Chap. 10
Moments of Inertia
Chapter Outline
Definitions of Moments of Inertia for Areas
Parallel-Axis Theorem for an Area
Radius of Gyration of an Area
Moments of Inertia for Composite Areas
Product of Inertia for an Area
Moments of Inertia for an Area about
Inclined Axes
Mohr’s Circle for Moments of Inertia
Mass Moment of Inertia
Definition
moment of inertia about x-axis
(second moment)
I x = ∫ y 2 dA
A
polar moment of inertia
(units: m4, mm4,…)
I y = ∫ x 2 dA
A
J O = ∫ r 2 dA = I x + I y
A
3
Parallel-Axis Theorem
I x = I x ' + Ad y
2
I y = I y ' + Ad x
2
J O = J C + Ad
2
4
Radius of Gyration
kx =
Ix
A
ky =
Iy
A
kO =
JO
A
By integration
5
6
7
8
9
10
11
p.519, 10-8.
Determine the moment of inertia of the shaded area about the x and y axes
For x :
12
p.521, 10-24
Determine the moment of inertia of the area about the x and y
axes
13
Composite Areas
Ex.10-5,
compute the moment of inertia of the composite area about the x
axis
Circle
I x = I x ' + Ad y
Rectangle
2
1
= π (25) 4 + π (25) 2 (75) 2
4
= 11.4 × 106 mm 4
2
I x = I x ' + Ad y
1
= (100)(150) 3 + (100)(150)(75) 2
12
= 112.5 × 106 mm 4
∴ I x = (112 .5 − 11.4) × 10 6 = 101 × 10 6 mm 4
1
(100)(150)3
3
p.526, 10-32.
Determine the moment of inertia of the composite area about the x
axis.
18
p.528, 10-52
Determine the beam’s moment of inertia Iy about the centroidal y axes
20
Mass Moment of Inertia
a property that measures the resistance of the body
to angular acceleration
I = ∫ r 2 dm
m
units: kg․m2 or slug․ft2
22
Mass Moment of Inertia
I = ∫ r 2 ρdV
(variable density ρ)
I = ρ ∫ r 2 dV
(ρ is a constant)
V
V
dV = dxdydz
23
24
25
Mass Moment of Inertia
Parallel-Axis Theorem
I = ∫m r 2 dm = ∫m [(d + x' ) 2 + y '2 ]dm
0
= ∫m ( x' + y ' )dm + 2d ∫m x' dm + d 2 ∫m dm
2
2
IG: moment of inertia about the axis
passing through mass center G
Ι = Ι G + md 2
Radius of Gyration
I
I = mk or k =
m
2
26
27
28
29
30
p. 555, 10-100
Determine the mass moment of inertia of the pendulum about an
axis perpendicular to the page and passing through point O.The
slender rod has a mass of 10 kg and the sphere has a mass of 15
kg.
p.564, 10-103.
Determine the mass moment of inertia of the over hung crank about3
the x’ axis. The material is steel having a density of ρ = 7.85 Mg/m .