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Gyroscope and Precession

A gyroscope is simply a wheel mounted in such a way that it can rotate freely about three normal axes. The first axis is the central axis (shaft) of the wheel about which it can spin; the other two axes allow the central axis to orient itself in any direction.

The general gyroscopic motion, though interesting, is quite involved. We shall instead study only one aspect of it, i.e. precession. To do that, we take a simpler picture of a gyroscope; consider a wheel fixed to a shaft and free to rotate about the axis of the shaft; the other end of the shaft is pivoted freely on a rigid support with the help of a ball and socket arrangement. The shaft is initially assumed to be in horizontal position.

First, suppose the wheel is not spinning (i.e. rotating about its shaft). When released, gyroscope will obviously fall downward under its weight. Consider the co-ordinate system as shown in fig. O is origin and weight Mg acts at CM C of the wheel, towards –ve Z-axis. Hence the torque , due to weight F = – Mg k, about O is given by

= r × F = – Mgl ( i × k) = Mgl j

where l is length of the shaft. Due to this torque, gyroscope rotates about Y-axis and falls downwards in XZ-plane. The torque produces a differential change in angular momentum of the system about Y-axis, given by

d L = dt = Mgl dt j

Now consider the situation when gyroscope is initially spinning very fast with angular velocity ω about the shaft. Thus, it has an initial angular momentum along X-axis due to its spin (shaft is initially horizontal).

L = I ω i

where I is MI of the wheel about its CM axis (i.e. shaft).

The shaft is then released so that torque due to the weight begins to act. Note while initial spin angular momentum L is along X-axis (shaft), the torque produces a change d L in angular momentum normal to shaft (i.e. along Y-axis). Hence, if L is very large (fast spin), then differential change d L in time dt is perpendicular to L. Consequently, magnitude of L does not change, only its direction changes; L vector simply rotates about Z-axis by angle Ø in time dt such that

L d Ø = dL

Thus, the gyroscope starts to rotate about X-axis with small angular velocity Ω given by

The above motion is called precession of gyroscope and Ω denotes the rate of precession.

If ω is large, the shaft stays practically horizontal and the gyroscope precesses with small angular velocity Ω about Z-axis. Though precession is caused by weight Mg of the gyroscope, rate of precession is indeed independent of M:

where I = MK², k is radius of gyration of the wheel.

It is striking to note that a spinning gyroscope does not fall under its weight but precesses about Z-axis maintaining horizontal position.