Centros de Massa Figuras simples

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y
A
x
Circular area
Ix = r414
C
pi
Iy = r414pi
= pi
r
r2
–
–
y
A
x
Semicircular area
Ix = r418
C
pi
Iy = r418pi
=
pi
2
r
4
3
r
pi
r2
—
—–
–
–
1
2–
Ix = r4116 pi
Iy = r4116 pi
–
I ( – sin 2 )x = r414–
b
h
y A = bh
x
Rectangular area
Ix = bh3112
Iy = hb3112
C
—
—
b
h C
A = bh
x
Triangular area
Ix = bh3136h13
1
2
–
–
—
C
y
x
A = r214–
Quarter circle area
4
3
r
pi
—
4
3
r
pi
pi
—
C
L = r2–
Quarter and semicircle arcs
2r
pi
pi
—
r sin
r
r
—–––
C
y
x
A =θ
θ
θ
θ
θ
r2
Circular sector area
θ θ
1
2–I ( + sin 2 )y = r414– θ θ2
3–
r sin
r
—–––
C
y
x
L = 2θ
θ
θ
θ
θ
r
Circular arc segment
b
h
a A
x
Trapezoidal area
C
= (a + b)h12
1
3
2
a + b
a+ b h———–
–
b
b
C
C
a
A ab=
Exparabolic area
Parabolic area
a3–4
a
b310—
a25—
1
3—
A ab= 43—
L
C
=
rr
rpi
Ca
A= ab23
b
3
8
–
–
– a
3
5
Semiparabolic area
b
Geometric Properties of Line and Area Elements
Centroid Location Centroid Location Area Moment of Inertia
–
––
G
Sphere
x
y
z
r
Ixx = Iyy = Izz = mr 225–
V = r 343pi–
V = h
h
r 213pi–
G
Hemisphere
x
y
z
r
Ixx = Iyy = 0.259mr2 Izz = mr225–
V = r 323pi–
r3–8
h
–2
h
–2
h
–4
–2
–2
G
Thin Circular disk
x
y
z
z'
r
Ixx = Iyy = mr2 Izz = mr212– Iz'z' = mr
23
2–
1
4–
G
Cone 
x
y
z
r
Ixx = Iyy = m380––
Ixx = mb2 Iyy = ma2
Ixx = Iyy = m 2 Ix'x' = Iy'y' = m 2 Iz'z' = 0
G
Cylinder
x
y
z
r
Izz = mr2Ixx = Iyy = m(3r2 + h2) 
Izz = mr2(4r2 + h2) 
1
12––
3
10––
G
Thin ring
x
y
r
Ixx = Iyy = mr2 Izz = mr212–
z
G
Slender Rodx'
y
y'
z
x
Thin plate
x
y
V = hr 2pi
z
a
b
1
12––
1
12––
1
12–– Izz = m(a2 + b2)112––
1
3–
G
1
2––
Center of Gravity and Mass Moment of Inertia of Homogeneous Solids