Moment of Inertia
EF 202 - Week 15
EF 202, Module 4, Lecture 3
Terminology
• Moment of inertia = second mass
moment
• Instead of multiplying mass by
distance to the first power (which
gives the first mass moment), we
multiply it by distance to the second
power.
EF 202, Module 4, Lecture 3
2
Definitions
Moment of inertia of a mass, m, about the x
axis:
Ix 
 y
2
 z dm
2
m
Moment of inertia of a mass, m, about the y
axis:
Iy 
 x
Iz 
 x
2
 z dm
2
m

Moment of inertia of a mass, m, about the z
axis:

2
m
 y dm
2
EF 202, Module 4, Lecture 3
3
Transfer Theorem - 1
• We can “transfer” the moment of
inertia from one axis to another,
provided that the two axes are
parallel.
•
In other words, if we know the moment
of inertia about one axis, we can
compute it about any other axis
parallel to the first axis.
EF 202, Module 4, Lecture 3
4
Transfer Theorem - 2
•
If the moment of inertia of a mass m
about an axis x’ through the mass
center is IGx’, and the distance from the
x’ axis to the (parallel) axis x is dy, then
the moment of inertia of the mass
about the x axis is
Ix  IGx'  md
2
y
transfer term
EF 202, Module 4, Lecture 3
5
Transfer Theorem - 3
•
The moment of inertia to which the
transfer term is added is always the
one for an axis through the mass
center.
•
The moment of inertia about an axis
through the mass center is smaller
than the moment of inertia about any
other parallel axis.
EF 202, Module 4, Lecture 3
6
Transfer Theorem - 4
•
We can transfer from any axis to a
parallel axis through the mass center
by subtracting the transfer term.
IGx'  Ix  md
2
y
EF 202, Module 4, Lecture 3
7
Radius of Gyration
•
•
By definition, the radius of gyration of a
mass m about the x axis is
Ix
kx 
m
Given the mass and the radius of
gyration,
Ix  mk

2
x
EF 202, Module 4, Lecture 3
8
Composite Masses
•
Since the moment of inertia is an
integral, and since the integral over a
sum of several masses equals the
sum of the integrals over the
individual masses, we can find the
moment inertia of a composite mass
by adding the moments of inertia of
its parts.
EF 202, Module 4, Lecture 3
9
Rods weigh 3 lb/ft. Find IA.
EF 202, Module 4, Lecture 3
10
Density is 200 kg/m3. Find Iz.
EF 202, Module 4, Lecture 3
11
Rod: 3 kg/m
Plate: 12 kg/m2
Find IG.
EF 202, Module 4, Lecture 3
12