ENGR 215 ~ Dynamics
Section 17.1
Moment of Inertia
• The translational aspects of motion are
described by the equation:
F  ma
• The rotational aspects of motion are described
by the equation:
M  I
Golf Ball Drop Demo
Moment of Inertia
• Just as mass is a measure of a body’s
resistance to acceleration inertia is a measure of
a body’s resistance of a body to angular
acceleration.
What is the purpose of the flywheel on
this 1942 John Deere Model G tractor?
Definition of Moment of Inertia
• We define the moment of inertia as the integral
of the “second moment” about an axis of all the
elements of mass dm which compose a body?
• r = moment arm or perpendicular distance to the
axis of rotation
I   r dm
2
m
Definition of Moment of Inertia
I   r dm
2
m
I   r  dV
2
V
Differential Volume
dV  dx dy dz
• To obtain the moment
of inertia we will only
consider symmetrical
bodies having surface
which can be
generated by revolving
a curve about an axis.
Shell Elements
• If the shell having a
height z, radius r, and a
thickness dy is chosen
then the volume is given
by:
dV  2 r z dy
• We need z as a function
of y to integrate.
Disk Elements
• If the disk having a
radius r and a thickness
dz is chosen then the
volume is given by:
dV   y dz
2
• We need y as a function
of z to integrate.
Lecture Example 1: Find the moment of inertia about the
z- axis
Lecture Example 2: Find the moment of inertia about the
y-axis
Parallel Axis Theorem
I  I G  md 2
I G  moment of inertia about the z' plane
passing through the center of mass G
m  mass of the body
d  perpendicu lar distance between parallel axes
Lecture Example 3: Find the Moment of Inertia for a
slender rod length, L about its center of mass.
Lecture Example 4: Find the Moment of Inertia for a
slender rod length, L about its end.
Lecture Example 5: Back to the golf ball drop!
Radius of Gyration
• Occasionally, the moment of inertia of a body
about a specified axis is reported as the radius
of gyration, k.
I  mk
2
I
k
m
Lecture Example 6: The
pendulum consists of two slender
rods AB and OC which have a
mass of 3 kg/m. The thin plate has
a mass of 12 kg/m2. Determine the
center of mass of the pendulum
and the moment of inertia about an
axis perpendicular to the page at
Point G.