In certain cases it is very difficult to maintain an equilibrium—try to walk across a tightrope. At the same time, nobody rewards a person sitting in a rocking-chair with applause. But he is also maintaining his equilibrium.

What is the difference between these two examples? In which case is equilibrium maintained “by itself”?

The condition for equilibrium seems to be obvious. For a body not to be displaced from its position, the forces exerted on it must balance; in other words, the sum of these forces must be equal to zero. This condition is really necessary for the equilibrium of a body, but is it sufficient?

A side-view of a hill easily built out of cardboard paper is depicted in Figure 1. A ball will behave in different ways depending on the part of the hill where it is placed. A force which makes it roll down will be exerted on the ball at any point on the slope of the hill. This active force is gravity, or rather its projection on the tangent to the section of the hill passing through the point which is of interest to us. It is therefore clear that the more gentle the slope, the smaller the force acting on the ball.

We are interested above all in the points at which the force of gravity is completely balanced by the reaction of the support, and hence the resultant force acting on the ball is equal to zero. This condition will be fulfilled at the top of the hill and at its lowest points-the hollows.

The tangents are horizontal at these points, and the resultant forces acting on the ball are equal to zero.

However, in spite of the fact that the resultant force is equal to zero at the top, we won’t be able to put a ball there, but even if we could, we would immediately detect the accessory cause of our success—friction. A small push or a light puff will overcome the frictional force, and the ball will leave its place and roll down.

For a smooth ball on a smooth hill, only the low points of the hollows will be positions of equilibrium. If a push or an air stream displaces the ball from such a position, it will return there by itself.

A body in a hollow (a hole or a depression) is undoubtedly in equilibrium. If we deflect it from such a position, a force returns it back. The picture is different at the top of the hill: if a body has left such a position, the force exerted on it tends to take it further away rather than bring it back. Consequently, the resultant force equal to zero, is a necessary but not a sufficient condition for stable equilibrium.

The equilibrium of a ball on a hill can also be regarded from another point of view. The hollows correspond to minima, and the top to maxima of potential energy. The law of conservation of energy prevents a change in positions for which the potential energy is minimum. Such a change would make the kinetic energy negative, which however, is impossible. The situation is entirely different at the top. A departure from these points entails a decrease in potential energy, and hence not a decrease, but an increase in kinetic energy.

Thus, in a position of equilibrium, the potential energy must assume a minimum value with respect to its values at neighbouring points.

The deeper the hole, the greater will be the stability. Since we know the law of conservation of energy, we can immediately say under what conditions a body will roll out of a depression. For this it is necessary to impart to the body the kinetic energy which would be enough for raising it to the edge of the hole. The deeper the hole, the greater will be the kinetic energy needed for disturbing the stable equilibrium.