The Essentials to the Mathematical world

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Adrian Lloyd
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The Essentials to the Mathematical world There is

the journey to you are starting, never give into

2 The Essentials to the Mathematical world There is nothing that is unachievable, any person can start the journey to you are starting, never give into hopelessness and always push on because nothing is more valuable than knowledge. - Neil degrasse Tyson to Joshua Antonio Marsili (after his first college Lecture)

3 In most of my slide show notes you will notice that there are occasionally highlighted text. This text is sometimes a vocabulary word, or an important aspect of a physical concept. This chapter will introduce to you the key concepts of the experimental science physics.

it is best to learn to: Ask the appropriate questions Design experiments to answer questions Draw appropriate

4 What is Physics? The Experimental science: Observe Nature Discover patterns/principles in Nature To Work and understand physics it is best to learn to: Ask the appropriate questions Design experiments to answer questions Draw appropriate conclusions Physics IS NOT MATH!! Math is simply the language or tool with which we translate our observations in nature. Having a supercomputer is great, but if you do not know how to use it, it becomes useless.

5 What is Physics? Physics requires Critical thinking above all else: Math is a tool, but without meaning it is useless! Systematic problem solving Apply Physics principles to various practical problems Learning principles is not physics, learning how to apply them in specific situations is. Observations in nature require tools called Models: A model is a simplified version of a complex physical system

6 Models Consider a baseball thrown through air The ball is not perfectly spherical nor perfectly rigid. There exists air resistance that may influence motion The earth rotates beneath it The ball s weight changes slightly at its distance from Earth varies. To include all this makes the analysis pretty hopeless Instead we create a simplified version of the problem! We neglect the size and shape of the ball and represent it as a particle. We neglect air resistance and treat the ball as if it is in a vacuum. We forget about Earth s rotation And make the weight of the ball constant Now we have a problem simple enough to deal with! We must remember that the validity of our predictions is limited by the validity of the model!

Standards and Units How do we view measurable things? Physical Quantity is any number used to describe an observation of a physical phenomena quantitatively.

7 Standards and Units How do we view measurable things? Physical Quantity is any number used to describe an observation of a physical phenomena quantitatively. All measurements must be compared to some reference standard. If a hallway is 13.0 meters long, we mean that it is 13 times as long as 1.0 meter. We call such a reference standard a Unit of the quantity being measured. When we use a number to describe a P.Q. we must specify the unit we are using, the number 13.0 by itself means nothing

Standards and Measurements To make precise measurements we need definitions of units that do not change The International System (SI) is the modern equivalent of the metric system used throughout

8 Standards and Measurements To make precise measurements we need definitions of units that do not change The International System (SI) is the modern equivalent of the metric system used throughout science. The SI system is based upon the Metric system Based on the metre-kilogram-second system of units (MKS) Establishes 20 prefixes Defines 22 named units, and many more unnamed derived units The motivation for the SI was the diversity of units that had sprung up within the CGS systems and the lack of coordination between the various disciplines that used them. In 1960 the International System of measurement (SI) was finally defined, this is a modified version of the original metric system.

Standards and Measurements All the Units in the SI system measure only a single Dimension.

Units of Measurement: Time is measured in Seconds abbreviated as (s).

Length is measured in Metres abbreviated as (m).

9 Standards and Measurements All the Units in the SI system measure only a single Dimension. Dimension is a single type of physical quantity Grams for example measures mass Standard Units of Measurement: Time is measured in Seconds abbreviated as (s). 1 Second is defined using the emission of radiation by a cesium atom. Length is measured in Metres abbreviated as (m). 1 Metre is defined as the distance light travels in 1/ seconds Mass is measured in kilograms abbreviated as (kg). 1 Kilogram is the mass of a particular platinum Iridium cylinder

Electric Current is measured in Ampere abbreviated as (A).

10 Standards and Measurements Standard Units of Measurement: Temperature is measured in Kelvin abbreviated as (K). Temperature scale using as its 0 (null) point absolute zero, the temp. at which all thermal motion ceases. Electric Current is measured in Ampere abbreviated as (A). 1 Ampere (Amp) is the rate of movement of 1 elementary charge per 1 second Mole and Candela are 2 other SI base units which we will not be using. The Fundamental Units are simply the base units in the metric System (Units without a prefix) These units are combined to define other Physical Quantities.

