Environmental Engineering Reference
In-Depth Information
This particular solution corresponds to fixing
t
=
0 when the angular displacement
is at a maximum.
As an aside, we note that a measurement of the frequency of small oscillations,
f
, (the period is just 1
/f
) provides a method to determine the moment of inertia of
a body about the axis of rotation. However, that strategy runs into problems if we
want to measure the moment of inertia about an axis through the centre of mass.
Then there is no torque and the body doesn't behave like a pendulum. In the next
section we will show how to circumvent that particular problem when we prove
that the moment of inertia for rotations about one axis is sufficient to determine
the moment of inertia about any parallel axis.
4.3.1 The parallel-axis theorem
Consider a rigid body, as illustrated in Figure 4.7. Two rotation axes are shown.
Axis
C
is chosen so that it goes through the centre of mass, axis
O
is parallel to
axis
C
. We imagine the body as being made up of slices perpendicular to the two
axes. In any of these slices the position of
C
relative to
O
is given by the position
vector
a
. We first consider an element of mass d
m
in one of the slices. We can
write
r
=
r
C
+
a
(4.35)
since the moment of inertia is constructed from
r
2
we take the scalar product of
this equation with itself to obtain
r
2
r
C
+
a
2
.
=
r
·
r
=
2
r
C
·
a
+
(4.36)
The element d
m
thus makes a contribution to the moment of inertia for rotations
about
O
of
r
2
d
m
(r
C
+
a
2
)
d
m.
d
I
=
=
2
r
C
·
a
+
(4.37)
O
C
a
r
r
C
d
m
Figure 4.7
Geometry for the proof of the parallel-axis theorem.
