. Scaling Vectors

III. Resultant Vector Think about this: A car traveling 40 mph shoots a ball out of canon attached to it in the opposite direction at the same speed. From an observer on the side of the road, what appears to happen to the ball? Mythbusters Video

III. Resultant Vector What is it? Equilibrant Vector Sum of vectors Same magnitude of resultant, but opposite direction

III. Resultant Vector How do I add vectors to determine a resultant? Draw the first vector starting at the origin Step up a new origin at the head of the first vector Draw the second vector at the end of the first vector The resultant is drawn from the starting position of the first vector to the head of the last vector

III. Resultant Vector Examples: 3 m @ 0o + 2 m @ 0o

III. Resultant Vector Examples: 2. 3 N @ 180o + 2 N @ 0o

III. Resultant Vector Examples: 3. 3 m @ 0o + 2 m @ 90o

III. Resultant Vector Examples: 4. 3 u @ 180o + 2 u @ 270o

III. Resultant Vector Examples: 5. 25 km @ 0o + 20 km @ 270o

IV. Vector’s X and Y Components Vector Resolution: breaking a vector into its horizontal and vertical components A Ax = A cosθ Ay Ay = A sinθ Ax Complete problems on pg. 6 on under Sect. VIII: Vectors

V. Vector Scale Diagram

V. Vector Scale Diagram