Objects falling from table It is an everyday experience:
You have placed an object a tiny little bit too much over the edge of a table and it falls down. This is the case when the center of gravity is above the floor and not above the table.
But what if the center of gravity of the object is exactly on the boundary of the table? Can we predict whether the object falls or stays on the table?
How to predict whether the object falls or not? How I should proceed?
 A: If you approach this problem mathematically and tried to calculate the time it takes for an object to tip over as a function of the overhang distance $x$ of the center of mass over the edge, you will find that $t \rightarrow \infty$ as $x \rightarrow 0$.
So does this mean that the object will tip but after an infinite time, or that it won't tip at all. Practically it makes no difference. Mathematically it is the second. It will never tip in finite time.
A: If the centre of gravity of the object is vertically above the edge of the table then the object is in equilibrium. However, this equilibrium position is unstable (like a pencil balanced on its point) because a small tilt of the object will lower the centre of gravity, which will then cause the tilt to increase. This is a positive feedback loop.
However, if the centre of gravity of the object is vertically below the edge of the table then the object is in a stable equilibrium position. A small tilt of the object will now raise the centre of gravity, which will then cause the tilt to reduce. This is a negative feedback loop.
A: Just a thought: if the object were to tilt even the slightest bit (perhaps it and/or the surface are a bit pliable?) then the CG would move further away from the table (if only ever so sligjtly) - and this is an unrecoverable process.
However, I know there is theory stating that there is an "undefined state" about the direction of bounce of a ball rolling along a floor towards a wall, if the wall and floor meet in a curve that precisely matches that of the ball. That's probably related to your issue, but I'm no scientist.
A: A string can be seen as a boundary on either side. If center of gravity is above the string, the object will stay on the string in equilibrium. Therefore if center of gravity is above the boundary, object will stay on the table.
A: In theory, it takes an infinite amount of time for the object to tip over. In practice though, there will always be a chance the object falls down, due to classical statistical mechanics. It's very hard to predict though when. When the object falls down depends on several conditions and qualities of the object. As there are its mass and form, its surface, the surface of the table, and the medium in which the object finds itself. Let's assume this medium is air (as you envision in your question). Let's further assume that all parts (atoms, molecules) of the object find themselves above the surface of the table, that the air has room temperature (about 20 degrees Celcius), and that there are no overall air movements (a strong wind could easily make the object tip over).
What can cause the object to fall down?
The only forces that can make this happen are the gravitational force and the force exerted by the air on the object.
The gravitational force on both sides of the object will be constantly varying, due to the fact that atoms are not at rest. Or better put: The CM is constantly varying around its mean value.
On top of that, there is the force exerted on the object by the medium and the table resulting. This force will vary slightly around the mean value (which is zero), again, due to the the random motion of atoms and molecules in the air and table. The net mean force excerted by the table is zero too, except (in this realistic case) in a very small "hinge strip". The point where this force grips the object varies randomly.
So it is possible that there is a net force pulling or pushing on the "open" side of the object to provide torque.
So a torque can be provided thanks to statistical mechanics. As might be clear from previous answers, even the smallest torque can make the object fall down. That is, in the absence of friction between the table and the object.
For an "everyday-sized" (mass and form) object, as the one you bring up, on an "everyday-table" the resisting force emanating from the contact between object and table will be too big too make the object fall.
But if the conditions are right, the object can fall down in a finite time. For example when the temperature is raised and the shape of the object is such that a small force can give a sufficient torque, for example if open piece has the form of a long thin strip and the piece on the table has the form of a cube. On top of that we can make the resisting friction force as small as possible, by crafting both object and table in a certain way.
As I already stated, it will be very hard to predict when the object falls down. Even when you perform an experiment with an object and table and prepare the air, the object, and the table in an optimal way (for a downfall to occur), then the most you can do is calculate the chance that the object will fall down in a certain time. Only for a theoretical object you can calculate that it will never fall down (at t approaching infinity).