System International One benefit of this system is that larger and smaller units are related to the Fundamental Units by multiples of 10 or 1/10. These are usually expressed in powers of ten notation.

11 System International One benefit of this system is that larger and smaller units are related to the Fundamental Units by multiples of 10 or 1/10. These are usually expressed in powers of ten notation. 1 km = 10 3 m, or 1 cm = 10-2 m Larger and smaller units are derived by adding a prefix to the name of a fundamental unit. For Example: 1 kilometer = 1 km = 10 3 metres = 10 3 m 1 kilogram = 1 kg = 10 3 grams = 10 3 g 1 kilowatt = 1 kw = 10 3 Watts = 10 3 W Note that kilo represent the same size despite the different fundamental unit. I am providing you with a table of the different prefixes in the Metric/SI system. Please Retain this for the entire school year!

12 Order of Magnitude Examples 1 x m A galaxy 1 x 10 9 m Our Sun 1 x 10 7 m Earth 1 x 10-5 m Red blood cells 1 x 10-9 m A single atom 1 x m Atomic Nucleus

Unit Consistency Equations express relations among physical quantities which are represented by algebraic symbols. Most algebraic symbols for P.Q.

For example: d might represent a distance of 10 m t might represent a time of 5 s v might represent a speed of 2 m/s An equation must ALWAYS be

13 Unit Consistency Equations express relations among physical quantities which are represented by algebraic symbols. Most algebraic symbols for P.Q. denote both a number and a unit. For example: d might represent a distance of 10 m t might represent a time of 5 s v might represent a speed of 2 m/s An equation must ALWAYS be Dimensionally consistent You cant add Metres and seconds Two terms may be added or equated ONLY if they have the same units For example: Length = speed x time = m s Length = (2 m ) (5 s) s The unit of 1/s cancels out the unit s leaving the unit of metres for length s

14 Unit Consistency To apply Unit Consistency you need to treat units as mathematical variables. When performing math operations, the units undergo the same operations. For Example: The area of a square: L w = 2.0 m 5.0 m = 10.0 m m = 10 m 2 The units for area are always metres squared. Acceleration of an ideal body: a = v t = 20.0 m s m = s s s = 4.0 m s 2 The final speed an object reaches after accelerating. v = v o + at = m s = m s + m s 2 s This is why making sure units match is so important! Kilometeres (km) will not cancel with metres (m), (km) must be converted to (m) for the math to work!

15 Unit Conversion Unit conversion is the process of changing the units in which a measurement was taken. This does NOT change the actual P.Q. measured, rather what the P.Q. is measured with. The distance between, Carmen and Kyle does not change but we can measure it in different units. This distance can be expressed as: 1 mile kilometres 1609 metres 5260 feet

16 Quantitative Analysis 1.2 (Multiple Choice) If you express your height both in inches and in centimeters, which unit will give the larger number for your height? A. The result in centimetres is larger, since you are comparing your height to a unit of length smaller than the inch. B. Your height in inches is larger, since an inch is larger than a centimeter. C. Either approach will give the same number, since your height does not depend on the unit used to express it.

17 Unit Consistency Unit Consistency is important when converting between units of the same dimension. The key idea here is that we can express the same P.Q. in two different units through an equality. For Example: 1000 metres = 1 km This does not mean the # 1 is equal to the # Rather that 1 km represents the same measured P.Q. as 1000 m Expressed as an equality: 1000 m 1 km = 1 = 1 km 1000 m

18 Unit Conversion When converting between units we can use a method called the chain rule The chain rule uses conversion factors to change between unit of the same dimension. The ratio of units, such as (1000 g / 1 kg) is called a conversion factor The units are treated as variables canceling out along the chain Using the Chain Rule: # unit # unit 1 # unit # unit # unit = # unit # unit # unit 1 # unit # unit Converting from 15 miles to metres: 15 mile km 1000 m 1 1 mile 1 km = 15 mile km 1000 m 1 1 mile 1 km = m Appendix E, and the Front Cover of your book has tables of conversion factors.

