Environmental Engineering Reference
In-Depth Information
for rotations about an axis at one end. In the case that the axis passes through the
centre, we need only change the limits of the integral:
+
l/
2
x
2
M
l
1
12
Ml
2
.
=
=
I
d
x
−
l/
2
Example 4.3.2
Compare the moments of inertia of a uniform thin circular ring of
mass M and radius R with a thin uniform disc of the same mass and radius. In both
cases the rotation axis is the axis of rotational symmetry, i.e. perpendicular to the
plane of the ring and disc.
Solution 4.3.2
For the ring all the mass lies at the same distance R from the rota-
tion axis and so we can write down the result straightaway as I
MR
2
. For the
disc, the smart way to proceed is to realise that a disc can be built out of a series
of rings. The mass of a ring of thickness dr and radius r is
=
M
πR
2
2
πr
d
r
2
M
R
2
r
d
r.
d
m
=
=
Note that this is just the mass per unit area
πR
2
multiplied by the area of the ring,
see Figure 4.4(b). The moment of inertia is then
R
2
R
2
M
1
2
MR
2
.
r
2
d
m
r
3
d
r
=
=
=
I
0
The moment of inertia of the disc is smaller than that of the ring with the same mass
even though the spatial extent of the two objects is the same. This is to be expected,
since the d
2
term in Eq. (4.27) means that matter far from the axis of rotation has a
greater contribution to the moment of inertia than matter close to the rotation axis.
Notice that in each of our calculations above the moment of inertia was of the
form
Mk
2
,
I
=
(4.29)
where
M
is the mass of the body and
k
is a length of the order of the spatial extent
of the body known as the radius of gyration. We can therefore specify the moment
of inertia by giving the mass and the radius of gyration (for a particular shape and
axis of rotation). Table 4.1 gives the radii of gyration for some simple objects. For
more complicated shapes the calculations become difficult, but a rough estimate
of the moment of inertia may be obtained by setting
k
equal to the approximate
size of the object (and that can be done without too much ambiguity provided the
object is not too elongated).
Example 4.3.3
An atomic nucleus with 150 nucleons and a size of around
6
.
4fm
may be produced in a 'high-spin' state following a nuclear fusion reaction. In such
a state the nucleus rotates at ω
10
21
s
−
1
. Estimate the angular momentum of the
nucleus about an axis through its centre assuming the nucleus to be a rigid body.
∼
