Environmental Engineering Reference
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as well as the result
k
×
r j
=
d j v j ,
(4.22)
where v j is a unit vector in the direction of v j . You should take a moment to make
sure that you can obtain Eq. (4.22) using the geometry shown in Figure 4.3.
The total angular momentum along the z
axis is obtained upon summing the
contributions from each particle:
N
N
ω
m j d j
L z =
m j d j v j
=
(4.23)
j
=
1
j
=
1
and we have made use of Eq. (4.19). The term in brackets is a property of the
rigid body and the axis of rotation. It is known as the moment of inertia about the
rotation axis, and is given the symbol I .
Via Eq. (4.23) we have succeeded in achieving our goal of relating the angular
momentum to the spin. Fixing the rotation axis means that we can focus our
attention on the z -component of L but we ought not to forget that L is really a
vector quantity. Likewise, the angular speed that appears on the right-hand side of
Eq. (4.23) should be viewed as the z -component of the angular velocity. As far as
this chapter is concerned we shall not really need to appreciate the vector nature of
the angular velocity but it will turn out to play an important role later in the topic,
especially in Chapters 8 and 10. Now is a good time for us to take the trouble to
define the angular velocity
. Of course it must be defined so that its z -component
is equal to the angular speed in the case of fixed-axis rotation, for that is what
appears in Eq. (4.23). We choose to define
ω
such that at any instant it points in
the same direction as the axis of rotation of the body and with a magnitude equal to
the angular speed at that instant. This definition allows for the possibility that the
body wobbles around or its angular speed changes: the direction and modulus of
ω
ω
changes accordingly. Notice that since we require ω z =
ω for rotations about the
z -axis, in the sense illustrated in Figure 4.3, then we have defined that the direction
of
should be parallel to the instantaneous axis of rotation as indicated by the right
hand rule, i.e. curl the fingers of your right hand as if following the circular path
of one of the particles in the body, then your thumb points in the direction of
ω
.
Returning to the case of rotations about the z -axis, we will save ink and write
ω
L
=
L z , i.e. we shall write Eq. (4.23) as
L
=
Iω.
(4.24)
Of course, the complicated bit is hidden in the symbol for the moment of inertia:
N
m j d j
I
=
(4.25)
j = 1
and we turn our efforts next to showing how to compute it.
As was the case with the calculation of the centre of mass of a macroscopic
object it is impossible to calculate this sum over all particles. We must instead
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