Wheels Within Wheels

At the beginning of this week, my parents returned home from their 40th anniversary trip around France. As a souvenir, Sally asked me to pick up one of the great wheels of regional Tomme cheese from the Annecy farmers' market. While delivering it to their hotel, feeling the weight of it got me thinking about timekeeping of all things.

I imagined letting the cheese roll in a large dish, with each crossing representing one "tick". This is similar to how a pendulum clock works, but with an important difference: Rolling is subject to rotational inertia. I was curious how that would change the frequency, but before I get to that, I should explain what rotational inertia is.

Inertia refers to an object's tendency to keep moving in the way it's currently moving. Changing an object's motion requires overcoming its inertia. For linear motion, the inertia is simply the mass: It's a lot easier to catch a baseball than a bowling ball. Rotational inertia is the same idea, but the movement is around a central axis. A top stays upright because its rotational inertia keeps the spinning in the same direction.

Rotational inertia depends on the mass, but also how far that mass is from the rotational axis. You can look here for the inertias of some common shapes, but they generally follow the same form:

where q is a factor between 0 and 1. For the animation below, I chose a hollow ring, a solid disk (like the cheese), a solid sphere, and finally, a frictionless object that only slides without rolling. These have q = 1, 1/2, 2/5, and 0 respectively.

Each of these objects moves inside a larger circular container. The position in this container can be defined by the angle off the center. Aside from the frictionless object, the others roll without slipping, meaning we can relate their rotation to the position in the larger ring:

where L is the arc-length from the bottom of the ring to the object, and the rs and θs correspond to the radii and angles of the large ring and the object.

I'll spare you the details, but the acceleration for these objects winds up being

and so the angular position is

What this says is that as q gets bigger, the frequency (equal to the factor multiplying t) decreases. Looking below, we see the objects with larger q do indeed lag behind the others.