Environmental Engineering Reference
In-Depth Information
a body about a fixed axis are plentiful: a CD on a CD player rotates about a fixed
axis through the centre of the CD; a yo-yo as it falls rotates about an axis whose
direction is fixed, even though the yo-yo is accelerating downwards; the rear wheel
of a bike shows fixed-axis rotation as long as the bike is travelling in a straight
line (when the cyclist takes a bend the direction of the rotation axis is no longer
fixed but changes as the direction of the bike's motion changes). We shall widen
our brief to include rotations about non-fixed axes in Chapter 10.
z
w
d
j
m
j
r
j
y
x
Figure 4.3
Fixed-axis rotation.
We therefore focus our attention upon the rotation of a rigid body about an axis
thatwedefinetobethe
z
-axis. Figure 4.3 illustrates the geometry. Particle
j
has
position vector
r
j
,mass
m
j
and is rotating about the
z
-axis with angular speed
ω
. Since the body is rigid we can be sure that the value of
ω
is the same for all
particles. Furthermore, each particle executes a circular orbit of radius
d
j
and we
can use Eq. (1.24) to write
v
j
=
ωd
j
,
(4.19)
where
v
j
is the speed of the
j
th
particle. This particle has angular momentum
L
j
=
r
j
×
m
j
v
j
=
m
j
r
j
×
v
j
,
where
L
j
is a vector perpendicular to the plane containing
r
j
and
v
j
. At this stage
we are only dealing with rotation about a fixed axis (the
z
-axis), in that case it is not
usually necessary to compute the
x
and
y
components of
L
j
. Thus we will consider
only the projection of
L
j
on
k
, the Cartesian basis vector in the
z
-direction, i.e.
L
jz
=
k
·
L
j
=
m
j
(
k
×
r
j
)
·
v
j
=
m
j
d
j
v
j
.
(4.20)
We have used the identity (for any vectors
a
,
b
and
c
):
(
a
×
b
)
·
c
=
a
·
(
b
×
c
)
(4.21)
