2. Physical Quantities and VectorsRev.pptx

Last Meeting …

 _{What is measurement?}

 _{Why are units important?}  _{Unit conversion?}

Chapter 2 PHYSICAL QUANTITIES and

VECTORS

 _{1. Physical Quantities: Classification}

 _{2. Physical Quantities: Scientific Notation}  _{3. Scalar vs Vector Quantities}

 _{4. Vector notation}

 _{5 Vector properties}  _{6. Vector components}

Chapter objectives

 _{1. Classify physical quantities and express in}

scientific notation

 _{2. Differentiate scalar quantity from vector}

quantity

 _{3. Define the components of a vector and use}

them in calculations

 _{4. Solve vector related problems graphically}

PHYSICAL QUANTITIES

 _{Classification of Physical Quantities}

A Fundamental Quantity

 can not be derived from other physical quantities.  It can only be matched up to a standard using an

instrument or some other techniques e.g. Time, Length

Derived quantities

 are obtained by manipulating the fundamental physical

quantities and;

 are measured either by theoretical predictions or actual

PHYSICAL QUANTITIES

When the need arises, we express quantities and measurements in terms of

a x 10

(Scientific Notation)

Where a is the measured value, condensed to a single significant 1-9 value at the left of the

decimal point, and b is the correct consequential exponent.

e.g

. 15,000 = 1.5

x 10

PHYSICAL QUANTITIES

SCIENTIFIC NOTATION

 _{Is also done so we can write exponent instead}

of prefixes.

 _{We use:}

PHYSICAL QUANTITIES



Consider a particle moving along a straight

line, we notice that it can only move in two

directions.



We can assign its motion to be

positive

in

one direction and

negative

in the other.

For a particle moving in three dimensions,

assigning signs is no longer enough to

indicate direction of motion.

VECTORS

 _{Vectors are quantities with magnitude and another}

attribute:

DIRECTION.

 _{These attributes are needed to complete the description} of the quantity, which is represented by a VECTOR.

 _{Examples are displacement, velocity and acceleration.}

Not all are vectors…

 _{Not all physical quantities involve} direction.

 _{Temperature, pressure, energy, mass and} time for example do not "point" in the

spatial sense.

 _{We call them scalars, and we deal with} them by rules of ordinary algebra.

 _{A single value with a sign (as in a}

Scalar

Scalar

• Quantity completely described by magnitude only • _{Ex. distance, time, temperature}

Vector

Vector

VECTORS

Magnitude of a

vector is a number

assigned to

determine quantity

Direction of a

vector is a number

that gives which

way the vector will

go

 _{In general}

 _Vector

VECTOR NOTATIONS

 _{Boldface Letter}

 _{Letter with arrow above}

 _{(graphical) Straight line with arrowhead}

 _{VECTOR magnitude representation:}

 _{When dealing with just the magnitude of a vector in print, an}

italic letter will be used: A  _or

A

A

WRITING VECTORS



There are

3 ways

of writing vectors:

1.

AXIAL NOTATIONS

2.

NEWS NOTATION

AXIAL NOTATION



write the magnitude and the angle it creates with

NEWS NOTATION



write the magnitude and

the appropriate direction

based on the NEWS

y

• _{rectangular coordinate}

system (Axial notation)

• _{geographic reference frame}

(NEWS) x

N

W E

S

1 unit, 45 1 unit, NE 1 unit, 180 _{1 unit, west}

1 unit, 330  _{1 unit, 60 E of S}

UNIT VECTORS IN CCS



A unit vector is a

dimensionless vector

with unit magnitude.

 _{Unit vectors are}

usually written in the form

Vector

Properties of Vectors

►_{Equality of Two Vectors}

 _{Two vectors are equal if they have the same magnitude and}

the same direction

►_{Movement of vectors in a diagram}

 _{Any vector can be moved parallel to itself without being}

affected

►_{Negative Vectors}

 _{Two vectors are negative if they have the same magnitude but}

are 180° apart (opposite directions) ► _{A = -B}

►_{Resultant Vector}

RESULTANT

►_{is the vector sum of two or}

more vectors. It is the result of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R.

 _{We usually write vectors in terms of their magnitude and direction.}

 _{But alternatively, we can write it in terms of their components.}

A = 5m, 60



N of E

Components of a VECTOR

►

A

component

is a

part

►

It is useful to use

rectangular

components

 These are the

►_{Consider the vector on the right:} ►_{The x-component of a vector is the}

projection along the x-axis

►_{The y-component of a vector is the}

projection along the y-axis

►_Then,

 = 60

A

A

x y

Components of a VECTOR

cos

x

A



A



sin

y

A



A



x y

A

A





A

Take note of where the

angle is measured from!

