CHAPTER 1 INTRODUCTION AND MATHEMATICAL CONCEPTS.

CHAPTER 1

INTRODUCTION AND

MATHEMATICAL CONCEPTS

1.1 The Nature of Physics

Laws of physics:Describe heat generated by burning matchDetermine star speedAssist police with radar

Galileo and NewtonLaws have roots in rocketry and space

travel

1.1 The Nature of Physics

Physics is the core of:X-raysTelecommunicationLasersElectronics

1.2 Units

SI CGS BE

Length Meter (m) Centimeter(cm)

Foot (ft)

Mass Kilogram (kg)

Gram (g) Slug (sl)

Time Second (s) Second (s) Second (s)

System

1.2 Units

Base Units used with laws to define additional units for quantities ForceEnergy

Derived Units combinations of base units

1.3 The Role of Units in Problem Solving

Conversion of Units3.281 ft = 1 m

Ex. 1

Express 979 m in ft

979 m 3.821 ft = 3212 ft

1 m

1.3 The Role of Units in Problem Solving

If units do not combine algebraically to give desired results conversion is not correctEx. 2 Express 65 mi/hr in m/s65 mi5280 ft hr m 29 meter1 hr 1 mi 3600 s 3.821 ft sec

Only quantities with same units can be added or subtracted

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

1.4 Trigonometry

sinØ= ho/h

cosØ= ha/h

tanØ= ho/ha

h ho

haø

1.4 Trigonometry

Ex: Trig

50°

ho

ha= 67.2 m

ho= ??tanØ= ho/ha

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

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)

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

1.5 The Nature of Physical Quantities: Scalars & Vectors

Scalar QuantityOne 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

1.5 The Nature of Physical Quantities: Scalars & Vectors

Vector QuantityOne that deals inherently with both

magnitude and directionArrows used to show direction

Direction of arrow = Direction of vectorLength of arrow is proportional to magnitude

All forces are vectorsForce = push/pullMagnitude measured in Newtons

1.5 The Nature of Physical Quantities: Scalars & Vectors

Main DifferenceScalars do not have direction; vectors do

Negative and positive signs do not always indicate a vector quantityVector has physical direction (east, west)Temperatures have (+) and (-) , but no

direction not a vector

1.6 Vector Addition & Subtraction

AdditionWhen adding vectors you must take both

magnitude and direction into account

1.6 Vector Addition & Subtraction

Colinear2 or more vectors that point in the same

directionArrange head-to-tail and add length of total

displacement Gives the resultant vectorR = A + B

Only works with this type of vector addition

A B

R

1.6 Vector Addition & Subtraction

Perpendicular2 vectors with a 90° angle between themArrange head to tail and use pythagorean

theoremR2 = A2 + B2

RA

B

1.6 Vector Addition & Subtraction

Not colinear, not perpendicularMust add graphically

Draw the components head-to-tail proportionally & accurately

Measure the resultant

1.6 Vector Addition & Subtraction

SubtractionMultiply one of the vectors by –1 to reverse

directionAdd like before

1.7 The Components of a Vector

Vector ComponentsComponents 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 + yR

yx

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 )

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

1.8 Addition of Vectors by Means of Components

Components are most convenient and accurate way to add vectors

If C= A + BThen Cx = Ax + Bx and Cy= Ay + By

Ax

BxAy

ByC

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

1.8 Addition of Vectors by Means of Components

Vector x Component y ComponentA 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 = 135.6 m Ay + By = Cy = 76 m

1.8 Addition of Vectors by Means of Components

A= 145 m

B= 105 m

20.0°

35.0°

Ay

AxBx

By

C

1.8 Addition of Vectors by Means of Components

C²= Cx² + Cy² = 155m

Ø=Tan-1(76m/135.7m) = 29°

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

Vocabulary

Pythagorean Theorem- h^2=ho^2+ha^2Scalar 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.

Mathematical StepsDraw vectors (sketch)Add Graphically (for estimation)Make a chartFind Components (Horizontal and

Vertical)Check your signsAdd columns of the chart togetherDraw the resulting componentsDraw the resultantUse Trig and the Pythagorean Theorem

to get angle and total length