Environmental Engineering Reference
In-Depth Information
10
Rigid Body Motion
In Chapter 4, when we considered the motion of rigid bodies we always made the
assumption that the axis of rotation was fixed. This simplification allowed us to
deal only with the components of the angular momentum and the angular velocity
along the rotation axis. Now we will treat the more general situation in which
the axis of rotation may not have a fixed direction in space; this will generally
bring into play all three components of
L
and
ω
. To motivate the discussion let
us first look at a simple example: a light rigid rod with a mass
m
at either end,
rotating with the midpoint fixed and with the rod making a fixed angle
θ
with the
x
3
axis of a Cartesian coordinate system (see Figure 10.1). The two masses each
describe circular motion of the same frequency about the
x
3
axis, hence the masses
have an equal angular velocity
is
defined always to be parallel to the axis of rotation, as discussed in Section 4.3.
Now let us compute the total angular momentum. We ignore the contribution of
the rod, assuming its mass to be negligible, and sum the angular momenta of the
two masses to obtain
ω
, which is parallel to the
x
3
axis. Recall that
ω
L
=
m
r
×
v
+
m(
−
r
)
×
(
−
v
)
=
2
m
r
×
v
.
(10.1)
L
is perpendicular to both
r
and
v
and is composed of a component
2
mr
2
ω
sin
2
θ,
=
L
3
(10.2)
which is parallel to
ω
and a component
2
mr
2
ω
sin
θ
cos
θ,
L
⊥
=
(10.3)
which rotates about the
x
3
axis with angular speed
ω
. We have used the relationship
v
=
ωr
sin
θ
to write Eq. (10.2) and Eq. (10.3) in terms of
ω
. Note that the very
