Environmental Engineering Reference
In-Depth Information
e
3
A
w
w
t
e
2
O
e
1
Figure 10.12 The angular velocity of the free symmetric top as observed in the body-fixed
frame. The vector
ω
precesses about the
e
3
direction.
We are thus led to a solution in which the
vector sweeps out a circle in the plane
defined by the body-fixed vectors
e
1
and
e
2
as shown in Figure 10.12. The total
angular velocity
ω
ω
in the body-fixed frame is thus
ω
=
A
cos
(t
+
φ)
e
1
+
A
sin
(t
+
φ)
e
2
+
ω
t
e
3
.
(10.66)
Note that the magnitude of
ω
is a constant since
ω
2
ω
1
+
ω
2
+
ω
3
=
A
2
ω
t
.
=
+
(10.67)
In the body-fixed frame
precesses about the symmetry axis with frequency
.It
is important to keep in mind that although we are working in the frame of reference
of the body,
ω
describes the rotation of the body as seen in the lab frame. However,
because the lab and body-fixed basis vectors do not coincide, the components of
ω
ω
are different in the two frames.
10.6.2 The lab frame
The motion of the free symmetric top is described by Eq. (10.64) in the
body-fixed frame. In this frame
, which gives the instantaneous angular velocity
of the body in the lab, precesses about the body-fixed symmetry axis. Admittedly
this is a bit of a mind bender! Can we understand what the motion looks like in
the lab? To do so we will identify some conserved quantities that will turn out
to simplify the job of making the transformation between the body-fixed and the
lab frames. In order to discuss this transformation mathematically we must first
specify the co-ordinate system to be used in the lab frame. We choose Cartesian
co-ordinates defined by the basis vectors
i
,
j
,
k
. Since there is no external torque,
L
is a constant vector when viewed from the lab frame and so it is convenient
to fix our co-ordinate system to be aligned with the direction of
L
. We therefore
define
k
such that
ω
L
=
L
k
,
(10.68)
where
L
is constant. However, in the body-fixed frame the components of
L
are
not constant, since the body, and hence the vectors
e
1
,
e
2
,
e
3
, rotate with respect
