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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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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APPLICATION OF VECTOR ADDITION

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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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VECTOR OPERATIONS'Section ()(+

Scalar 8ultiplication

and i$ision

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VECTOR ADDITION %SIN# EITER TE PARALLELO#RA LA. OR TRIAN#LE

Parallelogra& !a'

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>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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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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+or e=a&ple9

F A += i +6 j or F' = += i /− +6 0 j

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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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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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You can also represent a "2 $ector 'ith a &agnitude and angle.

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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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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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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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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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#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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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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#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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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

Let Learning Continue

13874_lecture notes 2.1 - 2.2_2.4

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