STATICS (Gr. root sta, to stand), the science of the equilibrium or balancing of forces; on a body or system of bodies, has gradually advanced from the days of Archimedes to the vast developments it has now acquired. Singularly enough, though most of its simpler theorems are very generally known, are almost popular, in fact, there is no science in which elementary teaching is so defective. The ordinary proofs of its fundamental principles, such as the parallelogram of forces, the principle if the lever, etc., are usually founded on the supposition that a body in equilibrium is absolutely at rest. Now, any one who knows,that the earth rotates about its axis, that it revolves about the sun, that the Sun is in motion relatively to the so-called fixed stars; that they are in all probability, in motion about something else is itself in motion, etc., will at once see that there is no such thing as absolute rest, and that relative rest or motion, unchanged with reference to surrounding bodies, is all that we mean by equilibrium. He will then, at once, see that the foundations of statics are to be sought in the Laws of Notion (q.v.). And, in fact, Newton's second law of motion gives us the, necessary and sufficient conditions of equilibrium of a single particle under the action of any forces; while his third law, with the annexed scholium, gives these conditions for any body or system of bodies whatever.
, The simplest statement of the conditions of equilibrium of a rigid body which can be given, is that furnished by this scholium of Newton's, which is now known by the name of the principle of energy (see FORCE) or work (q.v.). It is as follows: A rigid body is in equilibrium if, and is not equilibrium unless, in any small displacement whatever, no work is done on the whole by the forces to which it is subject. In the case of what are called the mechanical powers (q.v.), this is equivalent to the statement that work expended on a machine is wholly given hack by the machine—or that the work done by the power is equal to the work spent in overcoming the resistance.
It is shown in the geometrical science of kinematics that any motion whatever of a rigid body can be reduced to three displacements in any three rectangular directions, together with three rotations about any three rectangular axes—so that the equilibrium of a rigid body is secured if no work be done on the whole in any of these six displace ments. There are thus six conditions of equilibrium for a rigid body under tire action of any forces—and these are reduced to three (two displacements and one rotation), i•the forces are confined to one plane; and to one (a displacement) if the forces act all in one line.
Equilibrium may he stable, unstable, or neutral. It is said to be stable if the body, when slightly displaced in any way from its position of equilibrium, and left free, tends to return to that positioin It is unstable if there is any displacement possible which will leave the body in a position in which it tends to fa/I./tut/1er away from its position of equilibrium. It is neutral if the body, when displaced, is still in equilibrium. It is easily shown, but we cannot spare space for the proof, that a position of stable equilib rium is, in general, that in which the potential energy (see FORCE) of the body is n mini mum—of unstable equilibrium, where it is a maximum (for some one direction of displacement at least)—of neutral equilibrium, where the potential energy remains unchanged by any small displacement. Thus a perfect sphere, of uniform material, is in neutral equilibrium on a horizontal plane—while an oblate spheroid, with its aids of rotation vertical, is in stable equilibrium: and a prolate spheroid, with its axis vertical, is in unstable equilibrium on the plane. Similar statements hold for other than rigid bodies. Thus, a chain, or a mass of fluid, is in stable equilibrium when its potential energy is least, i.e., when its center of gravity is as low as possible. This simple state ment is sufficient for the mathematical solution of either question.