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I^ G
is the “mass moment of inertia” for a body about an axis passing through the body’s mass center, G.I^ G
is defined as:
I^ G
∫^
(^2) r dm
Units:
kg-m
2
or
slug-ft
2
I^ G
is used for several kinds of rigid body rotation problems, including:
(a)
F=ma analysis moment equation (
M
G^
= I
αG
(b)
Rotational kinetic energy ( T = ½ I
ωG
2
(c)
Angular momentum ( H
G^
= I
ωG
I^ G
is the resistance of the body to angular acceleration. That is, for a given net moment or torque on a body, thelarger a body’s I
, the lower will be its angularG
acceleration,
I^ G
also affects a body’s angular momentum, and how a
Mass Moment of Inertia, body stores kinetic energy in rotation.
I
G
Mass
Moment
of Inertia,
I
G^
(cont’d)
I^ G
for a body depends on the body’s mass and the location of the mass.The greater the distance the mass is from the axis ofrotation, the larger I
G^
will be.
For example, flywheels have a heavy outer flange thatlocates as much mass as possible at a greater distancefrom the hub.If I is needed about an axis other than G, it may becalculated from the “parallel axis theorem.”
I G
’s for Common Shapes
Thin^ I
G^
2
=^
mR 1 2
y
x
R
Disk: G
y
x
R
G
IG
2
=^
mR 2 5
Sphere:
x
y G
P
L
L 2 Slender IP
2
=^
mL 1 3
IG
2
=^
mL (^112)
About P(end of rod)
About G(center of rod)
Rod:
IG
2
2
=^
m(a
(^112)
x
y
G
a
b
Rectangular
Plate:
y
G
R
ThinRing:
IG
2
= mR
(All mass is atthe same radius, R)
x
Radius of Gyration, k
, for Complex ShapesG^
Some problems with a fairly complex shape, such as adrum or multi-flanged pulley, will give the body’s mass mand a radius of gyration, k
, that you use to calculate IG
.G
If given these, calculate I
G^
from:
I^ G
= mk
(^2) G
As illustrated below, using k
G^
in this way is effectively
modeling the complex shape as a thin ring.
G^
R
G
k^ G IG
= mk
(^2) G
Radius of Gyration, k
G
Some problems involving a complex shape with mass,m, and an outer radius, R, will give a “radius of gyration”,k^
, that can be used to determine I
for that shape.
The
equation, I
= mk , indicates that the complex shape
G is being modeled dynamically by a thin ring with mass, m,and a radius, k.
G
G^
G G
2
ComplexShape:
Thin Ring,k^
Model:G
