vectors and scalars quatities

8/17/2019 Vectors and Scalars Quatities

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SCALAR AND VECTOR

QUANTITIESRAPHAEL V. PEREZ, CpE


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SCALAR AND VECTOR

QUANTITIESDefine Scalar and Vector


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SCALAR QUANTITYhas only magnitde. !"nly the

measre # $antity%

VECTOR QUANTITY

has &oth magnitde and direction.


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SCALAR VECTORdistance

displacement

workpower

acceleration

ol!me

press!re

elocit"

speed

wei#$t

mass

%orce

resistance


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SCALAR AND VECTOR

QUANTITIES'HA( )S RES*L(A+(

VEC("R


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RES*L(A+( VEC("R is the is the-ector that reslts from adding t/o or

more -ectors together.

-1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5

1

1.5

2

2.5

3

3.5

4

y

V e c t o

r 1 = A

V

e c t o

r 2

= B

R E S U L

T A N T V E C

T O

R = R


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(he goal of this topic is tofind the 0A1+)(*DE "2

(HE RES*L(A+( VEC("R!R%, and the VEC("R

A+1LE !3%


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(HERE ARE (HREE (ECH+)4*ES ("

2)+D (HE RES*L(A+( VEC("R A+D(HE VEC("R A+1LE5

6. 1RAPH)CAL 0E(H"D 7 yo need thetechnical tools li8e sharp pencil, rler,protractor and the paper !graphing or

&ond% to sho/ the -ectors graphically.(he otpt is the connection of -ectorsis li8e a polygon.


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R ɵ


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SA0PLE PR"9LE0

6. A man /al8s at:; meters East

and


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Soltion5 'rite the gi-en

facts1i-en5

A = :; meters East9 =


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graph the -ectors from the origin

!head to tail%

head!arro/head%

VEC("R

tail

0A 1 + ) ( *

D E


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!head to tail%

NOTE: 1 GRID = 10 METERS

B = 3 0

M E T E R S ,

N O R T

A = !0 METERS, EAST

USE RULER TOMEASURE AND

TO DRAW A

LINE

θ = 37° N of E

R = "

0 M E T E

R S


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SA0PLE PR"9LE0

>. 1i-en5

A = ? 8m ,East9 = @ 8m, +EC = 8m,


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!head to tail%

NOTE: 1 GRID = 10 #$

%OSSIBLE GRA%

>. 1i-en5A = ? 8m ,East9 = @ 8m, +EC = 8m,


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ASSI)N*ENT

*se graphical 0ethod to find the magnitde of theresltant displacement and the -ector angle

6. 1i-en5A= 6. 1i-en5

0= ?. cm, +'+= >.? cm, SE"= 6.< cm, +E

R = 3 =


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+, T$e -"t$a#orean T$eorem

(he Pythagorean theorem is a sefl methodfor determining the reslt of adding t/o !andonly t/o% -ectors that ma8e a right angle to

each other. (he method is not applica&le foradding more than t/o -ectors or for adding-ectors that are not at ;degrees to eachother. (he Pythagorean theorem is a

mathematical e$ation that relates the lengthof the sides of a right triangle to the length ofthe hypotense of a right triangle.


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• .orm!la/

Res!ltant ector

• 'here A and 9 are -ectors can form aright angle and are also primarydirections.

Vector Angle

the ans/er is 50t$e R12 0(1 0opposite1 o% 0ad3acent1

•


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SA0PLE PR"9LE0

6. A man /al8s at :; meters East

and


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B = 3 0 M E

T E R S , N O R T

A = !0 METERS, EAST

sketchyourpro!e"

θ

θ =

θ =use calculator

R = "

0 M E T E

R S

= "0 $


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• 1. ____ is an example of a scalar quantity #$ %e!oc&ty $ 'orce c$ %o!u"e ($ #cce!er#t&o)

• 2. ___ is an example of a vector quantity #$ "#ss $ 'orce

c$ %o!u"e ($ (e)s&ty


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• 3. A scalar quantity: #$ #!*#ys h#s "#ss $ &s # +u#)t&ty th#t &s co"p!ete!y spec&,e( y&ts "#-)&tu(e c$ sho*s (&rect&o)

($ (oes )ot h#%e u)&ts

• 4. A vector quantity #$ c#) e # (&"e)s&o)!ess +u#)t&ty $ spec&,es o)!y "#-)&tu(e c$ spec&,es o)!y (&rect&o) ($ spec&,es oth # "#-)&tu(e #)( # (&rect&o)


