Imagine a large wheel with a smaller wheel fixed inside it, both sharing the same center. When this composite wheel rolls along a surface without slipping, it appears that both the outer and inner wheels cover the same linear distance in one full rotation. This seems counterintuitive—the inner wheel has a smaller circumference, so how can both trace the same path length?
Understanding the Mechanics
The explanation lies in how each wheel interacts with the surface:
- Outer Wheel: In contact with the ground, it rolls without slipping. The distance it covers in one full rotation equals its circumference.
- Inner Wheel: It rotates with the outer wheel but doesn’t touch the ground. Instead, it’s dragged along, combining rotation with slipping. Its motion is not true rolling.
This difference becomes evident in the paths traced by points on each wheel. A point on the outer edge follows a cycloidal path. In contrast, a point on the inner wheel traces a curtate cycloid—a shorter, looping trajectory due to its proximity to the center.
The Illusion of Equal Distance in the Video
In the old video often used to illustrate this idea, a single wheel features two concentric circles drawn on it—one large, one small. As the wheel rolls without slipping, only the outer edge touches the ground. Both drawn circles rotate together, but neither rolls; they’re simply carried along.
Visually, both circles appear to move the same distance across the screen. This creates the illusion of equal travel. In reality, the smaller circle slips more than the larger one due to its tighter arc around the center. As you may have noticed yourself, this so-called paradox arises from a misunderstanding of how rotation and slipping function when only the outer rim is in contact with the surface.
A Mathematical Perspective
Mathematically, the paradox shows that a one-to-one correspondence between points on two rotating paths doesn’t imply equal arc lengths. While each point on the smaller circle aligns with a point on the larger one, their trajectories differ due to the nature of their motion.
Conclusion
Aristotle’s Wheel Paradox isn’t a true paradox, but a reminder that intuition can mislead when dealing with motion and geometry. The apparent equal travel of both wheels results not from identical rolling behavior, but from the interplay between rotation, slipping, and perception.