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**INTRODUCTION AND MATHEMATICAL CONCEPTS**

CHAPTER 1 INTRODUCTION AND MATHEMATICAL CONCEPTS

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**1.1 The Nature of Physics Laws of physics: Galileo and Newton**

Describe heat generated by burning match Determine star speed Assist police with radar Galileo and Newton Laws have roots in rocketry and space travel

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**1.1 The Nature of Physics Physics is the core of: X-rays**

Telecommunication Lasers Electronics

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**1.2 Units SI CGS BE Length Meter (m) Centimeter(cm) Foot (ft) Mass**

System SI CGS BE Length Meter (m) Centimeter(cm) Foot (ft) Mass Kilogram (kg) Gram (g) Slug (sl) Time Second (s)

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1.2 Units Base Units used with laws to define additional units for quantities Force Energy Derived Units combinations of base units

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**1.3 The Role of Units in Problem Solving**

Conversion of Units 3.281 ft = 1 m Ex. 1 Express 979 m in ft 979 m ft = ft 1 m

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**1.3 The Role of Units in Problem Solving**

If units do not combine algebraically to give desired results conversion is not correct Ex. 2 Express 65 mi/hr in m/s 65 mi 5280 ft hr m meter 1 hr 1 mi s ft sec Only quantities with same units can be added or subtracted

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**1.3 The Role of Units in Problem Solving**

Dimensional Analysis Dimension= physical nature of a quantity and type of unit used to specify it Ex: Distance Length {L} used to check validity of equation

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1.4 Trigonometry sinØ= ho/h cosØ= ha/h tanØ= ho/ha h ho ø ha

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**1.4 Trigonometry Ex: Trig ho ha= 67.2 m ho= ?? tanØ= ho/ha**

ho= (ha)(tanØ) = (67.2m)(tan50°) = 80m ho 50° ha= 67.2 m

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**1.4 Trigonometry Inverse Functions**

used to find angle if two sides are known Ø= Sin-1(ho/h) Ø=Cos-1(ha/h) Ø=Tan-1(ho/ha)

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**1.4 Trigonometry Pythagorean Theorem**

Square length of hypotenuse of Right Triangle is equal to sum of square of lengths of other two sides h2= ho2 + ha2

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**1.5 The Nature of Physical Quantities: Scalars & Vectors**

Scalar Quantity One that can be described by a single number (including units) giving its size or magnitude Answers “How much is there?” Ex: Volume, Time, Temperature, Mass

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**1.5 The Nature of Physical Quantities: Scalars & Vectors**

Vector Quantity One that deals inherently with both magnitude and direction Arrows used to show direction Direction of arrow = Direction of vector Length of arrow is proportional to magnitude All forces are vectors Force = push/pull Magnitude measured in Newtons

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**1.5 The Nature of Physical Quantities: Scalars & Vectors**

Main Difference Scalars do not have direction; vectors do Negative and positive signs do not always indicate a vector quantity Vector has physical direction (east, west) Temperatures have (+) and (-) , but no direction not a vector

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**1.6 Vector Addition & Subtraction**

When adding vectors you must take both magnitude and direction into account

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**1.6 Vector Addition & Subtraction**

Colinear 2 or more vectors that point in the same direction Arrange head-to-tail and add length of total displacement Gives the resultant vector R = A + B Only works with this type of vector addition A B R

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**1.6 Vector Addition & Subtraction**

Perpendicular 2 vectors with a 90° angle between them Arrange head to tail and use pythagorean theorem R2 = A2 + B2 R A B

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**1.6 Vector Addition & Subtraction**

Not colinear, not perpendicular Must add graphically Draw the components head-to-tail proportionally & accurately Measure the resultant

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**1.6 Vector Addition & Subtraction**

Multiply one of the vectors by –1 to reverse direction Add like before

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**1.7 The Components of a Vector**

Vector Components Components of vector can be used in place of the vector itself in any calculation in which it is convenient to do so Components are any two vectors that add up vectorally to the original vector R= x + y R y x

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**1.7 The Components of a Vector**

In two dimensions the vector components of a vector A are two perpendicular vectors Ax and Ay that are parallel to the x & y axes Add together so that A = Ax + Ay Do not have to be x & y, but it is easier to use them ( especially with trig )

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**1.7 The Components of a Vector**

For a vector to be zero, all its components must be zero Two vectors are equal if, and only if, they have the same magnitude and direction If they are equal, their components are equal

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**1.8 Addition of Vectors by Means of Components**

Components are most convenient and accurate way to add vectors If C= A + B Then Cx = Ax + Bx and Cy= Ay + By By C Bx Ay Ax

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**1.8 Addition of Vectors by Means of Components**

Example: A jogger runs 145 m in a direction of 20.0° east of north (displacement vector A) and then 105 m in a direction 35.0° south of east (displacement vector B). Determine the magnitude and direction of the resultant vector C for these two displacments

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**1.8 Addition of Vectors by Means of Components**

Vector x Component y Component A Ax=(145m)(sin20.0°) Ay= (145m)(coz20.0°) =49.6m =136m B Bx=(105m)(cos35.0°) By= -(105m)(sin35.0°) =86.0m = -60.2m C Ax + Bx = Cx = m Ay + By = Cy = 76 m

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**1.8 Addition of Vectors by Means of Components**

Bx Ax 35.0° B= 105 m Ay A= 145 m By 20.0° C

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**1.8 Addition of Vectors by Means of Components**

C²= Cx² + Cy² = 155m Ø=Tan-1(76m/135.7m) = 29°

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Vocabulary Base SI units- units for length (m), mass (kg), and time (s). Derived units- units that are combinations of the base units. Trigonometry- Sinθ = ho/h Cosθ = ha/h Tanθ = ho/ha

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**Vocabulary Pythagorean Theorem- h^2=ho^2+ha^2**

Scalar quantity- A single number giving its size or magnitude. Vector quantity- A quantity that deals inherently with both magnitude and direction. Resultant vector- the total of the vectors. Vector components- two perpendicular vectors Ax and Ay that are parallel to the x and y axes, respectively, and add together vectorially so that A=Ax+Ay.

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Mathematical Steps Draw vectors (sketch) Add Graphically (for estimation) Make a chart Find Components (Horizontal and Vertical) Check your signs Add columns of the chart together Draw the resulting components Draw the resultant Use Trig and the Pythagorean Theorem to get angle and total length

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Chapter 1. Vectors and Coordinate Systems

Chapter 1. Vectors and Coordinate Systems

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