13883_lecture notes 2.7 - 2.8 (1)

8/18/2019 13883_Lecture Notes 2.7 - 2.8 (1)

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8/18/2019 13883_Lecture Notes 2.7 - 2.8 (1)

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Mechanics for Engineers: Statics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap

© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.

POSITION VECTORS & FORCE VECTORS

In-Class Activities:

• Che& Ho'e(or&

• Reading )ui*

• !ppliations+Rele%ane

• rite Position-etors

• rite a ore -etor

along a line• Conept )ui*

• /roup Proble'

• !ttention )ui*

Today’s Objectives:

Students (ill be able to 0

a Represent a position %etor inCartesian oordinate 2or', 2ro' gi%engeo'etr.

b Represent a 2ore %etor diretedalong a line.

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REAIN! "#I$1. 4he position %etor r PQ is obtained b

! Coordinates o2 ) 'inus oordinates o2 the origin.

B Coordinates o2 P 'inus oordinates o2 ).

C Coordinates o2 ) 'inus oordinates o2 P.

5 Coordinates o2 the origin 'inus oordinates o2 P.

#. ! 2ore o2 'agnitude , direted along aunit %etor U , is gi%en b F 6 777777 .

! 8U

B U +

C + U

5 9 U

E : U

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APP%ICATIONS

4his ship;s 'ooring line,onneted to the bo(, anbe represented as aCartesian %etor.

hat are the 2ores in the'ooring line and ho( do(e 2ind their diretions<

h (ould (e (ant to&no( these things

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APP%ICATIONS 8ontinued

4his a(ning is held up b three hains. hat are the 2oresin the hains and ho( do (e 2ind their diretions< h(ould (e (ant to &no( these things

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POSITION VECTOR

Consider t(o points, ! and B, in 3=5 spae."et their oordinates be 8>!, Y!, ?! and 8>B, YB, ?B,

respeti%el.

! position %etor is de2ined as

a 2i@ed %etor that loates apoint in spae relati%e toanother point.

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POSITION VECTOR

4he position %etor direted 2ro' ! to B, r AB , is de2ined as

r AB 6 A8 >B : >! i 9 8 YB : Y! j 9 8 ?B : ?! k '

Please note that B is the ending point and ! is the startingpoint. !"!YS subtrat the tailD oordinates 2ro' the tipDoordinates

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FORCE VECTOR IRECTE A%ON! A %INESection '()*

a ind the position %etor, r AB , along t(o points

on that line.

b ind the unit %etor desribing the line;sdiretion, u AB 6 8r AB +r!B.

Fultipl the unit %etor b the 'agnitude o2 the2ore, F = u AB .

I2 a 2ore is direted along a line,then (e an represent the 2ore%etor in Cartesian oordinatesb using a unit %etor and the2ore;s 'agnitude. So (e needto0

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E+A,P%E

Plan:1. ind the position %etor r AC and its unit %etor u AC .

#. Gbtain the 2ore %etor as F AC 6 #$ u AC .

!iven: 4he #$ 2orealong the able !C.

Find: 4he 2ore F AC in the

Cartesian %etor 2or'.

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E+A,P%E 8ontinued

8e an also 2ind r AC b subtrating

the oordinates o2 ! 2ro' theoordinates o2 C.

r!C 6 A## 9 3# 9 8=J#1+# 6 '

o( u AC 6 r AC +r!C and F AC 6 #$ u AC 6 #$ 8r AC +r!C

So F AC 6 #$A 8# i 9 3 j − J k +

6 A1#$ i 9 1L$ j = 3J$ k

!s per the 2igure, (hen relating ! to

C, (e (ill ha%e to go # ' in the @=diretion, 3 ' in the =diretion,and =J ' in the *=diretion. Hene,

r AC 6 A# i 9 3 j − J k '.

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CONCEPT "#I$

1. P and " are t(o points in a 3=5 spae. Ho( are the

position %etors r PQ and r QP related<! r PQ = r QP B r PQ = - r QP

C r PQ 6 1+r QP 5 r PQ = 2 r QP

#. I2 F and r are 2ore and position %etors, respeti%el, in SIunits, (hat are the units o2 the e@pression 8r M 8F + <

! e(ton B 5i'ensionless

C Feter 5 e(ton = Feter

E 4he e@pression is algebraiall illegal.

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!RO#P PRO%E, SO%VIN!

Plan:

1 ind the 2ores along !B and !C in the Cartesian %etor 2or'.# !dd the t(o 2ores to get the resultant 2ore, F R.

3 5eter'ine the 'agnitude and the oordinate angles o2 F R.

!iven: 4(o 2ores are ating on

a 2lag pole as sho(n inthe 2igure. B 6 $$

and C 6 NJ$

Find: 4he 'agnitude and theoordinate diretion

angles o2 the resultant2ore.

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!RO#P PRO%E, SO%VIN! 8ontinued

F AC 6 NJ$ 8r AC + r!C

F AC 6 NJ$ 83 i 9 # j : J k +

F AC 6 A#$ i 9 1J$ j : L$ k

F AB 6 $$ 8r AB + r!B

F AB 6 $$ 8 # i : 3 j : J k +

F AB 6 8#$$ i : 3$$ j : J$$ k

r AB 6 A# i − 3 j − J k 'r AC 6 A3 i 9 # j − J k '

r!B 6 A## 9 8=3# 9 8=J#1+# 6 '

r!C 6 A3# 9 ## 9 8=J#1+# 6 '

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R 6 A$# 9 8=1$# 9 8=1$L$#1+#

6 11.J

R 6 11N

F R 6 F AB 9 F AC

6 A$ i : 1$ j : 1$L$ k

!RO#P PRO%E, SO%VIN! 8ontinued

α 6 os=18$+11.J 6 JL.$O

β 6 os=18:1$+11.J 6 J.LO

γ 6 os=18:1$L$+11.J 6 1NJ.LO

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ATTENTION "#I$

1. 4(o points in 3:5 spae ha%e oordinates o2 P 81, #, 3 and

) 8, N, J 'eters. 4he position %etor r QP is gi%en b

! A3 i 9 3 j 9 3 k '

B A: 3 i : 3 j : 3 k '

C AN i 9 j 9 k '

5 A: 3 i 9 3 j 9 3 k ' E A i 9 N j 9 J k '

#. ! 2ore %etor, F , direted along a line de2ined b P) is gi%en b

! 8F + r PQ B r PQ +rP)

C 8r PQ +rP)5 8rP) +r PQ

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End of the Lecture

Let Learning Continue

13883_lecture notes 2.7 - 2.8 (1)

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