The green table has a small rod that extends from its center of mass, hanging from which is a red plumb bob. When the table sits level, the plumb bob hangs above the center of its base. As you tilt the table as shown, the point above which the plumb bob hangs moves closer to the edge of the base that is still in contact with the surface of the demonstration table. As long as the plumb bob hangs within the area of the base, the table is in stable equilibrium and remains upright. When the plumb bob hangs above the line of contact of the base with the surface, the table is, at least in one direction, in unstable equilibrium, but still remains upright. As soon as the plumb bob moves past the line of contact of the table with the surface, the table is no longer in equilibrium, and it tips over. A foam rubber block set where the top of the table will fall (not shown), cushions its fall and prevents damage.

The center of mass of an object is the weighted mean displacement of all of its mass points from some reference point, usually the origin of the coordinate system in which the object sits. This is the sum of the products of all the mass points with their distances from the origin, divided by the total mass of the object (r_cm = Σm_ir_i/M). If we set the center of mass of the object at the origin, we find that this sum equals zero. This tells us that, assuming that we could exert a force on the object at its center of mass, this would result in no torque on the object, but if we applied a force to the object at any other point, we would exert a torque on the object about its center of mass.

If the object is relatively small and close to the earth, then gravity acts uniformly on the object. That is, if the object is of uniform density, gravity exerts the same downward force on all of its mass points, and the sum of all of these forces is equal to a single downward force, F = Mg, acting at the center of mass of the object. The torque due to each of these forces, about the center of mass of the object, is τ = m_ir_i × g, and the sum of the torques is τ = (Σm_ir_i) × g. If we set the center of mass of the object at the origin, this sum equals zero. Thus if we apply a force at the center of mass that is equal to and opposite that of gravity, we can support the object without exerting a torque on it about its center of mass. This point at which the resultant force of gravity acts is also called the center of gravity. If we take the force of gravity as being uniform over the entire object, then the center of gravity and the center of mass are the same point. The green table shown above is not of uniform density, but it is symmetrical, and its center of mass (and center of gravity) is at a point within the vertical frame member along the line of the rod from which the plumb bob hangs.

When the table sits level, the demonstration table balances the force of gravity to support the weight of the table. The weight of the table is distributed over its base, at each point of which the demonstration table exerts an upward force to balance that of gravity. At any point not directly below the center of mass of the table, such a force should exert a torque about the center of mass, but each torque is balanced by an opposing torque from the corresponding point on the opposite side of the center of mass. Similarly, any torque that gravity could exert at the center of mass about any point on the base, is balanced by an opposing torque that it exerts at a corresponding point on the opposite side of the center of mass.

If you tilt the table by raising either end, this raises the center of mass of the table, thus raising its potential energy. Gravity, acting at the center of mass of the table, has a moment about the edge of the base of the table opposite the side you lifted, and exerts a torque opposite to the torque you exerted to tilt the table. If you then let go of the table, gravity returns it to its original position, lowering its center of mass to its original height. This would happen for any direction in which you tilted the table. The table is in stable equilibrium. (See demonstration 32.20 -- Neutral, stable and unstable equilibrium.)

As noted above, as you tilt the table farther and farther, the vertical line from the center of mass moves closer and closer to the edge of the base on which the table rests. At the same time, the center of mass rises, and the moment gravity has to exert a torque about the edge of the base decreases. At the point where the center of mass is exactly above the edge on which the table rests, the center of mass is at its maximum height, and since the line along which gravity acts now goes directly through the edge, gravity can exert no torque about the edge. If the table were balancing in this position, displacing it slightly to either side would lower its center of mass. Gravity would have a small moment by which to exert a torque about the edge, toward whichever side to which the table was displaced. This would increase the displacement, and the table would fall. The table would be in unstable equilibrium. If you tilt the table, then, until the plumb bob hangs exactly over the edge of the base, the table still rests, however lightly, against the jack. If you continue to tilt the table until the plumb bob just crosses the edge of the base, gravity can now exert a small torque about the edge, tilting the table farther in that direction, and the table falls.

As long as the vertical line that goes through the center of mass is within the base of the table, the table is in stable equilibrium and remains upright. If you tilt the table in any direction to the point where the vertical line through the center of mass is directly over an edge of the base of the table, the table is in unstable equilibrium (in at least one direction). As soon as you tilt the table far enough that the vertical line through the center of mass lies outside the area of its base, the table is no longer in equilibrium, and it falls over.