Environmental Engineering Reference
In-Depth Information
Combining this with Eq. (10.75) gives
d
e
3
d
t
=
L
I
sin
(
cos
j
−
sin
i
).
(10.78)
This can be compared with the time derivative of Eq. (10.75):
d
e
3
d
t
=
d
d
t
sin
(
cos
j
−
sin
i
)
(10.79)
to give
d
d
t
L
I
,
=
(10.80)
provided that
=
0. Thus we have shown that
ω
and
e
3
both precess about
L
in
the lab frame at a constant frequency
d
d
t
=
L
I
.
ω
p
≡
(10.81)
If
0, then we are back to the situation of fixed-axis rotation about a principal
axis (the symmetry axis), and
e
3
,
L
,and
=
0, Eq. (10.81)
takes on a more revealing form when we write it in terms of the top frequency
ω
t
. Using Eq. (10.69) we have
ω
are all parallel. For
=
L
·
e
3
=
ω
t
I
3
=
L
cos
,
(10.82)
which together with Eq. (10.81) implies that
I
3
I
cos
ω
t
.
ω
p
=
(10.83)
Thus, in the lab frame, a free symmetric top spins about the symmetry axis (with
angular speed
ω
t
) while the symmetry axis precesses (with angular speed
ω
p
)
about the fixed
L
vector. This mode of motion is often referred to as 'wobbling'
because of the rotating orientation of the symmetry axis. The relationship between
the co-planar vectors
L
,
and
e
3
is presented in Figure 10.14 for the case that
the top is prolate, i.e
I
3
<I
. The precession of
ω
around
e
3
in the body-fixed
frame describes what is labelled as the body cone. In the lab frame
ω
ω
precesses
about
L
to produce the space cone. The space and body cones intersect along
a line defined by the vector
and as the motion progresses the body cone rolls
around the space cone. For an oblate top (
I
3
>I
), the diagram is similar, except
that
ω
lies on the other side of
L
and the space cone sits inside of the body cone
4
.
We now have all the bits and pieces that we need to fully describe the translational
and rotational motion of a free symmetric top. Remember that in this context 'free'
means free of a net external torque, but there may well be external forces that
produce no net torque. Let us examine an example of free rotation that caught
ω
4
You should be able to prove this.
