2
4 Moments of Inertia
I
yy
=
B
=
∑
i
m
i
(
z
2
i
+
x
2
i
)
,
I
zz
=
C
=
∑
i
m
i
(
x
2
i
+
y
2
i
)
.
(4.2)
The
moment of inertia
of the system about the origin
O
is
I
O
=
∑
i
m
i
(
x
2
i
+
y
2
i
+
z
2
i
)
.
(4.3)
The
products of inertia
of the system about the axes
xy
,
yz
and
zx
are
I
yz
=
D
=
∑
i
m
i
y
i
z
i
,
I
zx
=
E
=
∑
i
m
i
z
i
x
i
,
I
xy
=
F
=
∑
i
m
i
x
i
y
i
.
(4.4)
Between the different moments of inertia one can write the relations
I
O
=
I
x
0
y
+
I
y
0
z
+
I
z
0
x
=
1
2
(
I
xx
+
I
yy
+
I
zz
)
,
and
I
xx
=
I
y
0
z
+
I
z
0
x
.
For a continuous domain
D
, the previous relations become
I
xOy
=
Z
D
z
2
dm
,
I
yOz
=
Z
D
x
2
dm
,
I
zOx
=
Z
D
y
2
dm
,
I
xx
=
Z
D
(
y
2
+
z
2
)
dm
,
I
yy
=
Z
D
(
x
2
+
z
2
)
dm
,
I
zz
=
Z
D
(
x
2
+
y
2
)
dm
,
I
O
=
Z
D
(
x
2
+
y
2
+
z
2
)
dm
,
I
xy
=
Z
D
xydm
,
I
xz
=
Z
D
xzdm
,
I
yz
=
Z
D
yzdm
.
(4.5)
The infinitesimal mass element
dm
can have the values
dm
=
ρ
v
dV
,
dm
=
ρ
A
dA
,
dm
=
ρ
l
dl
,
where
ρ
v
,
ρ
A
and
ρ
l
are the volume density, area density and length density.
4.2 Moment of Inertia of a Rigid Body
For a rigid body with mass
m
, density
ρ
, and volume
V
, as shown in Fig. 4.2, the
moments of inertia are defined as follows