Moment of inertia

1. Moment of Inertia

2.  CO 3
Apply the conditions of equilibrium to various practical problems
involving different force
system.
 CO 4
Choose appropriate theorems, principles or formulae to solve
problems of mechanics.
 CO 5
Solve problems involving rigid bodies, applying the properties of
distributed areas and masses
Course Outcomes: After completion of the course the student will
be able to:
Module 3_Moment of Inertia 2

3. Module 3_Moment of Inertia 3
What is a Moment of Inertia?
 It is a measure of an object’s resistance to changes to its rotation.
 Also defined as the capacity of a cross-section to resist bending.
 It must be specified with respect to a chosen axis of rotation.
 It is usually quantified in m4 or kgm2

4. Area Moments of Inertia
• Moment of inertia also called second moment of area of plane figure with
respect to x and y axis in its plane is defined respectively by
• It is the integral of the second moment of an elemental area about
reference axis.
• Moment of Inertia of an area is purely a mathematical property of the area

5. Polar Moment of Inertia
• Moment of inertial of physical body is called mass moment of inertia
• Moment of inertia about an axis perpendicular to the plane of area.
Denoted by J or Izz

6. Radius of Gyration
Module 3_Moment of Inertia 6

7. Module 3_Moment of Inertia 7
Radius of gyration
A
A B
A
A B
k

8. Module 3_Moment of Inertia 8
PARALLEL AXIS THEOREM
•It states that if the moment of inertia of a plane area about an axis in the plane of area through the C.G. of
the plane area be represented by IG, then the moment of the inertia of the given plane area about a parallel
axis AB in the plane of area at a distance h from the C.G. of the area is given by
IAB = IG + Ah2.
where IAB = Moment of inertia of the given area about AB
IG = Moment of inertia of the given area about C.G.
A = Area of the section
h = Distance between the C.G. of the section and the axis AB.
•It is a transfer theorem which is used to transfer moment of inertia from one axis to another axis
•These 2 axis should be parallel to each other and one of these axis should be centriodal axis

9. Module 3_Moment of Inertia 9

10. Module 3_Moment of Inertia 10
PERPENDICULAR AXIS THEOREM
Theorem of the perpendicular axis states that if IXX and IYY be the moment of inertia of a plane section
about two mutually perpendicular axis X-X and Y-Y in the plane of the section, then the moment of
inertia of the section IZZ about the axis Z-Z, perpendicular to the plane and passing through the
intersection of X-X and Y-Y is given by
IZZ = IXX + IYY.
The moment of inertia IZZ is also known as polar moment of inertia.

11. Module 3_Moment of Inertia 11

12. Moment of inertia of rectangle
Y
O
X
𝑮
𝒃
𝒉
Y
X
𝑰𝒙𝒙 =
𝒃 ⅆ𝟑
𝟏𝟐
𝑰𝒚𝒚 =
ⅆ𝒃𝟑
𝟏𝟐
12

13. Moment of inertia of rectangle about base
Y
O
X
𝑮
𝒃
𝒉
Y
X
𝑰𝑨𝑩 =
𝒃 ⅆ𝟑
𝟑
A B
13

14. Moment of inertia of a triangle
V
O B
𝑮
𝑩
𝑯
• 𝑰𝑽 =
𝑩𝑯𝟑
𝟒
• 𝑰𝑮 =
𝑩𝑯𝟑
𝟑𝟔
• 𝑰𝑨𝑩 =
𝑩𝑯𝟑
𝟏𝟐
𝒉
A
14

15. Moment of inertia of a circle
Y
O
X
G
𝑫
𝑫
𝑰𝒙𝒙 =
𝝅𝑫𝟒
𝟔𝟒
𝑰𝒚𝒚 =
𝝅𝑫𝟒
𝟔𝟒
Y
X
15

16. Moment of inertia of semicircle
Y
O
X
𝑹
Y
X
• 𝑰𝑨𝑩 =
𝝅𝑫𝟒
𝟏𝟐𝟖
• 𝑰𝑿𝑿 =
𝟎. 𝟏𝟏𝑹𝟒
• 𝑰𝒀𝒀 =
𝝅𝑫𝟒
𝟏𝟐𝟖
B
A
16
G

17. Moment of inertia of quarter circle
Y
O
X
G
𝑹
𝑹
X
Y
• 𝑰𝑨𝑩 =
𝝅𝑫𝟒
𝟐𝟓𝟔
• 𝑰𝑿𝑿 =
𝟎. 𝟎𝟓𝟓𝑹𝟒
• 𝑰𝒀𝒀 =
𝟎. 𝟎𝟓𝟓𝑹𝟒
17

18. Module 3_Moment of Inertia 18

19. 1. Calculate the moment of inertia of the angle section having the dimensions as shown about X
and Y axis shown.
19
10 cm
10 cm
2 cm
2 cm
8 cm
10 cm
2 cm
1
2
2 cm
G2
G1
MOMENT OF INERTIA OF COMPOSITE AREAS

