Contact Area of 2 spheres

The contact area is theoretically equal to zero.

Question the accuracy of the adjective "incompressible".

Not exactly zero - as I remember my geometry, they would make contact at one and only one point. And of course "points" in space have zero area, but in real life no sphere is perfectly round, so the contact area would be equal to the flatness in the sphere at that point. For a perfectly round sphere I suspect you have to go to the atomic level - so perhaps one atom's width?

An "incompressible" material is also a not "real life" notion so that the statement that area is zero corresponds to the way the question was formulated !

Then the contact area is a single point, of area zero.

Maybe he meant to say it was an "incomprehensible" material.

This was meant to be a mental exercise, not a real life problem.It was poorly worded, I agree.

Now that that is settled,consider if the spheres each had an infinite radius.

Would the contact area also be infinite?

Infinitely large or small?

Are these spheres made of 'Unobtanium', or transparent Al?

Infinitely large.Made of UnunPentium.

Imagine a world with no hypothetical situations.

Nope, still only a single point.

Would not a segment of a circle with an infinite radius be a straight line?

How could one prove it was not a straight line if the radius was infinite?

In your formulation is not precise: - external contact - area =0 which ever the radius is - internal contact - area is full spheres area since equal radius means identical spheres. I considered an "external contact" but if you think about "internal contact" the area is not zero. In case of external contact of equal spheres or internal contact of different spheres the area is only due to a surface deformation which was first analyzed by Hertz and which presumes a force applying the spheres one on the other. The are is function of the compressibility of the material given by the Young modulus which is present in the Hertz equations at power -1. Since you mention an incompressible material the modulus is infinite and the contact area is zero which ever the force is. This is a forum for engineers what you discuss is philosophy and some other places are may be more practical for such discussions. If the radius is infinite then the spheres are not anymore spheres but planes being part of a semi-infinite body and the contact area depends on the limits of this body in the contact plane. I have the feeling that you are not familiar to the notion of limit in mathematics.

As I described previously, this is a mental construct.Are you saying that to be engineering related it must be buildable in the real world, and that things that can only be imagined should not be contemplated, except in a philiosophical format?What about Einstein's theories, that are still being tested today.Were his theories philiosophy before they were proven?I suspect you have a very narrow definiton of engineering, and a very broad definition of philosophy.Consider quantum mechanics:This discipline is very must like philosophy.

Back to the matter of the spheres:Remember, I said the spheres are making contact.Are you saying the contact area is also infinite(as in the case of parallel lines) or that they will never touch, which is against the original described question ?Also, how do you have 2 exactly equal spheres with internal contact(concentric) and not overlapping?

If these HYPOTHETICAL spheres were constructed of an electrically very conductive material, and an electrical potential difference existed between them, that there would be no current flow from one to the other (excluding arcing or corona discharge) ,since the contact area is zero? Is that what you are saying?

What laws does this violate? I am willing to learn.

Enlighten me, please.

Dear Sir,

I am full of respect for those who want at any price be right. What I notice is that in order to be right and obtain a validation you mix ideal and real aspects.

Ok I shall not try to convince you since you will not accept arguments which ever they are. I decided to make you HAPPY:

You are right the incompressible spheres with a contact (according to my comments should be an external one) and even without any force applying one on the other have a contact area with an order of magnitude of a string!

I recognize that considering engineering as a real life bound profession I made an error for which I apologize. I recognize that my understanding of philosophy (may I suggest you use the spelling checker when you use words you are not accustomed to) is not at the right level since I am and remain a simple engineer with, as all engineers bound to reality, a limited capability to see the depth of such notions.

Since I am not at a level high enough to pursue this hypothetical discussion it will be my last comment, you may feel free to make any other comment but you will not any more get an answer from me. I react every time when I meet a will to be right at any cost.

Be happy you are the winner!!!!!!!!

Very faithfully

Nick Name

PS.: Since english is not my mother/father language I use the spelling checker to manifest my respect to it.

