Contact area of ideal sphere resting on flat surface

[Chubby Hubby]

Greetings All,
I have a rather odd question which has been bothering me. If you have a perfectly round sphere sitting on a perfectly flat plane, what is the area of surface contact between the two? Is there an actual value, or is it something which can't be calculated. I'm assuming the diameter of the sphere doesn't matter?
TIA, CH

 

[BiGyElLoWhAt]

The diameter of the sphere would matter. It would have to do with the weight of the sphere and the tensile strength of the material constructing it. Theoretically, for a perfectly rigid sphere, they contact at 1 point. That's not the case in actuality. There is some force applied on the bottom of the sphere that causes it to flatten a bit.
It'd be pretty rough to calculate (if you can even do it analytically), so it may or may not have to be done numerically.

 

[BiGyElLoWhAt]

Think about it like a tire, it makes very little contact when it's full of air, because the structural integrity of the tire is very high. If you start letting air out, more and more of the surface of the tire contacts the ground.

 

[berkeman]

Greetings All,
I have a rather odd question which has been bothering me. If you have a perfectly round sphere sitting on a perfectly flat plane, what is the area of surface contact between the two? Is there an actual value, or is it something which can't be calculated. I'm assuming the diameter of the sphere doesn't matter?
TIA, CH

I would think that the compressive stress from the weight of the ball would create some strain/deformation to give you a small flat spot. You could probably calculate the diameter of the flat spot from the density and Young's modulus of the ball's material...

https://en.wikipedia.org/wiki/Young's_modulus

EDIT -- Oops, too slow typing!

 

[Chubby Hubby]

What if there was no force to flatten it?

 

[BiGyElLoWhAt]

Well, at the very least you have weight. Unless it's massless. If you're in space, I would guess that it would approach the 1 point limit.

 

[russ_watters]

The way the OP is worded ("perfectly round", "perfectly flat"), the answer is just a math/geometry answer: they touch at a single point and form a tangent.

But no real-world sphere is perfectly round, hard and weightless and no real-world plane is perfectly flat and hard. So there is a contact patch (area) between all real spheres and planes.

 

[Chubby Hubby]

Thanks, Russ. I assume a tangent point can't be defined by a dimension? (Sorry, totally math impaired...)

 

[Nidum]

Contact Mechanics

 

[Mech_Engineer]

For a geometrically perfect point, it will not have a dimension associated with it. See here: https://en.wikipedia.org/wiki/Point_(geometry)

In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.

When you begin taking into account real-world mechanics and forces, Contact Mechanics will apply as Nidum pointed out.

Contact mechanics is the study of the deformation of solids that touch each other at one or more points.[1][2] The physical and mathematical formulation of the subject is built upon the mechanics of materials andcontinuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact.

 

[Chubby Hubby]

Thanks. Once I realized it was only a point where they contact each other, then I understood you really can't give it a dimension. I still can't get my head around the idea that although they are touching, that area can't be defined. I guess I don't have a good imagination...

 

[Mech_Engineer]

Remember a "perfect sphere" and "perfect flat plane" don't exist in the real world, so in reality there will be a small contact that can have a defined area (due to deformation). This is what the topic of Contact Mechanics is all about. The "point contact" only exists in a theoretical geometric world, for everything in the real world you would use properties of the bodies and forces to estimate their contact area; of course for very rigid bodies such as hardened metals or ceramics this might be a very small area.

 

[berkeman]

If you're in space, I would guess that it would approach the 1 point limit

There still will be gravitational attraction between the sphere and plane in space...

 

[Merlin3189]

Do solids ever touch? I thought the atoms repelled each other electrostatically and the "contact" area was caused by the electrostatic forces rearranging the atoms in the solids to keep them apart.

 

[Mech_Engineer]

Oh boy this is going down a rabbit hole...

 

[BiGyElLoWhAt]

There still will be gravitational attraction between the sphere and plane in space...

Aha, yes, but they "approach" 1 point =D