Download - 13874_Lecture Notes 2.1 - 2.2_2.4
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
FORCE VECTORS, VECTOR OPERATIONS & ADDITION COPLANAR FORCES
In-Class activities:
• Chec% Ho&e'or%
• Reading (ui)
• pplication o* dding +orces
•
Parallelogra& !a'• Resolution o* a ,ector -sing
• Cartesian ,ector otation /C,0
• ddition -sing C,
• ttention (ui)
Todays O!"ective:
Students 'ill be able to
a0 Resol$e a "2 $ector into co&ponents.
b0 dd "2 $ectors using Cartesian $ector notations.
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
READIN# $%I
1. 4hich one o* the *ollo'ing is a scalar 5uantit67
0 +orce B0 Position C0 8ass 0 ,elocit6
". +or $ector addition9 6ou ha$e to use :::::: la'.
0 e'ton;s Second
B0 the arith&etic
C0 Pascal;s
0 the parallelogra&
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
APPLICATION OF VECTOR ADDITION
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
SCALARS AND VECTORS'Section ()*+
Scalars Vectors
E=a&ples 8ass9 ,olu&e +orce9 ,elocit6
Characteristics It has a &agnitude It has a &agnitude
/positi$e or negati$e0 and direction
ddition rule Si&ple arith&etic Parallelogra& la'
Special otation one Bold *ont9 a line9 an
arro' or a >carrot?
In these Po'erPoint presentations9 a $ector 5uantit6 is represented like this /in !old9 italics9 and green0.
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
VECTOR OPERATIONS'Section ()(+
Scalar 8ultiplication
and i$ision
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
VECTOR ADDITION %SIN# EITER TE PARALLELO#RA LA. OR TRIAN#LE
Parallelogra& !a'
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
>Resolution? o* a $ector is brea%ing up a $ector into
co&ponents.
It is %ind o* li%e using the parallelogra& la' in re$erse.
RESOL%TION OF A VECTOR
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
ADDITION OF A S/STE OF COPLANARFORCES 'Section ()0+
• Each co&ponent o* the $ector is sho'n as a &agnitude
and a direction.
• unit $ectors? i and j to designate the =
and 62a=es.
• 4e @resol$e; $ectors into co&ponents using the = and 62
a=is coordinate s6ste&.
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
+or e=a&ple9
F A += i +6 j or F' = += i /− +6 0 j
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
ADDITION OF SEVERAL VECTORS
• Step 3 is to *ind the &agnitudeand angle o* the resultant$ector.
• Step " is to add all the =2co&ponents together9
*ollo'ed b6 adding all the 62co&ponents together.
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
Brea% the three $ectors into co&ponents9 then add the&.
FR A F1 F2 F3
A +1= i +16 j − +"= i +"6 j +3= i − +36 j
A /+1= − +"= +3=0 i /+16 +"6 − +360 j
A /+R=0 i /+R60 j
n e=a&ple o* the process
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
You can also represent a "2 $ector 'ith a &agnitude and angle.
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
E1APLE
Plan:
a0 Resol$e the *orces into their =26 co&ponents.
b0 dd the respecti$e co&ponents to get the resultant $ector.
c0 +ind &agnitude and angle *ro& the resultant co&ponents.
#iven:
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
E1APLE /continued0
F1 A D# i 3## j
F2 A DF G# cos /G0 i G# sin /G0 j
A DF 31J." i 31J." j
F3 A D /30 L## i /G0 L## j
A D 3L# i GJ# j
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
E1APLE /continued0
Su&&ing up all the i and j co&ponents respecti$el69 'e get9
FR A D /# F 31J." 3L#0 i /3## 31J." GJ#0 j
A D G1.J# i 1#MJ j
x
y
φ
FR
-sing &agnitude and direction
+R A //G1.J#0" /1#MJ0
"01"
A 1#MM
φ A tan21
/1#MJG1.J#0 A JN.J
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
CONCEPT $%I
1. Can 6ou resol$e a "2 $ector along t'o directions9
'hich are not at M# to each other7 0 Yes9 but not uni5uel6.
B0 o.
C0 Yes9 uni5uel6.
". Can 6ou resol$e a "2 $ector along three directions/sa6 at #9 L#9 and 1"#07
0 Yes9 but not uni5uel6.
B0 o.
C0 Yes9 uni5uel6.
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
#RO%P PRO2LE SOLVIN#
Plan:
a0 Resol$e the *orces into their = and 62co&ponents.
b0 dd the respecti$e co&ponents to get the resultant $ector.
c0 +ind &agnitude and angle *ro& the resultant co&ponents.
#iven:
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
F * A DJ## cos /L#0 i J## sin /L#0 j
A D G## i LM".J j
F ( A D2L## sin /G0 i L## cos /G0 j
A D 2G"G.3 i G"G.3 j
F 3 A D/1"130 L# i − /130 L# j
D L## i − "# j
#RO%P PRO2LE SOLVIN# /continued0
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
"#13. ll rights reser$ed.
#RO%P PRO2LE SOLVIN# /continued0
Su&&ing up all the i and j co&ponents9 respecti$el69 'e get9
F R A D /G## F G"G.3 L##0 i /LM".J G"G.3 F "#0 j
A D N.N i JLN.1 j
o' *ind the &agnitude and angle9
+R A //N.N0" /JLN.10
"0
4 A 1#G1
φ A tanF1
/ JLN.1 N.N 0 A L.G
+ro& positi$e =2a=is9 φ A L.G
x
y
φ
FR
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Mechanics for Engineers: Statics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South sia Pte !td
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ATTENTION $%I
1. Resol$e F along = and 6 a=es and 'rite it in $ector *or&. F A D :::::::::::
0 J# cos /3#0 i F J# sin /3#0 j
B0 J# sin /3#0 i J# cos /3#0 j
C0 J# sin /3#0 i F J# cos /3#0 j
0 J# cos /3#0 i J# sin /3#0 j
". eter&ine the &agnitude o* the resultant /F1 F (0 *orce in 'hen F1 A D 1# i "# j and F ( A D "#
i "# j .
0 3# B0 G# C0 #
0 L# E0 N#
3#
=
6
+ A J#
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Mechanics for Engineers: Statics, 13th SI Edition © Pearson Education South sia Pte !td
End of the Lecture
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