Given the vector

(Vector A) in the

illustration:

►

The previous equations are valid

only if



is

measured with respect to the x-axis

►

The components can be positive or negative and will

have the same units as the original vector

►

The components are the legs of the right triangle

whose hypotenuse is

A

 _{May still have to find θ with respect to the positive x-axis}

Components of a VECTOR

x y 1 2 y 2 x A A tan and A A

 _{The components can be positive or negative depending on the}

quadrant where the vector is located.

QUADRANT

I

II

III

IV

X

+

-

-

+

Y

+

+

-

Adding Vectors

►

When adding vectors,

their directions must

be taken into account

Vector Addition Techniques



Analytical



Graphical

A. Graphical Method

(using ruler and a protractor, draw the vectors graphically)

1. Parallelogram Method

most applicable for two vectors

2. Polygon Method

applicable for three or more vectors

Parallelogram Method

1. Draw the two vectors.

2. Construct a parallelogram.

3. Construct a line from the origin that will bisect the parallelogram.

3. Construct a line.

A



f

B

2. Construct a parallelogram.

R

Polygon Method

vector to the head of the last vector.

The line that connect the tail of the first vector to the head of last

A

f



B

R

1. Draw the two vectors. (head-to-tail)

A

B

C

R

D

R = A + B + C + D

R = D + A + C + B

D

A

C

B. Analytical Method

(utilizes Mathematical concepts in analyzing vectors)

1. Law of Sine and Cosines

Cosine Law: C

= A

+ B

– 2ABcosθ

Sine Law: A = B = C

sinα sinβ sinθ

2. Component Method

Adding Vectors Algebraically

►_{Choose a coordinate system and sketch the vectors}

►_{Find the x- and y-components of all the vectors}

►_{Add all the x-components}

 _{This gives Rx:}





x

v

►_{Add all the y-components}

 _{This gives Ry:}

►_{Use the Pythagorean Theorem to find the magnitude of}

the Resultant:

►_{Use the inverse tangent function to find the direction of}

R:

Adding Vectors Algebraically





v

R

R

R

R





Adding Vector by Components



simplifies the mathematics by

enabling us to add vectors in the

same directions



we avoid having to make careful

Component Method

2. Add the components along the x direction to form the x

component of the resultant vector

3. Similarly, add the y components to get the y component of the resultant vector

If we have many vectors (A, B, C, D,...N), and we need to

find the resultant we just need to "SUM" all the x's and

all the y's separately, as illustrated

In the diagram, A has magnitude 12 units and

B has magnitude 8 units. The x component of

A + B is about:

A

B

60º

45º

x y

Solution

A= 3 units east

B= 4 units 30 N of W C= 2 units 70 W of S

30

70

A

B

C

Get the vector sum

for C section

Solution

 _{Solve by component method:}

 _{Vector Components} X Y

 _{A = 3 0}

 _{B = 4 cos 30 = - 3.46 4 sin 30 = +2}  _{C 2 sin 70 = - 1.88 2 cos 70 = - 0.68}  _{Rx = - 2.34}

 _{Ry = +1.32} 

 _{= √ (-2.34)² + (1.32)² = 2.68}

 _{Note: since X = - and Y = +, so Vector is in II}

quadrant 2 2y xR R  2 y 2 x R R

R  

43 . 29 564 . 0 34 . 2 / 32 . 1

tan 1      

To best understand how the parallelogram method works, lets examine the two vectors below. The vectors have

magnitudes of 17 and 28 and the angle between them is 66°. Our goal is to use the parallelogram method to

determine the magnitude of the resultant.

• Step 1) Draw a parallelogram based on the two vectors

that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

 Step 2) We now have a parallelogram and know two

angles (opposite angles of parallelograms are congruent).

Step 3) We can also figure out the other pair of angles

since the other pair are congruent and all four angles must add up to 360.

Step 4) Draw the parallelograms diagonal. This diagonal is the resultant vector

 _{Use the law of cosines to determine the}

length of the resultant 



Cosine Law: C

= A

+ B

– 2ABcosθ

 _C2 = 282 + 172 - 2(28)(17) cos 114

A= 3 units east

B= 4 units 40 S of W C= 2 units 55 E of S

55

40

A

B

C

Get the vector sum

for V section

Solution

 _{Solve by component method:}

 _{Vector Components}

X Y

 _{A = 3 0}

 _{B = 4 cos 40 = - 3.06 4 sin 40 = -2.57}  _{C 2 sin 55 = + 1.64 2 cos 55 = - 1.15}  _{Rx = + 1.58}

 _{Ry = -3.72} 

 _{= √ (1.58)² + (-3.72)² = 4.04 Note: X} = + and

 _{Y = - so}

Vector

 _{is in IV}

quadrant 2 2y xR R  2 y 2 x R R

R  

67 35 . 2 58 . 1 / 72 . 3

tan 1      