2. Objectives:
Introduction of Moment of Inertia
Polar moment of inertia
Radious of Gyration
Moment of inertia of composite bodies
Parallel axis theorm
Perpendicular axis theorm
3. MOMENT OF INERTIA
The product of the elemental area and square of the
perpendicular distance between the centroids of area
and the axis of reference is the “Moment of Inertia”
about the reference axis.
Ixx = ∫dA. y2
Iyy = ∫dA. x2
dA
Y
x
y
x
5. Moment of Inertia of a Disk and a Ring
A wooden disk and a metal ring with the same diameter and equal mass roll with
different accelerations down an inclined plane. Now, if both are rolled along the lecture
bench by being given an equal impulse, the metal ring will continue to roll longer
than the disk because of its greater moment of inertia. However, when the ring
and disk are released simultaneously from the same height on an inclined plane, the
wooden disk reaches the bottom first due to its lesser moment of inertia.
6. Polar Moment of Inertia
• The polar moment of inertia is an important
parameter in problems involving torsion of
cylindrical shafts and rotations of slabs.
J0
r 2 dA
• The polar moment of inertia is related to the
rectangular moments of inertia,
J0
r 2 dA
Iy
Ix
x2
y 2 dA
x 2 dA
y 2 dA
7. Radius of Gyration
Frequently tabulated data related to moments
of inertia will be presented in terms of radius
of gyration.
I
mk
2
Where
m = Mass of the body
K= Radious of Gyration
or
k
I
m
9. Parallel Axis Theorem
• The moment of inertia about any axis parallel
to and at distance d away from the axis that
passes through the centre of mass is:
IO
I G md
2
• Where
o IG= moment of inertia for mass centre G
o m = mass of the body
o d = perpendicular distance between the parallel axes.
10. Perpendicular Axis Theorem
Iy= (1/12) Ma2
b
Ix = (1/12) Mb2
M Iz
Ix
Iy
a
Iz = (1/12) M(a2 + b2)
For flat objects the
rotational moment
of inertia of the
axes in the plane
is related to the
moment of inertia
perpendicular to
the plane.
