Precession of a Gyroscope

Precession is a very important example of torque in action. It is one of the most convincing demonstrations that torque is indeed a vector quantity.

Suppose one has a top, or a gyroscope, consisting of a spinning disk of mass $M$ and radius $R$ mounted on a frictionless axle resting on a tabletop. The axle is of length $D$. The disk is spinning at an angular velocity $\omega$, which is presumed to be large so that the angular momentum of the disk is large.

It is straightforward to compute the precession frequency of this gyroscope - the rate (in radians/sec) at which it sweeps out a cone around the vertical.

First we compute the torque about the pivot point (where the axle rests on the table) exerted by gravity acting on the system:

$$\vec{\tau} = \vec{r} \times \vec{F} = mgD \sin(\theta)$$ (5.14)

(into the page as drawn).

Second we compute the angular momentum of the system:

$$\vec{L} = I \vec{\omega} = \frac{1}{2} MR^2 \omega$$ (5.15)

(along the axis of rotation). Now note that the torque only changes the horizontal component of the angular momentum because it has no component parallel to the vertical component. This component is:

$$L_\perp = \vert\vec{L}\vert \sin(\theta)$$ (5.16)

rotation/gyroscope.2.eps

Now imagine looking down on the gyroscope from above so $L_\perp$ traces out a circle as the gyroscope precesses. In a time $\Delta t$ the angular momentum change is:

$$\Delta L = \Delta L_\perp = \tau \Delta t = L_\perp \Delta \phi$$ (5.17)

where $\Delta \phi$ is the angle $L_\perp$ sweeps out in that time.

Dividing the latter relation out:

$$\omega_p = \lim_{\Delta t \to 0} \frac{\Delta \phi}{\Delta t} = \frac{d\phi}{dt} = \frac{\tau}{L_\perp}$$

$$= \frac{M g D \sin(\theta)}{\frac{1}{2}MR^2 \omega \sin(\theta)}$$

$$= \frac{2 g D}{R^2 \omega }$$ (5.18)

where of course different spinning objects would have different relations from which $\tau$ or $L$ are computed but the idea would be the same.

Precession of this sort is the general solution of the differential equation:

$$\frac{d \vec{A}}{dt} = \vec{A} \times \vec{B}$$ (5.19)

where the change in $\vec{A}$ is always perpendicular to the plane containing $\vec{A}$ (changing) and $\vec{B}$ (presumed fixed). In future courses you'll study the mathematics of the solution in more detail (for example, when $\vec{A}$ and $\vec{B}$ are expressed in a cartesian coordinate system, where this becomes three differential equations coupling the three components according to a tensor form) and will learn about an additional sort of wobble the motion can have called nutation.

You will also study this in the specific context of magnetic resonance. Magnetic fields exert a torque on a magnetic moment. Magnetic moments are created by spinning charge. Thus a spinning charged particle in a static magnetic field has a continuous torque acting on it that causes it to precess around the magnetic field precisely like a top! Much magic can be worked on these precessing charged particles.