Feynman's Wobbling Plate

In his eccentric collection of autobiographical stories (see reference), Richard Feynman recounts: "I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling. I had nothing to do, so I start figuring out the motion of the rotating plate. I discovered that when the angle is very slight, the medallion rotates twice as fast as the wobble rate—two to one. It came out of a complicated equation! I went on to work out equations for wobbles. Then I thought about how the electron orbits start to move in relativity. Then there's the Dirac equation in electrodynamics. And then quantum electrodynamics. And before I knew it… the whole business that I got the Nobel prize for came from that piddling around with the wobbling plate." A replica of the Cornell plate is now part of an exhibit marking the centennial of the Nobel Prize.

Actually, Feynman misremembered (or was being mischievous): the factor of 2 actually goes the other way. In the Details section below, the motion of the plate is derived using Euler's equations for a rigid body.

In this Demonstration, the trajectory of the plate is shown in slow motion. The initial conditions to be chosen are θ, the inclination of the plate's symmetry axis to the vertical and (OverDot[ψ]), the rate of rotation about this axis. It is found that the wobble rate is actually slightly more than twice the rotation rate. The ratio approaches 2 as θ->0, but, at the same time, the wobble amplitude decreases.

THINGS TO TRY

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DETAILS

Euler's equations for free rotation of a rigid body are given by (Subscript[I, 1])(Subscript[(OverDot[ω]), 1])=((Subscript[I, 2])-(Subscript[I, 3]))(Subscript[ω, 2])(Subscript[ω, 3]); (Subscript[I, 2])(Subscript[(OverDot[ω]), 2])=((Subscript[I, 3])-(Subscript[I, 1]))(Subscript[ω, 3])(Subscript[ω, 1]); (Subscript[I, 3])(Subscript[(OverDot[ω]), 3])=((Subscript[I, 1])-(Subscript[I, 2]))(Subscript[ω, 1])(Subscript[ω, 2]). For a circular disk of radius R, mass M and negligible thickness, the principal moments of inertia are (Subscript[I, 1])=(Subscript[I, 2 ])=(1/4)M (R^2), (Subscript[I, 3 ])=(1/2)M (R^2)=2(Subscript[I, 1]). The third Euler equation reduces to (Subscript[(OverDot[ω]), 3])=0. Thus (Subscript[ω, 3])=const. This is the angular velocity of the plate about its axis of symmetry, and is taken as one of the initial conditions. The first two Euler equations are then satisfied by (Subscript[ω, 1])(t)=(Sqrt[(ω^2)-(Subscript[ω, 3]^2)])sin Ω t, (Subscript[ω, 2])(t)=(Sqrt[(ω^2)-(Subscript[ω, 3]^2)])cos Ω t, where Ω=(((Subscript[I, 1])-(Subscript[I, 3]))/(Subscript[I, 1]))(Subscript[ω, 3])=-(Subscript[ω, 3]). The motion of the rigid body in the space-fixed frame can be expressed in terms of the Euler angles θ, ϕ, ψ, using the relations (Subscript[ω, 1])=(OverDot[θ])cos ψ+(OverDot[ϕ]) sin θ sin ψ, (Subscript[ω, 2])=-(OverDot[θ])sin ψ+(OverDot[ϕ]) sin θ cos ψ, (Subscript[ω, 3])=(OverDot[ψ])+(OverDot[ϕ]) cos θ. If the axis of the plate is initially inclined by an angle θ from the vertical, with an angular velocity (Subscript[ω, 3]), then the motion of the plate is given by ϕ(t)=(((Subscript[I, 3])(Subscript[ω, 3]))/((Subscript[I, 1])cos θ))t=((2(Subscript[ω, 3]))/(cos θ))t, ψ(t)=Ω t=-(Subscript[ω, 3])t, θ(t)=θ=const. The rotation of the plate is represented by ψ(t), while its precession or "wobbling" is given by ϕ(t). As θ approaches 0, the wobbling frequency approaches twice the rotation frequency, but the wobbling amplitude also decreases.

References:

R. P. Feynman, Surely You're Joking, Mr. Feynman!, New York: W. W. Norton, 1985 pp. 157–158.

H. Goldstein, Classical Mechanics, Cambridge, MA: Addison-Wesley, 1953.

Contributed by: S. M. Blinder

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