20. Module 3_Moment of Inertia 20

21. Module 3_Moment of Inertia 21

22. Module 3_Moment of Inertia 22

23. Module 3_Moment of Inertia 23

24. Module 3_Moment of Inertia 24

25. Module 3_Moment of Inertia 25

26. Module 3_Moment of Inertia 26
Fig. shows a T-section of dimensions 10 × 10 × 2 cm. Determine the moment of inertia of the section about the
horizontal and vertical axes, passing through the center of gravity of the section. Also find the polar moment of
inertia of the given T-section.
Ixx = 314.221
Iyy =172
Izz = Ixx + Iyy = 486.221

27. Module 3_Moment of Inertia 27
Find the moment of inertia of the section shown in Fig. about the centroidal axis X-X perpendicular to the web.
2166.667

28. Module 3_Moment of Inertia 28

29. Module 3_Moment of Inertia 29
Determine the moments of inertia of the section about horizontal and vertical axes passing through the
centroid of the section.
28.84 × 106
8.4 × 106

30. Module 3_Moment of Inertia 30

31. Module 3_Moment of Inertia 31

32. Module 3_Moment of Inertia 32
Find the moment of inertia of the lamina with a circular hole of 30 mm diameter about the axis AB as shown in
Fig. [Ans. 638.3 × 103 mm4]

33. Module 3_Moment of Inertia 33

34. Module 3_Moment of Inertia 34

35. Module 3_Moment of Inertia 37
Mass Moment of Inertia

36. Module 3_Moment of Inertia 38
Mass Moment of Inertia of a Circular Disc.

37. Module 3_Moment of Inertia 39

38. Module 3_Moment of Inertia 40

39. Mass moment of inertia of a ring of Radius R
Let A be the cross-sectional area of the ring and 𝝆 be the mass density of the ring material
Consider an elemental length dl
Volume of this elemental length = A x dl
Mass of this elemental volume = density x volume = 𝝆 x A x dl
Second moment of this mass about ZZ axis = ⅆ𝒎 𝒙𝟐
= 𝝆 x A x dl x 𝑹𝟐
41

40. 𝑰𝒛𝒛 = 𝟎
𝟐𝝅𝑹
𝝆 x A x dl x 𝑹𝟐
= 𝑹𝟐𝝆 x A 𝒍 𝟎
𝟐𝝅𝑹
= 𝑹𝟐𝝆 x A x 𝟐𝝅𝑹
= 𝝆 x A x 𝟐𝝅𝑹x 𝑹𝟐
= m𝑹𝟐
𝑰𝒛𝒛= 𝑰𝑿𝑿 + 𝑰𝒀𝒀 = 2 𝑰𝑿𝑿
Therefore
𝑰𝑿𝑿 = 𝑰𝒀𝒀 =
m 𝑹𝟐
𝟐
42

41. Module 3_Moment of Inertia 43
Mass Moment of inertia of a Solid cylinder
Here we have to consider a few things:
•The solid cylinder has to be cut or split
into infinitesimally thin rings.
•Each ring consists of the thickness of dr
with length L.
•We have to sum up the moments of
infinitesimally these thin cylindrical
shells.

42. Module 3_Moment of Inertia 44

43. Module 3_Moment of Inertia 45

44. Theorem of Pappus’s and Guldinus
• Surface area and volume generated by rotating a curve and a plane
area about a non intersecting axis
• Another method for finding Centroid/Center of gravity.
• Surface of revolution
• The surface generated by rotating a plane curve about a fixed axis
• Body of revolution
• The body generated by rotating a plane area about a fixed axis
Module 3_Pappu's Theorem 46

45. Module 3_Pappu's Theorem 47
The first theorem of Pappus states that the surface area of a surface of
revolution generated by the revolution of a curve about an external axis is equal
to the product of the arc length of the generating curve and the distance
traveled by the curve's geometric centroid.
First theorem of Pappus

46. Module 3_Pappu's Theorem 48

47. Module 3_Pappu's Theorem 49
Second theorem of Pappus
The second theorem of Pappus states that the volume of a solid of
revolution generated by the revolution of a lamina about an external axis is
equal to the product of the area A of the lamina and the distance traveled by
the lamina's geometric centroid.

48. Module 3_Pappu's Theorem 50

49. Module 3_Moment of Inertia 51

50. • A small block of weight 1000 N as shown in Figure, is placed on a 30°
inclined plane with μ= 0.25. Determine the horizontal force to be
applied for impending motion down the plane
Module 3_Moment of Inertia 52

51. Module 3_Moment of Inertia 53
Find the centre of gravity of lamina from O.