Ask a Scholar: Touching Spheres

Vera B. Koutsoyannis

Dear Ask a Scholar,

Let’s say you have two spheres, each with a 2’’ diameter and they are just touching each other.  I’m trying to figure out the actual dimensions (in inches) where they touch.  I know mathematically speaking, it’s just a single point without dimensions, right?   What’s reality?

Answered by Vera B. Koutsoyannis, who teaches physics at Western High School in Davie, Florida, and is an adjunct professor of mathematics at Broward Community College.

Mathematically, if two spheres touch, the place of contact is one point.  If we consider two intersecting spheres, their intersection is a circle.  As the spheres are pulled apart, the circle becomes smaller, until the spheres are only just touching, when the area of the circle is just one point.  It can be expressed as the limit, as R approaches zero, of the area of intersection (πR2).

In real life, however, the area of contact depends on many different factors. 

The first I’ll consider is the difference in scale between the observer and the size of the spheres.  For human beings, the point of contact between a couple of smooth spheres, like volley balls, or round marbles, does seem like just a point, too small to measure.   But supposing that the spheres are as great as the Earth, whose curvature is difficult to perceive on the surface, then the “point of contact” may have a very large area, since at the scale of humans the Earth’s surface is a flat plane.  It would seem, then, that the area of contact between two spheres is directly proportional to their size.

Another factor is the relative “smoothness” of the surfaces.  Smooth surfaces have well-defined boundaries that show the point of contact precisely, whereas rough surfaces have fuzzy boundaries which tend to enlarge the point of contact, since more area between the spheres has material belonging to both spheres.

Factor three is whether the spheres are pressed together by some force, and resist flattening against each other with rigidity; the person is pushing two metal balls together, causing them to have a larger contact point than if they sit together barely touching.  If the two spheres carry opposite charge, the same effect of pressing together occurs, unless they are metal and quickly discharge on contact.

If the spheres are pressed together, there is a factor of elasticity to consider, since the more flexible the materials, the greater the contact area that is created. 

Another consideration is that where the spheres touch, the boundary has a topology as intricate as a fractal, again depending on (a) the detail we want, (b) the size of the observed detail, and (c) the roughness of the materials.  Please see the lucid blog explaining fractal borders by Mark Chu-Carroll: http://scienceblogs.com/goodmath/2007/07/fractal_borders_1.php.

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Also answered by Brody Dylan Johnson, who is an Associate Professor of Mathematics at Saint Louis University in St. Louis, Missouri.

In the ideal mathematical setting two solid spheres may contact only at a single point, but, in reality, there is no such perfect sphere.  The irregularities in the shapes of two solids generally lead to more complicated contact regions.  In sports, the contact regions between various objects are greatly affected by the elasticity of the objects. (A material with greater elasticity is easier to deform.)  For example, tennis balls undergo significant deformations when making contact with the court or racquet, leading to large contact areas in comparison to the surface area of the ball itself.  The same goes for the contact between bat and ball in baseball.  However, in other games, e.g., billiards or bowling, the materials are less elastic and the contact regions have smaller area in relation to the area of the balls themselves.  One could get some idea about the relative contact areas by coating one of the objects with ink or paint and observing the transfer from one object to the other when they make contact.  Bouncing a wet tennis ball on a dry surface will also illustrate this concept pretty well!

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