Environmental Engineering Reference
In-Depth Information
The moment of inertia for a planar object can therefore always be written in the
form
I
11
I
12
0
I
21
I
22
0
00
I
11
+
.
I
=
(10.26)
I
22
Provided we choose the axis of rotation so that it is parallel to
x
3
,i.e.
(
0
,
0
,ω)
,
we can use Eq. (10.26) to show that once again we are in a situation where the
motion is governed by a single moment of inertia,
I
33
=
ω
=
I
11
+
I
22
,since
L
=
I
I
33
ω
e
3
. Incidentally, we have also shown that the moment of inertia about
an axis perpendicular to the plane of a planar object can be expressed as the sum
of the moments of inertia about two perpendicular axes lying in the plane. This
result is known as the Perpendicular Axis Theorem.
Let us now take a look at another example in which symmetry helps in the
calculation of the moment of inertia tensor, and which gives us some results that
we will use later in the chapter. We will consider the rotation of a solid cube.
We will be interested in rotations about an axis through the centre of the cube and
about an axis along an edge of the cube. Remember that we must always choose the
origin of our co-ordinate system to lie on the rotation axis since our derivation of
Eq. (10.12) starts with Eq. (10.4), which is valid only for rotations about the origin.
However, once we have chosen an origin somewhere on the rotation axis, we are
then free to choose the directions of our co-ordinate axes to make the calculation
of
I
as simple as possible.
Example 10.2.1
Calculate the moment of inertia tensor for a uniform cube of mass
M and side b that is suitable for rotations about any axis through: (a) its centre;
(b) a corner.
ω
=
Solution 10.2.1
(a) To make the calculation easier, it makes sense to use the sym-
metry of the cube and to choose a body-fixed Cartesian co-ordinate system with
axes parallel to the edges of the cube, and the origin at the centre of the cube
(Figure 10.6). Recall that the moment of inertia tensor is defined relative to an ori-
gin and that, to be useful, the origin ought to lie on the intended axis of rotation.
With these choices, all of the products of inertia vanish and we have
I
11
=
I
22
=
I
33
.
Now
2
M
b
3
(x
2
+
M
b
3
x
3
x
3
)
I
11
=
d
V
=
d
V
V
V
M/b
3
since the density ρ
=
is uniform. Putting
d
V
=
d
x
1
d
x
2
d
x
3
and integrating
over x
3
we obtain
x
3
3
+
b
2
b
3
b
2
b
2
2
M
I
11
=
d
x
1
d
x
2
2
2
−
−
b
2
−
1
6
Mb
2
.
=