19 Unit Conversion (Cont.) When converting between measurements with multiple units such as 2.5 MPH or 25 m/s, the order of conversion does not matter. Converting From Miles per hour to meters per second. Converting miles first: 2.5 miles 1 hour 1609 m 1 miles 1 hours 60 min 1 min 60 sec = m 3600 sec = 1.12 m s Converting hours first: 2.5 miles 1 hour 1 hours 60 min 1 min 60 sec 1609 m 1 miles = m 3600 sec = 1.12 m s The order of multiplication does not matter.

20 Problem # 10 Chapter 1 While driving in an exotic foreign land, you see a speed limit sign on a highway that reads 180,000 Furlongs per fortnight. How many miles per hour is this? ( 1 furlong is miles, and a fortnight is 14 days)

21 Problem # 11 Chapter 1 You fill up your gas tank in Europe when the euro is worth $1.25 and gasoline costs 1.35 euros per liter. What is the cost of gasoline in dollars per gallon? How many your answer compare with the cost of gas in the USA?

22 Significant Figures (Lab use ONLY) All measurements have uncertainties The uncertainty δx, in a measurement is determined by the level of precision of the device used to make the measurement. Most ordinary rulers are only reliable to the nearest millimeter. Usually a measured quantity is generally reported along with the measured value x, in the format of: x ± δx (unit) A statistical measure of the uncertainty is associated with a quantity called the standard deviation of the data. The standard deviation, σ, is calculated using the following equation: σ = i=1 k (X i X ave ) 2 k 1

23 Significant Figures (Lab use ONLY) Significant figures keep track of uncertainty through small calculations: Any measurement you make is limited by its level of precision (resolution) To keep track of uncertainty remember these rules: 1. Nonzero digits are always significant. (1.12 mm) 2. All final zeroes after the decimal point are significant. (1.000 mm) 3. Zeroes between two other significant digits are always significant. ( mm) 4. Zeroes used solely for spacing the decimal point are not significant. (0.053 mm) In addition or subtraction, work with the numbers as they are written, but the final answer must have the same level of precision as the least precise number in the case below, precision to the tenths place m m m = m = m

24 Vector and Scalar Quantities Scalars are P.Q s. described by a number and unit. have only a Magnitude ( How much or how big ) Examples: time, temperature and mass Calculations with Scalar quantities follow general arithmetic. 6 kg + 3 kg = 9 kg 12 m = 4 m 3 s s Vectors are P.Q s. that have a Directional Quality have Direction (usually an angle) and Magnitude Example: Force, acceleration, velocity, momentum, displacement Adding vectors DOES NOT FOLLOW general arithmetic, it requires a different set of operations.

Vector Properties Vectors notation; A A Vector s Magnitude is represented by its letter in absolute value brackets. Magnitude of A = The magnitude of a vector is a scalar value.

25 Vector Properties Vectors notation; A A Vector s Magnitude is represented by its letter in absolute value brackets. Magnitude of A = The magnitude of a vector is a scalar value. A a) A head a) Vectors, are geometrically represented by a straight arrow. I. The arrow indicates the Direction II. III. The length indicates the Magnitude. The end with the arrow is called the head and the opposite end is the tail. tail θ

26 Vector Properties Vectors exist in Euclidean Space All Space is Isotropic (equal in all directions). Because of this vectors can be located anywhere in Euclidean space, as long as the orientation does not change. b) β A θ α B b) For Vectors to be equal they must meet the conditions: Vectors must be equal in direction: β θ = α Vectors must be equal in magnitude: B = A = C C A = B C

27 Vector Properties c) The angles of vectors are measured from the positive x axis. c) The negative of a vector is another vector of the same magnitude but opposite in direction. C β A θ C is the negative of A which is denoted as: C = A A θ The negative sign indicates a rotation of 180 o. θ o = β β C

28 Challenge Question 8 vectors are shown superimposed on a grid. A C D E B F G H a) List all of vectors that have the same magnitude as vector A b) List all of the vectors that have the same magnitude as vector B c) List all of the vectors that have the same magnitude as vector C d) List all of vectors that have the same magnitude as vector A e) List all of the vectors that have the same magnitude as vector B f) List all of the vectors that have the same magnitude as vector D