You do not make me happy by conceding that I may be right. An answer to my question is what I seek. There may be no right answer.I merely want a discussion of the possibilities entailed in the original question.I have not said that anyone was wrong with their answer, merely questioned their answer(s) in search of more information.I am sorry your ego is so fragile that it cannot bear scrutiny of your answers. You stated that you felt I was not familiar with the limits in mathematics, and I merely stated (in defense) that there is a very fuzzy line between philosophy and mathematics and engineering.Insofar as you hurled the first stone, the burden is upon you to accept the consequences.Since you will no longer reply to this forum, and insist on sulking like a child, I am sure your contributions will not be missed.But, on the off chance that you will read this, i wish you all the best that life has to offer, and I harbor no ill will, for you, merely sympathy.

Have a blessed day.

The leap to infinity doesn't help you in this instance because a point is infinitesimally small. You can make the engineering approximation that a very large radius implies a very small curvature - i.e., a flat surface, but that's operating in the real world. If you take two perfectly spherical (and incompressible) galaxies and nudge them against each other, the area of contact is still a single infinitesimal point.

Smaller than a Plank length squared?

If the material is incompressible, i.e. the modulus of elasticity is infinite, the contact area is zero no matter how large the radius of the spheres. If the material has a known modulus of elasticity, then there is a formula for calculating the contact area between the spheres.

Must they compress to make contact? Even the lightest contact compresses any material to some degree?

Can you give me the formula for calculation that you mention?

I am not disagreeing with what you say, I am certain it is true and valid, but I have trouble wrapping my mind around the concept of contact requiring compression or deformation of one or both of the objects.On the atomic level, nothing is solid.Everything is really a force field. If the spheres were made of a material with lots of valence electrons,could some of the electrons share orbits without interference, and thereby contact without compression?I do not know, I am trying to understand.

Thanks for your patience with my lack of understanding.

I don't know much about the behavior of electrons and protons. I suspect when we see the two spheres just touching, a physicist would say they are not touching at all because all of the electrons and protons have space around them so nothing is touching.

From an engineering point of view, if the two spheres are said to be making contact with zero force acting between them, the contact area is a point which has zero area.

If there is a force pushing the two spheres together, the contact area will be a circle whose radius expands as the force increases. Each sphere will have a flattened end where the two are in contact. The bearing stress will be approximately 1.5Fy, so the force resisted by the flattening of the spheres is about 1.5Fy*A. If the force is F, then A = F/(1.5*Fy) where F is the yield strength of the material. If the spheres are solid steel with a yield of 50,000 psi and the force pushing them together is 1,000#, then A = 1000/(1.5 * 50,000) = 0.0133 in². The radius of the contact circle is thus 0.065" or the diameter is 0.13", about 1/8".

If the force is only 100#, A = 0.00133 in² and the diameter is 0.04" or about 1/24".

The above is an approximation assuming the spheres are accurately machined to a large radius. If you require a more accurate estimate, it would be necessary to go into the theory of elasticity with stress functions etc., not something I intend to do for this discussion.

Thanks! That works for me.Just what I was looking for.I appreciate your patience.

"Theory of Elasticity" by Timoshenko and Goodier - Second Edition, p. 372 Article 125 "Pressure between Two Spherical Bodies in Contact" treats the subject in much more detail. Assuming Poisson's Ratio of 0.3 for two spheres of radius R₁ and R₂, with compressive force P, the following relationships are derived:

a = 1.109*{P*R₁*R₂/(E*(R₁ + R₂)}^1/3

q₀ = 3*P/(2*pi*a²) = 0.388{P*E²*(R₁ + R₂)²/(R₁²*R₂²)}^1/3

In the above,

a is the radius of the contact area

q₀ is the maximum pressure at center of surface of contact

E is the modulus of elasticity for the spheres (for steel this would be 29,000,000 psi)

Using the above relationships, the present problem can be solved using R₁ = R₂ = R

Hmmm.... two spheres each having an infinite radius. So, if they were touching, their combined span would be 4R or 4 times infinity. Sounds like we have a problem.