Section 20–3 The gyroscope

(Uniform precession / Oscillatory precession / Damping precession)

In this section, Feynman discusses uniform precession (steady precession), oscillatory precession (nutation), and damping precession. To be precise, the section could be titled as “The precessional motion of gyroscope.” The discussion could be shifted to the last section because the concept of precession is not intuitive and many students may have difficulties understanding it. In Surely You’re Joking, Mr. Feynman!, Feynman made a mistake in saying that a wobbling plate spins twice as fast as it wobbles, but the truth is the plate wobbles twice as fast as it spins (Chao, 1989).

1. Uniform precession:

“… we have not proved (and it is not true) that a uniform precession is the most general motion a spinning body can undergo as the result of a given torque (Feynman et al., 1963, section 20–3 The gyroscope).”

In Fig. 20–3, Feynman shows how a horizontal torque causes a top to precess in the sense that its spin axis rotates in a circular cone about the vertical. Using the equation, τ = dL/dt = Ω × L₀, he explains that the direction of the precessional motion is in the direction of the torque, or at right angles to the forces producing the torque. Alternatively, we can use the phrase “torque axis,” “spin axis,” and “precession axis” to describe the motion of the top. That is, the torque axis (τ) should be perpendicular to the spin axis (L) and precession axis (Ω) in accordance with τ = Ω × L₀. Thus, uniform precession of a spinning top can be defined as rotational motion of the spin axis around the precession axis (e.g., vertical axis) due to a changing torque (constant magnitude) in the horizontal plane.

The magnitude of an angular momentum vector ΔL is expressed as L₀Δθ and the time rate of change of the angular momentum is τ = ΔL/Δt = L₀Δθ/Δt = L₀Ω. We need to idealize a simple top (or gyroscope) as a rigid body that has a symmetry axis and frictionless pivot. As a good approximation, the applied torque should be small relative to the angular momentum of the rapidly spinning top for a simple precession. In a sense, Feynman was rather sloppy by asking students to take the directions of the various quantities into account to see that τ = Ω × L₀. For a detailed explanation, one may prefer the equation dL/dt = dL/dt + w ´ L that can be used to derive the Euler’s equations of motion for a rotating rigid body (Morin, 2003).

2. Oscillatory precession:

“The general motion involves also a “wobbling” about the mean precession. This “wobbling” is called nutation (Feynman et al., 1963, section 20–3 The gyroscope).”

Feynman says that the general motion of a gyroscope involves a wobbling that is about the mean precession. This wobbling is called nutation (Latin word for nodding) and the gyro does fall below the mean precession, but as soon as it falls, it keeps turning and returns again to the initial level. This motion may also be known as oscillatory precession because the spin axis oscillates about the mean precession angle as it rotates about the precession axis. Feynman adds that all the formulas in the world such as the equation (20.15) does not describe the general motion of a gyroscope because it is valid only for uniform precession. However, we can use Euler’s equations of a rigid body to describe the general motion in terms of the Euler angles.

The general motion of a gyroscope is a superposition of torque-induced (uniform) precession and torque-free (oscillatory) precession. In Tips on Physics, Feynman clarifies that: “…if you throw an object into space alone, like a plate or a coin, you see it doesn’t just turn around one axis. What it does is a combination of spinning around its main axis, and spinning around some other cockeyed axis in such a nice balance, that the net result is that the angular momentum is constant (Feynman et al., 2006, p. 124).” The constancy of angular momentum implies that the additional motion due to nutation is torque-free. Specifically, nutational motion depends on the initial angular displacement and angular velocity of the spin axis just like how the pendula motion depends on its initial height and velocity.

3. Damping precession:

“The answer is that the cycloidal motion of the end of the axis damps down to the average, steady motion of the center of the equivalent rolling circle (Feynman et al., 1963, section 20–3 The gyroscope).”

We can use the phrase damping precession to describe how the nutational motion of the spin axis is damped by frictional forces. According to Feynman, the nutational motion is too quick for the eye to follow, and it damps out quickly because of the friction in the gimbal bearings. In Tips on Physics, Feynman elaborates that “when the airplane quiets down and goes in a straight line for a while, you’ll find that the gyro doesn’t point north anymore, because of friction in the gimbals. The airplane has been turning, slowly, and there has been friction, small torques have been generated, the gyro has had precessional motions, and it is no longer pointing in exactly the same direction (Feynman et al, 2006, p. 98).” Thus, pilots need to reset the directional gyro against the compass regularly.

Feynman explains that the axis of the gyro will be eventually a little bit lower than it was at the start. We should clarify that the gravitational potential energy of the gyro is converted into the rotational kinetic energy and thermal energy (due to friction between the bearings and the pivot). In Tips on Physics, Feynman adds that “the earth is not rigid; it’s got liquid goop on the inside, and so, first of all, its period is different from that of a rigid body, and secondly, the motion is damped out so it should stop eventually - that’s why it’s so small. What makes it nutate at all, despite the damping, are various irregular effects which jiggle the earth, such as the sudden motions of winds, and ocean currents (Feynman et al., 2006, p. 125).” In essence, the Earth also precesses like a gyroscope and its precession is affected by frictional forces.

Questions for discussion:

1. How would you explain the torque on a top using Newton’s second law of motion?

2. Why does the downward force of gravity make the top to move sidewise?

3. How would you explain the axis of the gyro is a little bit lower than it was at the start when the precessional motion settles down?

The moral of the lesson: the top does fall a little bit in the sense that its spin axis is lowered a little bit to allow precession about the vertical axis; the lowering of the spin axis causes a conversion of gravitational potential energy to rotational kinetic energy.

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Morin, D. (2003). Introductory Classical Mechanics. Cambridge: Cambridge University Press.

5. Chao, F. B. (1989). Feynman's Dining Hall Dynamics. Physics Today, 42(2), 15.