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27
y
y
y
y
y
a
C
Figure Area & Centroid Area Moment of Inertia (Radius of Gyration) 2 Product of Inertia
C a
C
b
a
2a
a C
C
x
x
x
x
x
A = πa 2
I xc = I yc
= πa
xc = a
I x = I y = 5πa
yc = a 4
J = πa
2
A = π (a 2 – b 2 )
xc = a
yc = a
A = πa 2 /2
xc = a
yc = 4a/(3π)
2
A = a θ
2a
sinθ
xc
=
3 θ
y = 0
c
2⎛
sin2θ
⎞
A = a ⎜θ
− ⎟
⎝ 2 ⎠
3
2a
sin θ
xc
=
3 θ − sinθcosθ
y = 0
c
4
4
I x = I y = π
c c
5πa
I x = I y =
4
J = π
I
I
I
I
xc
yc
x
y
4
4
4 4 ( a − b ) 2
4
4 4 ( a − b )
4
2 ( 9π
− 64)
a
=
72π
4
= πa
8
4
= πa
8
4
= 5πa
8
4
2 2 πb
− πa
b −
4
Ix = a 4 (θ – sinθ cos θ)/4
Iy = a 4 (θ + sinθ cos θ)/4
I
I
x
y
Aa
=
4
Aa
=
4
2
2
⎡
3
2sin
θ cosθ
⎤
⎢1
−
⎥
⎣ 3θ
− 3sin
θ cosθ
⎦
⎡ 3
2sin
θ cosθ
⎤
⎢1
+
⎥
⎣ θ − sin θ cosθ
⎦
4
2 2
x = r
c yc
2 2
x = ry
2 2
p = a
r
r
r
2
x =
c c
2
x =
2
p =
r
r
r
2
x
r
2
yc
2
x
2
y
r
r
r
c
2
x
r
2
y
r
2
y
2
= a
= 5a
2
2
4
4
2 2 2
ry
= ( a + b )
2 2 2
ry
= ( 5a
+ b )
2 2 ( a + b ) 2
a
=
= a
2
2
2
= a 4
2
= 5a
4
a
=
2
4 2
a
=
4
2 ( 9π
− 64)
Housner, George W. & Donald E. Hudson, Applied Mechanics Dynamics, D. Van Nostrand Company, Inc., Princeton, NJ, 1959. Table reprinted by permission of G.W. Housner & D.E. Hudson.
2
x
r
r
36π
4
2
4
4
( θ − sin θ cosθ)
θ
( θ + sin θ cosθ)
θ
2 ⎡
3
a 2sin
θ cosθ
⎤
= ⎢1
−
⎥
4 ⎣ 3θ
− 3sinθ
cosθ⎦
2 ⎡
3
a 2sin
θ cosθ
⎤
= ⎢1
+
⎥
4 ⎣ θ − sinθ
cosθ⎦
I
I
xc yc
I
I
I
I
I
I
I
I
xy
xy
= 0
= Aa
2
y = 0
= Aa
2
= πa
xc c
xc yc
xy
xc
y
xy
xc
y
xy
= 0
= 2a
2
= 0
c
= 0
= 0
c
= 0
2
2 2 ( a − b )
3
STATICS (continued)