Environmental Engineering Reference
In-Depth Information
L
Space Cone
w
e
3
Body Cone
Θ
Figure 10.14
Space and body cones for a symmetric top.
the attention of Richard Feynman
5
while he was sitting the cafeteria of Cornell
University
6
. Feynman told his biographer, Jagdish Mehra that he was watching a
student playing with a plate, tossing it into the air and catching it. If you have
ever tried this you will know that it is important to give the plate some angular
momentum in order to keep its orientation stable and thereby make it easier to
catch. The plate would have been spinning about its symmetry axis, and as we
have shown, the symmetry axis would have been simultaneously precessing about
the angular momentum vector, making the plate wobble in flight. Feynman noticed
that an emblem printed on the plate rotated at about half the frequency of the
wobble, i.e.
ω
p
≈
2
ω
t
. We will obtain this result in the following example.
Example 10.6.1
Show that the precession frequency of Feynman's plate is twice
the top frequency, provided the plate doesn't wobble too much.
Solution 10.6.1
By saying that the plate does not wobble too much it is meant that
there is only a small angle between the
e
3
axis and the
L
vector. In fact, you might
like to use Figure 10.14 to help picture the wobbling motion by noting that the
shaded area directly specifies the spatial orientation of the plate. For small angles,
we can set
cos
≈
1
in Eq. (10.83) and obtain
I
3
I
ω
p
≈
ω
t
.
To make the calculation of the ratio of moments of inertia easier we treat the plate
as a perfectly flat disc. Since this is a planar object we can use the perpendicular
axis theorem to write
I
3
≡
I
33
=
I
11
+
I
22
=
2
I
hence, ω
p
≈
2
ω
t
.
5
Richard Phillips Feynman (1918 - 1988).
6
Jagdish Mehra,
The Beat of a Different Drum: The Life and Science of Richard Feynman
, Clarendon
Press.
