Lecture-4 (Dr. M Fadhali).Ppt [Compatibility Mode]2slides

8/2/2019 Lecture-4 (Dr. M Fadhali).Ppt [Compatibility Mode]2slides

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Two-Dimensional

Motion and Vectors

Scalars and Vectors

In Physics, quantities are described as eitherscalar quantities or vector

quantities .

A scalar quantityhas only a magnitude (numbers and units) but no

direction.

A vector quantityhas both a magnitude and a direction.


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Which of the follow are vectors?

distance

displacement

mass

weight

temperature

velocity

acceleration

No

Yes

No

Yes

No

Yes

Yes

Vectors...

There are two common ways of indicating that

something is a vector quantity:

Boldface notation:AA

Arrow notation:

AA =

AA

AA


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January 10, 2011 Physics 114A - Lecture 5 5/26

The Components of a VectorThe Components of a VectorLength, angle, and components can be

calculated from each other using trigonometry:

cosxA A q= sinyA A q=

2 2

x yA A A= +

1tan /y xA Aq

-=


2D Cartesian and Polar Coordinate Representations2D Cartesian and Polar Coordinate Representations


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Vector addition:

The sum of two vectors is another vector.

A = B + C

B

C A

B

C

Vector subtraction:

Vector subtraction can be defined in

terms of addition.B - C

B

C

B

-C

B - C

= B + (-1)C


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Unit Vectors:

A Unit VectorUnit Vector is a vectorhaving length 1 and no units.

It is used to specify adirection.

Unit vector uu points in thedirection ofUU.

Often denoted with ahat: uu =

UU

x

y

z

ii

jj

kk

l Useful examples are the cartesianunit vectors [ ii, j, k, j, k]

point in the direction of thex, yand zaxes.

R = rxi + ryj + rzk

Vector addition using components:

l Consider CC=AA + BB.

(a) CC = (Axii+ Ayjj) + (Bxii+ Byjj) = (Ax+ Bx)ii+

(Ay+ By)jj

(b) CC = (Cxii+ Cyjj)

l Comparing components of(a) and (b):

Cx= Ax+ Bx

Cy= Ay+ ByCC

BxAA

ByBB

Ax

Ay


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l Vector A = {0,2,1}

l Vector B = {3,0,2}

l Vector C = {1,-4,2}

What is the resultant vector, D, fromadding A+B+C?

(a)(a) {{33,,--44,,22}} (b)(b) {{44,,--22,,55}} (c)(c) {{55,,--22,,44}}

Example

D = (AXi+ AYj+ AZk) + (BXi+ BYj+ BZk) + (CXi+ CYj+ CZk)

= (AX + BX + CX)i+ (AY + BY+ CY)j+ (AZ + BZ + CZ)k

= (0 + 3 + 1)i+ (2 + 0 - 4)j+ (1 + 2 + 2)k

= {4,-2,5}


Multiplying VectorsMultiplying Vectors

x y z

x y z

A A i A j A k

B B i B j B k

= + +

= + +

r

r

x x y y z z

AB

A B A B A B A B

A B Cosq

= + +

=

r r

Dot Product (Scalar Product)

Cross Product (Vector Product)

( )

( )

( )

( )

y z z y

z x x z

x y y x

AB

x y z

x y z

A B A B A B i

A B A B j

A B A B k

A B Sin a b

i j k

A A A

B B B

q

= -

+ -

+ -

=

=

r r

(determinant)

Given two vectors:

Note that , ,

and .

A B A A B B

A B B A

^ ^

r r rr r

r rr r

AB is the magnitude of Btimes the projection of Aon B (or vice versa).

Note that AB = BA


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Describing Position in 3-Space A vector is used to establish the position of a particle of

interest. The position vector, r, locates the particle at some

point in time.

January 11, 2011 Physics 114A - Lecture 6 14/2414/24

The Displacement Vector

r xx yy= +r

2 1r r rD = -r r r

2 1

2 2 1 1 ( ) ( )

r r r

x x y y x x y y

xx y y

D = -

= + - +

= D + D

r r r


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Instantaneous Velocity in 3D

V = lim (r / t) as t 0 = dr / dt 3 Components : Vx = dx / dt, etc

Magnitude,

|V| = SQRT( Vx2 + Vy

2 + Vz2)

Average Velocity in 3-D

Vavg = (r2 r1)/(t2-t1)

= r / t

t is scalar so, V vector

parallel to vector


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Properties of VectorsProperties of Vectors


The Components of a VectorThe Components of a VectorWe can resolve vector into perpendicular components using

two-dimensional coordinate systems:

Polar Coordinates Cartesian Coordinates

cos25.0 (1.50 m)(0.906) 1.36 mxr r= = =

sin25.0 (1.50 m)(0.423) 0.634 my

r r= = =

2 2 2 2 2(1.36 m) (0.634 m) 2.25 m 1.50 mx yr r r= + = + = =

[ ]1 1tan (0.634 m) / (1.36 m) tan (0.466) 25.0q - -= = =

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Lecture-4 (Dr. M Fadhali).Ppt [Compatibility Mode]2slides