02 Chapter 2 – Force Vector

7/24/2019 02 Chapter 2 Force Vector

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Statics and Dynamics (1013)CHAPTER 2 FORCE VEC

Dr !"#d A$%&an 'in !a"insr

Ptr"%m En*inrin* D+artmnt

Fac%ty ", -"scincs and Ptr"%m

.ni/rsiti Tn""*i PETROAS (.TP)

Internal


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C#a+trO%tin

Scaars and Vct"rs

Vct"r O+rati"ns

Vct"r Additi"n ", F"rcs

Additi"n ", a Systm ", C"+anar F"rcs

Cartsian Vct"rs

Additi"n and S%'tracti"n ", Cartsian Vct"

P"siti"n Vct"rs

F"rc Vct"r Dirctd a"n* a in

D"t Pr"d%ct

Internal


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Scalars and Vectors

Scalars Vectors

%antity c#aractri$d 'y POS4T4VE and

E-AT4VE n%m'r

%antity #as '"t# a !A-4T.D

D4RECT4O

4ndicati"n 5 T6t B"" 5A HAD7R4TTE 5 A

4ndicati"n 5 B"" 5 A HAD7R4TTE 8

Scalars Vectors

%antity c#aractri$d 'y POS4T4VE and

E-AT4VE n%m'r

%antity #as '"t# a !A-4T.D

D4RECT4O

4ndicati"n 5 T6t B"" 5A HAD7R4TTE 5 A

Internal

Mn

%nit

DAn*r,arr"9 2

SArr"

ARROW(-ra+#ica R+rsntati"n ", a Vct"r)

Tai

A

1

20

O

P

Had

in ", Acti"n

R,rnc A6is


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Vector Operations

Internal

VMultiplication and

Division of a Vector by aScalar

P

Tr

R9A


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Vector Operation $cont%&

Internal

Rs"%ti"n ",

Vct"r

R R R

a

'

a

Rs%tant ",F"rc

E6tnd +ara in ,r"mt# #ad ", R t" ,"rm

c"m+"nnt

A

B

C"m+t

1 2 3


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Vct"r Additi"n ", F"rcs

Internal

F2

F3

F1

F2

F3

F1

F2

F1

F1< F2 F1< F2

1

2

3


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A++icati"n ", a& ", Sin and C"sin in dtma*nit%d and dircti"n ", /ct"r ,"r

Internal


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Additi"n ", Systm ", C"+anar F"rc

Internal

Scaar "tati"n

T" ' %sd "ny ,"r c"m+%tati"na +%r+"s "t ,"r *ra+#ica r+rsntati"ns in =*%rs

"tati"n ,"r r+rsntin* t# dircti"na sns ", t# rctan*%ar c"m+"

Cartsian Vct"r

idsi*nat t# direction 'dsi*nat t# direction Arr"ad &i ' dscri

positive/ negative sign d+"intin* dircti"nalong t

negative axis


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C"+anar F"rc Rs%tants

Internal

PROB(M

S"(P )

Scaar "tati"n

S"(P *'y

Carts

V

S"(P +


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(,AMP(

Plan:a) Rs"/ t# ,"rcs int" t#ir 68y c"m+"nnts

') Add t# rs+cti/ c"m+"nnts t" *t t#rs%tant /ct"r

c) Find ma*nit%d and an* ,r"m t# rs%tant

c"m+"nnts

-iven:T#r c"nc%rrnt,"rcs actin* "n atnt +"st

.ind: T# ma*nit%dand an* ", t#rs%tant ,"rci

j

Internal

Internal

# i


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#artesian Vector

Internal

Ri*#t HanddC""rdinat

Systm

Rctan*%arC"m+"nnts ", a

Vct"r

.nit Vct"r5

6

y

$

A$

A6

Ay

A/

A

S+ci=d t# dircti"n ", A

R+rsnt t# %nit /ct"r t#at #a/in* t#sam dircti"n as A r A is a a /ct"r &it#a ma*nit%d A? 0

Dimnsi"nss %nit /ct"r

A 6+rssd in trms ", ma*nit%d anddircti"n s+araty

A

A

1

P"siti/ scaar D=ns ma*nit%d ", A

Dimnsi"nss /ct"r D=ns dircti"n and sns ", A

7i '

