Chap. 10 Moments of Inertia

Chapter Outline

Definitions of Moments of Inertia for AreasParallel-Axis Theorem for an AreaRadius of Gyration of an AreaMoments of Inertia for Composite AreasProduct of Inertia for an AreaMoments of Inertia for an Area about Inclined AxesMohr’s Circle for Moments of InertiaMass Moment of Inertia

3

Definition

∫∫∫

+==

=

=

A yxO

Ay

Ax

IIdArJ

dAxI

dAyI

2

2

2

moment of inertia about x-axis(second moment)

polar moment of inertia(units: m4, mm4,…)

4

Parallel-Axis Theorem

2

2

2

AdJJ

AdII

AdII

CO

xyy

yxx

+=

+=

+=

'

'

5

Radius of Gyration

AIk x

x = AI

k yy = A

Jk OO =

By integration

6

7

8

9

10

11

12

p.519, 10-8. Determine the moment of inertia of the shaded area about the x and y axes

For x :

13

p.521, 10-24Determine the moment of inertia of the area about the x and y axes

Composite Areas

Ex.10-5, compute the moment of inertia of the composite area about the x axis

2yxx AdII += '

46

224

10411

75252541

mm×=

+=

.

)()()( ππ

2yxx AdII += '

46

23

105112

75150100150100121

mm×=

+=

.

))()(())((

Circle Rectangle

466 10101104115112 mmI x ×=×−=∴ )..(3(100)(150)

31

18

p.526, 10-32. Determine the moment of inertia of the composite area about the x axis.

20

p.528, 10-52 Determine the beam’s moment of inertia Iy about the centroidal y axes

22

Mass Moment of Inertia

a property that measures the resistance of the body to angular acceleration

∫= mdmrI 2

units: kg․m2 or slug․ft2

23

Mass Moment of Inertia

∫∫

=

=

V

V

dVrI

dVrI2

2

ρ

ρ (variable density ρ)

(ρ is a constant)

dV = dxdydz

24

25

26

Mass Moment of Inertia

Parallel-Axis Theorem

Radius of Gyration

∫+∫+∫ +=

∫ ++=∫=

mmm

mm

dmddmxddmyx

dmyxddmrI222

222

'2)''(

]')'[(

IG : moment of inertia about the axis passing through mass center G

0

2G mdΙΙ +=

mIkmkI 2 == or

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 about the x’ axis. The material is steel having a density of ρ = 7.85 Mg/m3.