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:

* * *

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!

* * *

About “Ask a Scholar”
Have a question Wikipedia can’t answer? We’ll match your question to a scholar with an answer.  Questions submitted to “Ask a Scholar” should call for educated judgment rather than facts that can be found easily with an internet search. We especially welcome questions that provide professors the occasion to draw erudite distinctions and incorporate mention of matters you had no idea were connected to the topic at hand. Simply email NAS or submit questions via Intellectual Takeout's Ask the Professor feature. We'll field your question to a scholar and get back to you with an answer as soon as possible.
Image: Pixabay, Public Domain 
  • Share

Most Commented

June 5, 2024


Subpoenas for All!

Ohio Northern University gnaws its teeth with an appetite for vindictive lawfare....

June 6, 2024


Backlash: Sometimes It Hurts So Good

We have undermined the leftist status quo in higher education for decades with the persistence of Morlocks. You really should be more alarmed about us than you are. Not that I’m going......

May 7, 2024


Biden Admin Is Weaponizing Title IX To Promote Fringe Sexual Politics

Earlier this month, the Office for Civil Rights in the Biden Education Department issued a new regulation on how schools must observe Title IX. This rule transf......

Most Read

May 15, 2015


Where Did We Get the Idea That Only White People Can Be Racist?

A look at the double standard that has arisen regarding racism, illustrated recently by the reaction to a black professor's biased comments on Twitter....

October 12, 2010


Ask a Scholar: What is the True Definition of Latino?

What does it mean to be Latino? Are only Latin American people Latino, or does the term apply to anyone whose language derived from Latin?...

May 7, 2012


Ask a Scholar: Declining the Second Term

Has there ever been a president who did not run for a second term by choice?...