9.5 Mass Moment of Inertia 237
given center of rotation. A measure of how hard it is to rotate something with respect to
center of rotation is calledmass moment of inertia. All of you will take a class in physics
and some of you may even take a dynamics class where you will learn in more depth about
the formal definition and formulation of mass moment of inertia. But for now, let us
consider the following simple situation as shown in Figure 9.2. For a single mass particlem,
located at a distancerfrom the axis of rotationzz, the mass moment of inertia is
defined by(9.7)
Now let us expand this problem to include a system of mass particles, as shown in Figure 9.3.
The mass moment of inertia for the system of masses shown about thezzaxis is now(9.8)
Similarly, we can obtain the mass moment of inertia for a body, such as a wheel or a shaft, by
summing the mass moment of inertia of each mass particle that makes up the body. As you take
calculus classes you will learn that you can use integrals instead of summations to evaluate the
mass moment of inertia of continuous objects. After all, the integral sign is nothing but a big
“S” sign, indicating summation.(9.9)
The mass moment of inertia of objects with various shapes can be determined from Equa-
tion (9.9). You will be able to perform this integration in another semester or two. Examples
of mass moment of inertia formulas for some typical bodies such as a cylinder, disk, sphere, and
a thin rectangular plate are given on the following page.Izz r
2
dmIzzr
2
1 m 1 r^2
2 m^2 r^2
3 m 3Izzr
2
mz
r
m
z
r 3
r 2 r 1
z
m 3
m 1
m 2
z
■Figure 9.2
The mass moment of inertia of a point mass.■Figure 9.3
Mass moment of inertia of a system consisting of three point masses.Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