02 chapter 2 – force vector
Post on 24-Feb-2018
224 Views
Preview:
TRANSCRIPT
-
7/24/2019 02 Chapter 2 Force Vector
1/20
Statics and Dynamics (1013)CHAPTER 2 FORCE VEC
Dr !"#d A$%&an 'in !a"insr
Ptr"%m En*inrin* D+artmnt
Fac%ty ", -"scincs and Ptr"%m
.ni/rsiti Tn""*i PETROAS (.TP)
Internal
-
7/24/2019 02 Chapter 2 Force Vector
2/20
C#a+trO%tin
Scaars and Vct"rs
Vct"r O+rati"ns
Vct"r Additi"n ", F"rcs
Additi"n ", a Systm ", C"+anar F"rcs
Cartsian Vct"rs
Additi"n and S%'tracti"n ", Cartsian Vct"
P"siti"n Vct"rs
F"rc Vct"r Dirctd a"n* a in
D"t Pr"d%ct
Internal
-
7/24/2019 02 Chapter 2 Force Vector
3/20
Scalars and Vectors
Scalars Vectors
%antity c#aractri$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
-
7/24/2019 02 Chapter 2 Force Vector
4/20
Vector Operations
Internal
VMultiplication and
Division of a Vector by aScalar
P
Tr
R9A
-
7/24/2019 02 Chapter 2 Force Vector
5/20
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
-
7/24/2019 02 Chapter 2 Force Vector
6/20
Vct"r Additi"n ", F"rcs
Internal
F2
F3
F1
F2
F3
F1
F2
F1
F1< F2 F1< F2
1
2
3
-
7/24/2019 02 Chapter 2 Force Vector
7/20
A++icati"n ", a& ", Sin and C"sin in dtma*nit%d and dircti"n ", /ct"r ,"r
Internal
-
7/24/2019 02 Chapter 2 Force Vector
8/20
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
-
7/24/2019 02 Chapter 2 Force Vector
9/20
C"+anar F"rc Rs%tants
Internal
PROB(M
S"(P )
Scaar "tati"n
S"(P *'y
Carts
V
S"(P +
-
7/24/2019 02 Chapter 2 Force Vector
10/20
(,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
-
7/24/2019 02 Chapter 2 Force Vector
11/20
#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 )
-
7/24/2019 02 Chapter 2 Force Vector
12/20
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
-
7/24/2019 02 Chapter 2 Force Vector
13/20
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
-
7/24/2019 02 Chapter 2 Force Vector
14/20
-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
-
7/24/2019 02 Chapter 2 Force Vector
15/20
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
-
7/24/2019 02 Chapter 2 Force Vector
16/20
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 !"#
-
7/24/2019 02 Chapter 2 Force Vector
17/20
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
-
7/24/2019 02 Chapter 2 Force Vector
18/20
(,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
-
7/24/2019 02 Chapter 2 Force Vector
19/20
(,AMP((c"ntin%d)
rAO9 L(081) i
-
7/24/2019 02 Chapter 2 Force Vector
20/20
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
top related