# Angular Momentum and Gyroscopic Precession

James D. Jones,
M. Casco Associates, 81 Turkey Ridge Road, East Stroudsburg, PA 18301, USA

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 I w. 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| = mgrsin a

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| = mgrsin a

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 Li to a new value Lf = Li + dL. A vector representation of this state of affairs is shown below. The small angle, d f, 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:

d f = dL/L = (mgrsin a)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 = (mgrsin a)/I w

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 wp is much smaller than w. This condition is met when I w 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.

Authors with theoretical explanations, interesting photos
and videos, GIF animations, JAVA applets are welcomed.