The Surface Area of Nothing

Redditor splongo asks two questions:

If a perfectly spherical ball is sitting on a perfectly flat surface, what is the size of the contact area? Would it not be infinitely small?


I'm at a loss for a completely coherent answer. I don't think that, as a physical reality, there can be a perfectly spherical or flat object. But as a theoretical model (e.g. the formula for a sphere), there can be such objects, and therefore intersection between them. Would the surface area of the intersection point be non-dimensional, or just incalculably small?

How would you answer splongo's questions?

Link -via The Agitator | Photo: katerha

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I'm told that mathematically, the contact area is infinitely small, essentially nil.

But in physics, the contact area is as "small as it is possible to be", i.e. one squared Planck length (equal to 1.6162×10?35 metres)

http://en.wikipedia.org/wiki/Planck_length

Both are impossible in reality, but mathematicians will come up with a different answer from physicists.
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"what is the size of the contact area?" 0 "Would it not be infinitely small?" Yes.

But of course this is a non-physical world. If you worked out the force on that point it would be infinite, so something would need to give. The ball would become non-spherical, or the surface non-flat (and more likely, both), until the force in the contact patch - that's the area where the two surfaces touch - is no longer strong enough to continue to deform the objects. Either that, or one of them breaks.
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