Environmental Engineering Reference
In-Depth Information
d
V
r
Figure 10.2
A continuous body with a small element of volume d
V
.
will find it convenient to make a transformation to a non-rotating inertial frame,
which we shall call the lab frame. Our previous discussion, in Section 8.2, of
transformations between rotating and non-rotating frames led to the result
d
r
d
t
d
r
d
t
=
rot
+
ω
×
r
,
(10.4)
where
r
is the position vector of a particle relative to an origin chosen to lie
somewhere on the axis of rotation.
Let us now obtain the angular momentum of the body by adding together the
contributions from its constituent particles, in a similar way to that used for the
previous example of two masses on a rod. Applying Eq. (10.4) when the rotating
frame is a body-fixed frame gives
d
r
d
t
v
=
=
ω
×
r
(10.5)
since, by definition,
d
r
d
t
rot
=
0
for any particle in the body-fixed frame. We now
construct the total angular momentum by summing over particles:
=
×
=
×
ω
×
L
m
α
r
α
v
α
m
α
r
α
(
r
α
) ,
(10.6)
where we have used the summation convention on repeated indices (see Section 8.2)
and
α
labels the particles. Rewriting the triple vector product using the identity
a
×
(
b
×
c
)
=
(
a
·
c
)
b
−
(
a
·
b
)
c
gives
m
α
[
r
α
ω
−
L
=
(
r
α
·
ω
)
r
α
]
.
(10.7)
Alternatively, we can write the
i
th
component of
L
in a Cartesian co-ordinate
system as
m
α
r
α
ω
i
r
αj
ω
j
r
αi
,
=
−
L
i
(10.8)
where there is also an implicit summation on the index
j
that comes from the scalar
product (see Eq. (8.7)). If you are uncomfortable with the summation convention,