%sdt#r"%*#"%t t#is '""

C t i V t ( t )


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Cartsian Vct"r (c"nt)

InternalInternal

CartsianVct"r

R+rsntati"n

!a*nit%d ",Cartsian

Vct"r

DCart

6

y

$

Azk

Axi

Ayj

A

k

ij

Ad/anta*s5 Sim+i=cati"n ", /ct"r

a*'ra S+arati"n 't&n

ma*nit%d and dircti"n ",ac# c"m+"nnt /ct"r

x

y

z

Azk

Axi

Ayj

A

A

A

Az

Ax

Ay

A is +"siti/ s%ar r""t ", t#s%m ", s%ars ", its

6

Axi


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Additi"n and S%'tracti"n ", Cartsian Vc

Internal

A

B

R

z

y

x

i

Additi"n ", Cartsian Vct"rs

R= A + B9

S%'tracti"n ", Cartsian Vct"rs

R= A -B9

C"nc%rrnt F"rc Systm

A*'raic s%ms ", t# rs+cti/ 6@y@$ "r i@'@0


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-RO1P PROB(M SOV23-

1) .sin* t# *"mtry and tri*"n"mtry@ rs"/and &rit F1and F2in t# Cartsian /ct"r ,"r

2) AddF1andF2t" *t FR

3) Dtrmin t# ma*nit%d and an*s @ @

-iven: T# scr& y iss%'ctd t" t&","rcs@ F1and F2

.ind: T# ma*nit%d ant# c""rdinatdircti"n an*s ", rs%tant ,"rc

Plan:

Internal

Internal

D t P d t


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Dot Product

Internal

Partic%ar mt#"d ,"r Gm%ti+yin* t&" /ct"rs .sd t" =nd t# an* 't&n t&" ins "r c"m+"nnts in A&ays r,r as scaar +r"d%ct ", a /ct"r

0IJI1K0

Laws of Operation Cartesian Vector Formulation

1. Commutative Law :A.B = B.A

2. Multiplication by Scalar :aA.B! = aA!.B = A.aB! = A.B!a

". #i$tributive Law :A. B+#! = A.B! + A.#!

#ot pro%uct &or eac' o& Carte$ian v

e.(. i.i = 1!1!co$ )= 1* i. = 1!i.i = 1 . = 1 -.-=1i. = ) i.- = ) -. = )

#ot pro%uct o& 2 (eneral vector$ AA.B = Axi+ Ayj+ Azk! . Bxi+ B

= AxBx+ AyBy+ AzBz

pp ca ons o o ro uc


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pp ca ons o o ro ucAn* ,"rm 't&n t&" /ct"rs "r

intrsctin* ins

Internal

0IJI1K0

A.B = AxBx+ AyBy+ AzBz

& A.B = )

Aperpen%icular to B

Applications of Dot Prouct !"#


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Applications of Dot Prouct !"#

/'e component$ o& vector parallel an% perpen%icular to a line

Internal

A parallel to or collinear wit$ t$e line aa%

0roection o& A onto t'e line

ector repre$entation o&

#irection$ o& t'e line:

Scalar proection o& A alon( a line i$ %etermine%

&rom t'e %ot pro%uct o& A an% t'e unit vector u

A perpenicular to aa%

1. #etermine &rom t'e %ot pro%u

2. i$ -nown by t'e 0yt'a(orean /'eorem


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(,AMP(

Plan:

1 Find rAO

2 Find t# an* 9 c"s81L(F rAO)(F rAO)M

3 Find t# +r"cti"n /ia FAO9 F uAO ("r F c"s

)

-iven: T# ,"rc actin* "nt# #"" at +"int A

.ind: T# an* 't&nt# ,"rc /ct"r andt# in AO@ and t#ma*nit%d ", t#+r"cti"n ", t# ,"rca"n* t# in AO

Internal

Internal


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(,AMP((c"ntin%d)

rAO9 L(081) i


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uAO9 rAOrAO 9 (13)i < (23)j < (23) k

FAO9 F uAO9 (N)(13) < ()(23) < (3)(23) 9N00

Or5 FAO 9 F c"s 9 1122 c"s (QNQ) 9 N00

(,AMP((c"ntin%d)

Internal

Internal

02 Chapter 2 – Force Vector

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