Wheel

SUMMARY: Wheels help humans perform work and travel by providing a mechanical advantage.

Circles are present in many places in nature and mathematicians studied them long before the common use of the wheel. A wheel is traditionally a cylinder rotating around an axle. Together, a wheel and an axle form a simple machine that can change direction and magnitude of forces. Wheels are widely used in transportation as gears, as handles and knobs, and for converting the energy of water, animals, or people into work. The notion of curvature is of interest to many mathematicians, scientists, engineers, and others. In geometry, wheels are often modeled as circles or as concentric circles. In addition to standard circles or cylinders, mathematicians have explored the properties of wheels of other shapes along with varying surfaces. Aristotle’s Wheel paradox, named for Aristotle of Stagira, is an interesting mathematical problem involving the paths traced by a wheel made of two concentric circles. It seems to imply that the circumferences of different sized circles are equal. This is one of many mathematical questions that arise from rotating concentric circles or exploring the curves generated by wheels.

History and Mechanical Advantage

Wheeled vehicles were invented about 6,500 years ago, but they were not used widely until the rise of large, organized, road-building societies. This discrepancy between the discovery and its wide adoption, because of the lack of infrastructure, is frequent in science. Using wheels as levers to change the magnitude of force for applications like grinding grain was more widespread in many societies. The force advantage that a wheel provides is equal to the radius of the wheel divided by the radius of the axle. For example, a ship’s capstan with the radius of eight feet and the axle radius of one foot multiplies the force of sailors using it by eight. This relationship is the reason that water wheels on small, weak streams that do not provide much force have to be larger than on fast-moving streams—a weak stream will not provide enough force to turn a small wheel. Rotating handles or knobs, grinders, drills, and old-fashioned water wells all use the wheel’s mechanical advantage.

Geometry and Physics of Rolling: Work Smart, Not Hard

Rolling vehicles on wheels save work compared to dragging the same weight along the ground. Friction between the ground and a dragged object occurs along the length of the path. The work needed to overcome this friction is proportional to the friction coefficient, which depends on the surfaces of the object and the path. On smooth surfaces, such as ice, the friction coefficient is lower than on rough surfaces, such as rock. Work is also proportional to the weight of the object and the length of the path. When an object is rolled, its weight presses the axles to the wheels. Instead of the object-road friction, the force to overcome is now the axle-wheel friction, which is also proportional to the weight. When a wheel turns around, the vehicle travels the distance equal to the wheel’s circumference. If the radius of the axle is one-tenth of the radius of the wheel, then the distance the axle slides within the wheel is one-tenth of the distance the vehicle travels and the required work is divided by 10. It is relatively easy to reduce axle-wheel friction many times by using smooth surfaces, oil, and ball bearings. Vehicles for heavier loads usually have more wheels to distribute the force of the load.

Reinventing the Wheel

Since wheels are essential to most human endeavors, there are many wheel-related sayings. “Reinventing the wheel” means “needlessly duplicating a well-known method.” Ironically, wheels themselves are being constantly reinvented. For example, roller bearings first appeared in drawings in Leonardo da Vinci’s notebooks in the sixteenth century but were patented and used widely only in the nineteenth century.

Though reinventing the wheel is virtually impossible, throughout the twentieth and twenty-first centuries, advancements to wheels led to movement advancements. For example, magnetic bearings reduce axle-wheel friction to essentially zero and, therefore, promise huge increases in machine efficiency; their development started in the 1980s. In the 1990s, mathematics and science museums began to feature bikes with square wheels that move smoothly over special surfaces consisting of “catenaries,” which are hyperbolic shapes resembling hanging lengths of chains. Shape-shifting wheels created by the Defense Advanced Research Projects Agency’s (DARPA) Ground X-Vehicle Technologies was created to improve the mobility and safety of military combat vehicles without additional armor. Different materials have been used as well. Though most automobiles use rubber wheels, they are prone to holes. Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) sought to mitigate this issue by created a self-healing rubber using reversible polymers. So long as the wheel exists, scientists, researchers, and inventors will continue to try to improve upon it.

Bibliography

Burrows, Leah. “Harvard Researchers Develop Tough, Self-Healing Rubber.” Harvard John A. Paulson School of Engineering and Applied Sciences, 14 Aug. 2017, seas.harvard.edu/news/2017/08/harvard-researchers-develop-tough-self-healing-rubber. Accessed 12 Oct. 2024.

Farris, Frank. “Wheels on Wheels on Wheels—Surprising Symmetry.” Mathematics Magazine 69, no. 3 (1996).

Gambino, Megan. “A Salute to the Wheel.” Smithsonian Magazine, 17 June 2009, www.smithsonianmag.com/science-nature/a-salute-to-the-wheel-31805121/. Accessed 12 Oct. 2024.

Goodstein, Madeline P. Wheels! Science Projects With Bicycles, Skateboards, and Skates. Berkeley Heights, NJ: Enslow Publishers, 2009.

“GXV-T Advances Radical Technology for Future Combat Vehicles.” Darpa, 22 June 2018, www.darpa.mil/news-events/2018-06-22. Accessed 12 Oct. 2024.

Helfand, Jessica. Reinventing the Wheel. New York: Princeton Architectural Press, 2002.