This is a problem that was solved by Hertz and is mathematically very well understood. The answer doesn't require the ball or the surface to be imperfect in their shape, just their stiffness.
To understand how it works, consider a soft rubber ball sitting on a soft rubber surface. Now allow both ball and surface to deform elastically - the contact area increases but the amount of rubber-air interface decreases, and hence the energy is lowered. (Note that if the ball is sitting on top of the surface, its centre of gravity moves down and its potential energy is also decreased - this helps, but is not required.) Of course the act of deforming the rubber increases its strain energy. Equilibrium is reached when the two energy terms are equal but opposite.
Now turn to splongo's ball and surface, which we're assuming to be much stiffer and hence to resist deformation in a way that rubber doesn't. Initially, the point of contact will indeed be just that - a point. However, the maths (which I shan't go into here) shows that at this point, the change in surface energy with a TINY amount of deformation tends to infinity. No matter how stiff the material, it WILL deform, and the contact area will be finite.
More information - and equations to calculate the contact area - can be found here: http://en.wikipedia.org/wiki/Contact_mechanics
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To understand how it works, consider a soft rubber ball sitting on a soft rubber surface. Now allow both ball and surface to deform elastically - the contact area increases but the amount of rubber-air interface decreases, and hence the energy is lowered. (Note that if the ball is sitting on top of the surface, its centre of gravity moves down and its potential energy is also decreased - this helps, but is not required.) Of course the act of deforming the rubber increases its strain energy. Equilibrium is reached when the two energy terms are equal but opposite.
Now turn to splongo's ball and surface, which we're assuming to be much stiffer and hence to resist deformation in a way that rubber doesn't. Initially, the point of contact will indeed be just that - a point. However, the maths (which I shan't go into here) shows that at this point, the change in surface energy with a TINY amount of deformation tends to infinity. No matter how stiff the material, it WILL deform, and the contact area will be finite.
More information - and equations to calculate the contact area - can be found here: http://en.wikipedia.org/wiki/Contact_mechanics
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