jump to first page 1 mechanics ap200 dr. c. w. ong meriam, j.l., kraige, l.g., “engineering...
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MechanicsAP200Dr. C. W. Ong
Meriam, J.L., Kraige, L.G., “Engineering Mechanics, Dynamics”, John Wiley
Course Work 30% (Exercises and tests) Examination 70%
Download materials http://ap.polyu.edu.hk/apacwong/
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Chapter 1 Vector
A vector has a length A and a direction (unit vector)
A
AeAA
Ae
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2D Cartesian coordinate system (one form of presentation)
22
yx
)ˆ sinx (cos ˆ ˆ ˆ
yx
A
AAAA
yAy AxAeAA
x
y
A
YA
XA
Pythagoras theorem
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3-D Cartesian coordinate system
ˆ cos ˆ cos ˆ cos
ˆˆˆˆ
zAyAxA
zAyAxAeAA zyxA
AAAAA zyx
22222 :Note
z
A
x
e
yAy
Ax
Az
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Addition of vectors y
x
YB
xB
B
A
YA
XA
BA
Oy)(x)(
yx
yx
yyxx
yx
yx
BABABA
B BB
A AA
Subtraction of vectors y )(x)( - yyxx BABABA
y
x
YB
xB
B
BA
YA
XA
A
O
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scale)a(cos ABBΑ
yyxx BABABA
Dot (scalar) product of two vectors
In 2-D Cartesian coordination system
Definition:
Note: AB cos= AB cos (A - B)
= AB(cos A cos B +sin A sin B)
= (Acos A)(Bcos B) + (Asin A) (Bsin B)
A
B
A
x
y
B
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BABABABA zzyyxx
y
B
A
BA
O
From cosine law:
cos2)()(
)( )()(
)180cos(2
222222
222
222
ABBBBAAA
BABABA
BABABA
zyxzyx
zzyyxx
In 3-D Cartesian coordinate system:
x
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Application of dot product: Projection
Area A
A
A’
n z
a
l= l a n
Projected area = 'A = l a cos z
area projected'cos
cos)11(ˆˆ ˆ
Aal
AznAzA
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Exercise: Find the projection of vector with a
length of 10 unit along .
A
n
A
n
60o30o
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Cross product
sin ABBA
A
B
A
B
C
i
k
j
rule) hand(right and lar toperpendicu is
product cross ofresult theofDirection
BA
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zBABABA
BABA
AB
AB
ABBA
xyyx
xyyx
BAAB
AB
ˆ)(
) cos sin - cos(sin
)sin(
sin
A
B
Ax
y
B
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and kAjAiAA zYX
kBjBiBB zYX
jABkBAiBA zxyxzy
iABjBAkAB zyzxyx
zy
y
BBB
AAA
kji
BA
x
zx
In Cartesian coordinate system:
k)BABA(j)BABA(i)BABA(BA
)BABA()BABA()BABA(
)BABA2BABA2BABA2(
)BABABABABABA(
)BABA2BABA2BABA2BABABA(
)BABABABABABABABABA(
BA
)BABABA()BBB)(AAA(AB
BA
)BABABA(1ABcos1AB
cos1ABsinABBA
xyyxzxxzyzzy
2yzzy
2xzzx
2xyyx
zzyyzzxxyyxx
2y
2z
2x
2z
2zy
2x
2y
2z
2x
2y
2x
zzyyzzxxyyxx2
z2
z2
y2
y2
x2
x
2z
2z
2y
2z
2x
2z
2zy
2yy
2x
2y
2z
2x
2y
2x
2x
2x
22
2zzyyxx
2z
2y
2x
2z
2y
2x
22
2zzyyxx2
2