Environmental Engineering Reference
In-Depth Information
x
3
d
m
−
a
a
x
2
Figure 10.4 Cross-section of a cylinder showing the symmetry with respect to a change of
sign of the
x
2
co-ordinate.
planar object an element of mass d
m
σ(
r
)
d
A,
where d
A
is an element of area.
The components of the moment of inertia are given by
=
d
Aσ(
r
)
[
r
2
δ
ij
−
I
ij
=
r
i
r
j
]
.
(10.24)
A
Since the object is flat and thin we can choose its position and orientation such that
it lies in the plane where
x
3
=
0, as indicated in Figure 10.5. This simplifies the
calculation of the moment of inertia tensor because two of the products of inertia,
I
13
and
I
23
, are automatically zero. Calculation of the diagonal elements is also
simplified by choosing
x
3
=
0:
d
Aσ(
r
)r
2
,
I
11
=
A
d
Aσ(
r
)r
1
,
I
22
=
A
d
Aσ(
r
)(r
1
+
r
2
)
I
33
=
=
I
11
+
I
22
.
(10.25)
A
x
2
d
m
r
x
1
Figure 10.5
Moment of inertia of a planar object. The
x
3
axis is out of the page.
