Environmental Engineering Reference
In-Depth Information
we can make things more transparent by putting the summations back in:
3
r
α
ω
i
−
.
L
i
=
m
α
r
αi
r
αj
ω
j
(10.9)
α
j
=
1
Notice that
L
depends linearly on the rotational frequency, i.e. if we multiply
ω
by
a scalar factor then
L
increases by the same factor. However,
L
is not generally
parallel to
. We can make the connection between the two vectors clearer by
rewriting Eq. (10.9) as
ω
m
α
r
α
δ
ij
−
r
αi
r
αj
ω
j
,
L
i
=
(10.10)
j
α
where we have introduced the Kronecker delta symbol,
δ
ij
, which is defined to be
=
=
δ
ij
0
(i
j),
=
1
(i
=
j).
(10.11)
One way to make sense of Eq. (10.10) is to view it as the multiplication of the
column vector
ω
, (whose elements are
ω
j
) by a matrix with elements
m
α
r
α
δ
ij
−
r
αi
r
αj
,
=
I
ij
(10.12)
α
i.e. we can write Eq. (10.10) as
L
i
=
I
ij
ω
j
.
(10.13)
To see that this equation really represents the multiplication of a vector by a matrix
you should remember that there is an implicit summation on the index
j
and that
this summation runs over the columns of the matrix (whose components are
I
ij
),
e.g.
L
2
=
I
23
ω
3
is the component of
L
in the
e
2
direction. Notice
also that
I
ij
has the dimensions of a moment of inertia, i.e. mass times the square
of a length. It links the angular velocity and the angular momentum, but crucially,
unlike the scalar moment of inertia used in Chapter 4,
I
ij
possesses directional
information. It is the co-ordinate representation of a geometric object known as the
moment of inertia tensor.
I
21
ω
1
+
I
22
ω
2
+
10.2 THE MOMENT OF INERTIA TENSOR
In the previous section we obtained the result
L
i
=
I
ij
ω
j
,
which expresses the fact that the components of the vectors
L
and
ω
are linked by
a3
×
3 matrix with elements
I
ij
. We can associate the matrix elements
I
ij
with the
