Chapter 2 A Mathematical Toolkit Physics Chapter 2 A Mathematical Toolkit

Chapter 2 A Mathematical Toolkit 2.1 The Measures of Science The Metric System and SI SI Prefixes Scientific Notation Converting Units (Factor-Label Method) 2.2 Measurement Uncertainties Accuracy and Precision Significant Digits 2.3 Visualizing Data Graphing Data Linear Relationships Nonlinear Relationships

2.1 The Measures of Science 2.1 The Measures of Science What is a measurement? A comparison between an unknown quantity and a standard What is a standard? What are some good properties of a standard?

The Metric System and SI The Metric System and SI What is the Metric System? Why should we use it? What is an SI base unit? What are derived units?

The Metric System and SI The Metric System and SI SI Base Units Base Quantity Base Unit Symbol Length Mass Time Temperature Amount of substance Electric current Luminous intensity

Length What is length? Straight line distance between two points Length What is length? Straight line distance between two points English unit? Inch, foot, yard, mile? SI base unit? Meter (m) The distance traveled by light in a vacuum in 1/299,792,458 second!

Mass What is mass? Amount of stuff English unit? Slug SI unit? Mass What is mass? Amount of stuff English unit? Slug SI unit? Kilogram (kg) Stored at NIST

Time What is time? The separation between the occurrence of two events. English unit? Second SI unit? Second (s) Frequency of radiation emitted by Cesium-133

Temperature What is temperature? How hot or cold something is. Temperature What is temperature? How hot or cold something is. Measure of the average kinetic energy of the particles in an object English unit? Fahrenheit SI unit? Kelvin (K) not degrees Celsius (ºC)

Other SI Units Mole (amount) Ampere (current) Other SI Units Mole (amount) Ampere (current) Candela (luminous intensity)

The Metric System and SI The Metric System and SI Common Derived Units: Speed (m/s) Density (g/mL) Volume 1L = 1000mL 1cm3 = 1 mL

SI Prefixes What is an SI prefix? SI Prefixes What is an SI prefix? How does it work with base units and derived units?

SI Prefixes Prefix Symbol Meaning Multiplier mega- M 1 million SI Prefixes Prefix Symbol Meaning Multiplier mega- M 1 million 1,000,000 kilo- k 1 thousand 1,000 deci- d 1 tenth 1/10 (0.1) centi- c 1 hundredth 1/100 (0.01) milli- m 1 thousandth 1/1000 (0.001) micro-  1 millionth 1/1,000,000 (10-6) nano- n 1 billionth 1/1,000,000,000 (10-9)

SI Units Let’s check our knowledge(1-3)

Scientific Notation M x 10n Used for very large or very small numbers Scientific Notation Used for very large or very small numbers Consists of two parts: Number between 1 and 10 Multiplied by A whole-number power of ten M x 10n

Scientific Notation Rules: To put it into your calculator Scientific Notation Rules: Exponents tell the number of spaces decimal is moved + means > 1 - means < 1 To put it into your calculator Always use the EE or Exp

Converting units using the Factor-Label Method Converting units using the Factor-Label Method In some (Physics) problems you may need to change the units of a measurement but not the size of the measurement. This can be easily done by use of the Factor-Label Method

Converting units using the Factor-Label Method Converting units using the Factor-Label Method In the Factor-Label Method units are cancelled out using conversion fractions Example: 3 km = ? Miles 1 mile = 1.62 km

Converting units using the Factor-Label Method Converting units using the Factor-Label Method 3 km = ? Miles 1 mile = 1.62 km 3 km (1 mile/1.62 km) = 1.85 miles So..3 km is the same length as 1.85 miles the distance is just measured in different units!

Scientific Notation and Converting Measurements Let’s check our knowledge(4,5)

2.2 Measurement Uncertainties 2.2 Measurement Uncertainties Precision the degree of exactness of a measurement How close measurements are to other values The smaller the measurement units the more precise a measurement (why?)

Measurement Uncertainties Measurement Uncertainties Accuracy How close the measurement is to the “correct” value Can a measurement be precise but not accurate? Can a measurement be accurate but not precise?

Making good measurements Making good measurements Parallax A difference in measurement readings caused by the angle at which an object is viewed Example? Reduce?

Significant Digits Significant Digits Significant Digits Significant Digits The valid digits in a measurement All the digits know for certain plus one estimate (uncertain) Example: volume of a solution

Significant Digits Rules for determining Significant Digits: All nonzero numbers are significant All zeros between two significant digits are significant All end zeros right of the decimal place are significant 1245 12.04 12000 12000.0 12.0030

Significant Digits Adding/Subtracting with Significant Digits Significant Digits Adding/Subtracting with Significant Digits Round answer to the least precise value involved Round the answer to the smallest shared decimal place in any value added/subtracted

Significant Digits Multiplying/Dividing with Significant Digits Significant Digits Multiplying/Dividing with Significant Digits Round answer to the least precise value involved Round the answer to the number with the smallest number of significant digits

Significant Digits and Measurements Significant Digits and Measurements Let’s check our knowledge (6-8)

2.3 Visualizing Data Which says more about the relationship between two variables? OR

Visualizing Data-Graphs Visualizing Data-Graphs What properties must all graphs posses?

Graphs What are the dependent and independent variables? Graphs What are the dependent and independent variables? What axis are they graphed on?

Graphs A graph visually shows the relationship between two variables. Graphs A graph visually shows the relationship between two variables. There are both linear and nonlinear relationships between variables.

Linear Graphs Linear Relationship Linear Graphs Linear Relationship When data points are graphed and a straight line can be drawn through them.

Linear Graphs Linear Relationship Equation: y = mx + b Linear Graphs Linear Relationship Equation: y = mx + b What does this mean? Is this a direct or inverse relationship?

Linear Graphs Direct Linear Relationship Equation: y = mx + b Linear Graphs Direct Linear Relationship Equation: y = mx + b As the independent variable increases so does the dependent variable Slope is positive

Linear Graphs Inverse Linear Relationship Equation: y = mx + b Linear Graphs Inverse Linear Relationship Equation: y = mx + b As the independent variable increases the dependent variable decreases Slope is negative

Nonlinear Graphs Nonlinear Relationship Nonlinear Graphs Nonlinear Relationship When data points are graphed and a curved line can be drawn through them.

Nonlinear Graphs Nonlinear Relationship Nonlinear Graphs Nonlinear Relationship We will be concerned about two types: Quadratic relationships Direct relationship Inverse relationship

Quadratic Graphs Quadratic relationship-direct Nonlinear Relationship Quadratic Graphs Nonlinear Relationship Quadratic relationship-direct Line curves upward What does it mean? Equation: y = ax2 + bx + c

Inverse Graphs Quadratic Inverse relationship Nonlinear Relationship Inverse Graphs Nonlinear Relationship Quadratic Inverse relationship Line curves downward What does it mean? Equation: y = a/x2 x2y = a

Types of Graphs What types of relationships are graphed?

Types of Graphs What types of relationship is graphed?

Types of Graphs What types of relationship is graphed?

Types of Graphs What types of relationship is graphed?

Types of Graphs What types of relationship is graphed?

Analyzing Data Graphs can also be used to determine the values of measurements not made in lab.

Extrapolation Extrapolation—determining values for points located outside the data points

Interpolation Interpolation—determining values for points located between the data points

Analyzing Data Let’s check our knowledge (9-15)