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GYROSCOPIC PRECESSION
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Rotating systems exhibit some behavior that appears strange when we
apply our intuition, developed for linear motion. The motion of the
gyroscope shown here is an example. The silver colored rotor inside
the gimbal rings is spinning rapidly about its axis. As we will see,
it is the presence of the little weight, hung on one end of the axis
of rotation of the spinning rotor, that causes the observed motion
about the vertical axis. Our intuition might suggest that the weighted
end of the rotor's axis would move down, not go around.

Let's begin this story with the idea that the angular velocity of a
rotating object is a vector quantity whose magnitude is the number of
radians per unit time, covered by the rotating object, and whose
direction is perpendicular to the plane of rotation. If the rotation
is counter clockwise the sense of the angular velocity vector will be
out of the plane of rotation. If clockwise, the sense will be into the
plane. We will symbolize angular velocity by the Greek letter omega
(w). In rotating systems angular velocity is analogous to the linear
velocity of translating systems.

Also, in rotating systems there is a quantity analogous to mass in
systems undergoing translational motion. That is the moment of inertia
(I). The moment of inertia of an object is a scalar quantity whose
magnitude depends on the geometry of the object and the axis of
rotation.

In linear motion the product of the scalar mass times the vector
velocity is defined as the linear momentum of the system. Likewise in
rotary motion, the product of the scalar moment of inertia and the
vector angular velocity is defined as the angular momentum (L).

Next we need the rotational analog of the force in a linear system.
That is a quantity called torque (t). The torque is the vector cross
product of a force applied about a particular axis, and the radius
from that axis to the point of application of the force. As such it is
perpendicular to the plane containing the radius and the force vector.
In linear motion, the rate of change of linear momentum is the force
applied to an object. In rotary motion, the rate of change of angular
momentum is the torque applied to an object.

Now we can analyze the motion of the gyroscope using the terms defined
above. The angular momentum is a vector along the axis of rotation, of
magnitude Iw. The torque produced by the weight hung from one end of
the axis of rotation is a vector perpendicular to the vertical plane
containing the axis of rotation of the rotor. Its magnitude, from the
cross product definition, is the distance along the axis of rotation
from its midpoint to the location where the weight is attached (r),
times the mass of the suspended weight (m), times the acceleration due
to gravity (g), times the sine of the angle between the axis of
rotation and the vertical (a).

|t| = mgrsina

Remember that the torque is the rate of change of the angular momentum
(dL/dt), so the magnitude of dL/dt is given by:

|dL/dt| = mgrsina

The direction of the torque vector is perpendicular to the vertical
plane containing the axis of rotation. Since torque is dL/dt then
dL/dt is horizontal and perpendicular to the angular momentum vector.
In a tiny time increment, dt, the angular momentum vector changes from
its initial value L[i] to a new value L[f] = L[i] + dL. A vector
representation of this state of affairs is shown below. The small
angle, df, is the change in orientation in the horizontal plane of the
angular momentum vector, and therefore of the axis of rotation.

Based on the vector triangle above, we see that:

df = dL/L = (mgrsina)dt/L

The rate of change of f is f/dt so if w[p] is the angular frequency of
the axis of rotation, called the precessional frequency:

w[p] = (mgrsina)/Iw

We have replaced L with its equivalent Iw in the expression above.

Now is the time to confess that there is an implicit assumption buried
in our reasoning. We have assumed that the angular momentum was all
due to the rotation of the rotor. In fact the precessional motion also
contributes to the total angular momentum. Our analysis is valid only
as long as w[p] is much smaller than w. This condition is met when Iw
is large compared to mgrsina. Otherwise the motion of the gyroscope is
much more complicated, as you might observe in an actual experiment
where the rotation of the rotor slows down over time. We can see that
as the rotor slows, the precessional frequency increases. At some
point when the precessional frequency exceeds a critical value, the
gyroscope will begin to wobble and eventually tumble in its gimbals.