Environmental Engineering Reference
In-Depth Information
w
x
3
x
2
x
1
b
Figure 10.6
A cube rotating about an axis through its centre.
Hence
6
Mb
2
100
010
001
1
.
I
=
(10.27)
(b) When the moment of inertia tensor is calculated about a corner the products
of inertia do not vanish. However the symmetry of the problem with respect to
the interchange of the coordinate axes still helps us, giving I
11
=
I
22
=
I
33
and
I
12
=
I
23
=
I
31
. Then
b
b
b
d
x
1
d
x
2
d
x
3
b
3
2
3
Mb
2
M(x
2
+
x
3
)
I
11
=
=
and
0
0
0
b
b
b
d
x
1
d
x
2
d
x
3
b
3
1
4
Mb
2
.
I
12
=−
Mx
1
x
2
=−
0
0
0
So that
12
Mb
2
2
3
1
4
1
4
−
−
8
−
3
−
3
=
1
.
Mb
2
1
4
2
3
1
4
I
=
−
−
−
38
−
3
−
3
−
38
1
4
1
4
2
3
−
−
Example 10.2.2
Calculate the angular momentum relative to the origin when the
cube of the previous example rotates about an edge parallel to the x
3
axis with an
angular speed ω.
Solution 10.2.2
Since the cube rotates about an edge we can use the moment of
inertia tensor from part (b) of the previous example to obtain
2
3
1
4
1
4
1
4
−
−
−
0
0
ω
Mb
2
1
4
2
3
1
4
=
Mb
2
ω
1
4
L
=
−
−
−
.
1
4
1
4
2
3
2
3
−
−
