Environmental Engineering Reference
In-Depth Information
You can easily demonstrate the relative stability of rotations about the principal
axes of a rigid body yourself. Choose an object with some clear symmetries so
that the principal axes are easily identifiable: a tennis racquet makes a good choice
if you can manage to spin it and catch it without injuring yourself, otherwise use
a topic that is secured with an elastic band so that it doesn't open. Figure 10.16
shows the principal axes of a tennis racquet. The matter distribution in a tennis
racquet is such that
I
3
<I
1
<I
2
. When the racquet is spun about the long (
x
3
) axis
stable rotation is clearly possible, as is the case when the racquet is spun in the air
in a direction perpendicular to the plane of the strings. However if you attempt to
spin the racquet about the
x
1
axis you will see a much more erratic behaviour that
makes it very difficult to catch.
x
3
x
1
COM
Figure 10.16 A tennis racquet with the
x
1
and
x
3
principal axes illustrated. The
x
2
axis
is through the centre-of-mass (COM) and into the page. Stable rotations are possible about
the
x
2
and
x
3
axes but rotation about
x
1
is unstable.
10.8 GYROSCOPES
10.8.1 Gyroscopic precession
We have worked hard to gain a thorough understanding of the motion of free
rigid bodies through the use of Euler's equations with the torque set to zero. In this
section we will 'raise the bar' just a little and explore the effect of a torque on a
rapidly spinning body. This is the physics of gyroscopes. Before we get to this, let
us just say a few words on the physical characteristics of the system. A gyroscope
(see Figure 10.17) typically consists of a flywheel on an axle which is fixed to a
supporting cage. Most of the mass of the gyroscope is contained within the flywheel
which is free to rotate on the axle. There is always some mechanism for setting
the flywheel spinning at high angular speed. This is typically a string wound about
the axle that can be tugged sharply, or there may be an electric motor that drives
the flywheel. The motion of the gyroscope is fascinating. It almost seems to defy
