Environmental Engineering Reference
In-Depth Information
To get the moment of inertia of the entire body for rotations about axis
O
we need to
integrate over the whole slice and then over all the slices that make up the body, i.e.
r
C
+
a
2
d
m,
=
·
+
I
2
r
C
a
r
C
d
m
r
C
d
m
a
2
d
m.
=
+
2
a
·
+
(4.38)
The term
r
C
d
m
vanishes due to the definition of the centre of mass. We
therefore have
a
2
r
C
d
m
I
=
+
d
m,
Ma
2
,
i
.
e
.
I
=
I
C
+
(4.39)
where
I
C
is the moment of inertia about axis
C
and
M
is the mass of the body. We
can see from Eq. (4.39) that
I
I
C
that is, the moment of inertia is a minimum
when the rotation axis goes through the centre of mass. Note that Eq. (4.39) says
nothing about axes that are
not
parallel to
C
.
≥
Example 4.3.4
Determine the moment of inertia of a thin uniform circular disc
of radius R and mass M about an axis that just touches the edge and which is
perpendicular to the plane of the disc (axis O in Figure 4.8).
1
2
MR
2
we
can use the parallel-axis theorem to find the moment of inertia about axis O:
Solution 4.3.4
Since we know that the moment of inertia about axis C is
3
2
MR
2
.
MR
2
I
O
=
I
C
+
=
C
O
R
Figure 4.8
Moment of inertia about the edge of a circular disc of radius
R
.
4.4 SLIDING AND ROLLING
The beauty of Eq. (4.13) is that it works even when the centre of mass of a system
of particles is accelerating, provided that we calculate the angular momentum and
torque about the centre of mass. We have also shown that the angular momentum
associated with fixed-axis rotation of a rigid body is determined by the angular
speed
ω
and the relevant moment of inertia (Eq. (4.6)). We can combine these two