29 Adding Vectors Displacement measures the change in an objects position, from its starting point to its ending point. d) Suppose a tennis ball undergoes a displacement A followed by a 2 nd displacement B. d) A B C = A + B The final resulting displacement C is the vector Sum or resultant of vectors A and B. C = A + B The The above is not the same operation as adding 2 scalar quantities. 5 = 2 + 3

30 Vector Properties e) C = A + B e) If we add the vectors A and B in reverse order we obtain the same resultant vector C! B A + B = B + A This means that the addition of vectors is commutative. A C = B + A Scalars and vectors cannot be added, but they can be multiplied: f) A 2A f) The vector 2A is in the same direction as vector A but twice as long. g) Multiplying a vector by a negative scalar changes its magnitude and direction. g) A 2A

31 Examples of Vector Addition The addition of the below vectors is shown to the right. A + B = C y A C B y x B C A x The difference of the below vectors is shown to the left. A C = B y The addition of the below vectors is shown to the right. (C is different than the others) A + B + C = D A B D C x

32 Example 1.5 On a cross country ski trip, you travel 1.00 km north and then 2.00 km east. a) How far are you and in what direction are you from your starting point? b) What are the magnitude and direction of your displacement? y 2 km 1 km displacement x

33 Conceptual Analysis 1.5 Two vectors, one with magnitude of 3 m and the other with magnitude of 4 m, are added together. The resultant vector could have a magnitude as small as: A. 1 m B. 3 m C. 4 m

34 For a right angle triangle, the exist certain functions that relate the different sides. The Opposite side is always the side the angle is facing. The Adjacent side is the side the angle touches. sin θ = cos θ = Definitions of Trig. Functions opposite side hypotenuse adjacent side hypotenuse y θ Hypotenuse Adjacent side Opposite side x We need a simple and precise method for adding vectors.

35 y Components of Vectors: Any vector can be represented as a Vector sum of two vectors; One parallel to the x axis A x One parallel to the y axis A y. A y and A x are referred to as component vectors of A because they are along the axis, such that: A = A y + A x A A y y A y The component vectors A A x By Definition each component vector lies along one of the two coordinate-axis directions. Directions of Vector Components: When a vector component points in the ( ) its magnitude is negative. When a vector component points in the (+) its magnitude is positive x A x x

36 Components of Vectors: If we know the magnitude and direction of A, we can calculate its components use trigonometry functions. y A A x A = cosθ A x = A cosθ A y A y A = sinθ A y = A sinθ These equations are valid provided that: The angle is measure counterclockwise The angle is measured from the positive x axis θ A x x

37 Components of Vectors: Observe B, its component vector B x is negative. y The cosine of an angle in the 2 nd quadrant is negative. B B y (+) Both B x and B x are negative B x ( ) θ x B = B x 2 + By 2 We can describe a vector completely by giving either its magnitude and direction or its x and y components. The direction of a vector is calculated by: θ = tan 1 B y B x

38 Example 1.6 y A crow Flies a distance of 500 m in a direction of 35 o. Raul travels from the same starting point 700 m at 235 o. Express these quantities in terms of displacement vectors and their components. 235 o 35 o A x B

39 Adding Vectors: Observe the addition to the right: y R x = A x + B x R y = A y + B y To add vectors A and B and obtain their vector sum (resultant) R we must B y R B A y A A x B x x

40 Example 1.7 Vector A has a magnitude of 50 cm and a direction of 30 o, and vector A has a magnitude of 35 cm and a direction of 110 o. Both angles are measured counter clockwise from the positive x axis. Use components to calculate the magnitude and direction of the vector sum: R = A + B

41 Problem 62, chapter 1 A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, then 3.50 km at 315 o and then an additional distance in an unknown direction. Her final position is 5.80 km at 0 o. Find the magnitude and direction of the third vector.

2053 College Physics. Chapter 1 Introduction

2053 College Physics. Chapter 1 Introduction 2053 College Physics Chapter 1 Introduction 1 Fundamental Quantities and Their Dimension Length [L] Mass [M] Time [T] other physical quantities can be constructed from these three 2 Systems of Measurement