Gyroscope

A simple gyroscope consists of a wheel fixed to a shaft and free to spin about the axis of shaft. If one end of the shaft of a no spinning gyroscope is placed on a support as shown in the figure below and the gyroscope released, the gyroscope falls by rotating downward about the tip of the support. Since the fall involves rotation, it is governed by Newton’s second law of angular motion given by the equation; τ = dL/dt

This equation tells us that the torque causing the downward rotation changes the angular momentum of L of the gyroscope from its initial value zero. The torque τ is due to the gravitational force Mg acting on the gyroscope’s centre of mass, which make we take to be at the centre of the wheel. The moment arm relative to the support tip, is R. The magnitude of τ is given by τ = MgR.

A rapid spinning gyroscope behaves differently. Assume it is released with the shaft angled slightly upward, it first rotates slightly downward but then, while it is spinning about its shaft, it begins to rotate horizontally about a vertical axis through the support point in a motion called precession. In this motion, the gyroscope does not fall because when it is released, the torque due to Mg must change not an initial angular momentum of zero but rather some already existing nonzero angular momentum due to the spin.

For a wheel of moment of inertia, I, rotating with angular velocity ω, and the angular momentum L is given by L = Iω.

We must keep in mind that dL= τdt , however for a rapid spinning gyroscope, the magnitude of L is fixed by equation L = Iω above. Thus the torque can change only the direction of L, not its magnitude. The precession rate Ω can be derived as below;

dL= τdt=MgR dt

dØ=dL/L= MgRdt/Iω where dØ is the incremental angle through which the shaft and L precess around the z axis as L changes by the incremental amount in an