Wheels Within Wheels

Ready to have your mind blown in 30 seconds? Check out the classic paradox of Aristotle’s Wheel. (Actually, it is more like Arist-NOT-le’s Wheel because it probably wasn’t his idea.) If you think you understand the resolution of the paradox, try resolving it again for wheels composed of discrete sets of points, representing the atoms in the wheel, for example.

Aristotle's Wheel
Source: Wolfram MathWorld.

The controlled use of the wheel, much like the controlled use of fire, is one of those developments in human history that is taken as a benchmark in our path to sentience. While both wheels and fire are simple enough that they arise spontaneously in nature (think of boulders rolling down a hill), their mechanisms are subtle enough that it takes a leap of insight to learn how to create them at will and use them as tools. Consider the big stone wheels of the car in “The Flintsones”. They are not so different to rocks that you might find lying around, but how different they look when attached to the frame of a car! Now they no longer seem like simple rocks, but an amazing device that allows us to travel around while expending less energy. Even those of us who do advanced quantum mechanics like to sit back occasionally and just bask in the elegance of basic physics, like rolling wheels.

The Flintstones
Fred’s sweet ride.

My personal tribute to the wheel is the following paradox, which came up in discussion while wandering drunkenly with friends through the hills of Hohenruppersdorf. Imagine a perfect wheel in a vacuum, of some radius and width, on a flat surface with some coefficient of friction. The wheel has some initial angular velocity, as might be imparted by applying a horizontal push to the top of the wheel. Suppose that the friction between the wheel and the ground is such that the wheel rolls without slipping.

A scientific diagram of the Rolling Stones
Source: The Internets

We now make the following observations:

(1) In the rest frame of the flat ground, the point of contact between the rolling wheel and the ground is always stationary. Since we are considering an ideal system, the rolling motion does not dissipate any energy. Therefore, the wheel will continue to roll forever on the flat surface with the same kinetic energy that it started with.

(2) It is obvious that there is some nonzero friction between the wheel and the ground; if the surface were completely frictionless, the wheel would not be able to roll at all without slipping. In fact, it would simply spin on the spot forever.

(3) Friction dissipates energy as heat. Therefore, the wheel should be losing energy as it rolls, causing it to eventually come to a stop. But this contradicts (1), implying a paradox.

Clearly, at least one of the above statements is wrong, under our assumptions. But which one is it? They all sound quite reasonable. Just for fun, I’m going to let you ponder it for at least a day before I post the answer. Enjoy!

Edit: The Solution (probably)!

Ok, enough suspense. As elkement pointed out in the comments, any realistic wheel must dissipate some energy and hence eventually stop; indeed, the very assumption of a perfectly flat surface with friction might be suspect, since friction originates in microscopic irregularities and the electromagnetic interactions of particles. However, even in a realistic scenario, a wheel that is skidding along the ground will dissipate vastly more energy than a wheel that is rolling along the ground. This is exactly why a circular wheel is much better than a square wheel. So in a realistic scenario, the question becomes instead: why is the energy dissipated by rolling friction so much less than for sliding friction?

The answer lies in distinguishing two types of friction: static friction and sliding friction. Imagine you have two heavy pillars which have fallen inward and are resting on each other. Taken as an ideal system, these two pillars do not dissipate any energy. Realistically, they will dissipate a tiny amount of energy due to micro-movements of their atoms under the stress of supporting each other, but we can ignore this. The main point is that, even though virtually no energy is dissipated, there must be significant static friction between the base of the pillars and the ground, in order to keep the bottoms of the pillars from sliding outwards and collapsing the structure. If the pillars were on low-friction surface like ice, then the structure would be unstable: the horizontal component of force exerted on the ground by the pillars would overcome the coefficient of static friction and the pillars would slide apart, dissipating lots of heat in the process due to the resulting sliding friction. But as long as the static friction is high enough to prevent slipping, there is no sliding friction and hence there is (almost) no dissipation of energy.


Pretty much exactly the same logic applies to the rolling wheel. Here, the suspect assertion is the statement (3). In the ideal case, the static friction that keeps the wheel from slipping does not dissipate energy (and realistically, only very little energy compared to when there is sliding). So we throw out (3). The ideal wheel rolls forever, even though there is constant (static) friction between it and the ground. Prove me wrong!


One thought on “Wheels Within Wheels

  1. I dare to respond (though I know it’s very easy to make a fool of oneself when tackling those deceptively simple puzzles 😉 I think energy is or better has to be dissipated to allow for rolling motion even if the point of contact is (sort of) stationary.
    In a realistic scenario the ground is not perfectly flat but flexible and slighted indented at the position of the wheel and at the track traversed already by the wheel. Thus the resulting forces the ground exerts on the wheel are asymmetric (hard to describe without an image) and that decelerates the wheel. Dissipation is due to deformation of the floor (and its returning to the normal state).
    Now I try to imagine what would happen if the ground would be perfectly flat and really rigid: I think it this case the initial push would never result in rolling motion, but in sliding. Again, hard to explain without a drawing (and there is probably a better way to explain anyway)… I am trying to figure out the torques: If the ground is slightly flexible the torques generated by the pushing force and the force effectively exerted by the warped ground would need to balance each other. However, if the ground is perfectly flat the lever arm of the “ground force” is zero because the point relevant for calculating the torques has moved from the asymmetric position directly beneath the center of gravity.
    Thus I think in order to get the wheel going at all the ground needs to be a bit flexible – hence some reasonable torque, hence the deformation and the dissipation.

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