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• 5. A oy pus!es a"ainst t!e #all #it! 5$ poun%s

of force. &!e #all %oes not move. &!e resultantforce is: #$ ./0 pou)(s $ 100 pou)(s c$ 0 pou)(s

($ .7/ pou)(s

• '. A man #al(s 3 miles nort! t!en turns ri"!t an%#al(s 4 miles east. &!e resultant %isplacement is: #$ 1 "&!e SW

$ 7 "&!es NE c$ / "&!es NE ($ / "&!es E


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• ). A man #al(s at 1$ m *ast+ ut !e returns ac(

at 1$ m at #est. &!e resultant %isplacement is: #$ 0 k" $ 20 k" c$ 10 k" ($ .10 k"

• ,. &!e %i-erence et#een spee% an% velocity is: #$ spee( h#s )o u)&ts $ spee( sho*s o)!y "#-)&tu(e *h&!e %e!oc&tyreprese)ts oth "#-)&tu(e 4stre)-th$ #)( (&rect&o)

c$ they use (&5ere)t u)&ts to represe)t the&r "#-)&tu(e ($ %e!oc&ty h#s # h&-her "#-)&tu(e


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• ). A plane yin" 5$$ /0!r %ue nort! !as a tail

#in% of 45 /0!r t!e resultant velocity is: #$ /6/ "&!eshour (ue south8 $ 6// "&!eshour )orth8 c$ /6/ "&!eshr (ue )orth8 ($ 6// MIhr (ue south

• ,. &!e %i-erence et#een spee% an% velocity is: #$ spee( h#s )o u)&ts $ spee( sho*s o)!y "#-)&tu(e *h&!e %e!oc&tyreprese)ts oth "#-)&tu(e 4stre)-th$ #)( (&rect&o)

c$ they use (&5ere)t u)&ts to represe)t the&r "#-)&tu(e ($ %e!oc&ty h#s # h&-her "#-)&tu(e


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• . &!e resultant ma"nitu%e of t#o vectors

#$ Is #!*#ys pos&t&%e $ 9#) )e%er e :ero c$ 9#) )e%er e )e-#t&%e ($ Is usu#!!y :ero

• 1$. !ic! of t!e follo#in" is not true. #$ %e!oc&ty c#) e )e-#t&%e $ %e!oc&ty &s # %ector

$ spee( &s # sc#!#r ($ spee( c#) e )e-#t&%e


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4, ANALYTICAL 0CO*-ONENT1 *ET5OD

Each -ector has t/o components 5

the Fcomponent and the ycomponent)f the -ectors are in secondary directions 5

!+', +E, S' or SE directions%

AF = A cos 3FA y = A sin 3F

/here5A = the gi-en -ector -ale3F = the gi-en angle from F aFis

AF = the F 7 component of -ector A

A y = y 7 component of -ector A


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Component formla for F and y5

A x = A cos θ x

A y = A sin θ x

Sm of F and y Components5


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2)+AL A+S'ER )S5

0agnitde of the Resltant Vector5

(he -ector Angle5

•


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Consider t$e si#n conentions for the Sm of F and yComponents

4adrant of 0agnitde

6 6 I

7 6 II

7 7 III

6 7 IV

8 Y7a9is 0Nort$ or So!t$1

8 :7 a9is 0;est or East1

4adrant of 0agnitde6 6 I

7 6 II

7 7 III

6 7 IV

8 Y7a9is 0Nort$ or So!t$1

8 :7 a9is 0;est or East1


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A


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Recall the SA0PLE PR"9LE0

1i-en5A = ? 8m ,East9 = @ 8m, +EC = 8m,


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S l ti n/ D t


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Sol!tion/ Draw a ta


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Sol!tion/ Comp!te %or t$e ma#nit!de and

ector an#le

T$e ma#nit!de

= ,.3) (m

(he -ector Angle5

+ of E

&ecase of the sign con-entionof the components and theangle /hich is !G,G% at G3!positi-e angle%

W>I9> IS NEAR IN OUR ?RE@IOUS DRAWIN IN RA?>I9AL MET>OD


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Actal happen on -ectors

!not needed to ra h%

NOTE: 1 GRID = 10 #$

%OSSIBLE GRA%

A = ? 8m ,East9 = @ 8m, +EC = 8m,


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!head to tail%


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R

& >

, 4 A

k

m ,

θ =').')°

& R Ʃ x, R Ʃ y' = &3(1)0!*, +(+!2*!'

-./r.t I


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.INAL ANS;ER


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4 N

ectors6

component

76compone

nt

A

<

9

Tot#!

A =

B =

C =


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A =

B =

C =

D = 7 N

ectors

6component

76component

A

<

9

D

vectors and scalars quatities

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