# statics

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## statics,

branch of mechanics**mechanics,**

branch of physics concerned with motion and the forces that tend to cause it; it includes study of the mechanical properties of matter, such as density, elasticity, and viscosity.

**.....**Click the link for more information. concerned with the maintenance of equilibrium in bodies by the interaction of forces upon them (see force

**force,**

commonly, a "push" or "pull," more properly defined in physics as a quantity that changes the motion, size, or shape of a body. Force is a vector quantity, having both magnitude and direction.

**.....**Click the link for more information. ). It incorporates the study of the center of gravity (see center of mass

**center of mass,**

the point at which all the mass of a body may be considered to be concentrated in analyzing its motion. The center of mass of a sphere of uniform density coincides with the center of the sphere.

**.....**Click the link for more information. ) and the moment

**moment,**

in physics and engineering, term designating the product of a quantity and a distance (or some power of the distance) to some point associated with that quantity.

**.....**Click the link for more information. of inertia. In a state of equilibrium all the forces acting on a body are exactly counterbalanced by equal and opposite forces, thus keeping the body at rest. The principles of statics are widely applied in the design and construction of buildings and machinery.

## Statics

The branch of mechanics that describes bodies which are acted upon by balanced forces and torques so that they remain at rest or in uniform motion. This includes point particles, rigid bodies, fluids, and deformable solids in general. Static point particles, however, are not very interesting, and special branches of mechanics are devoted to fluids and deformable solids. For example, hydrostatics is the study of static fluids, and elasticity and plasticity are two branches devoted to deformable bodies. Therefore this article will be limited to the discussion of the statics of rigid bodies in two- and three-space dimensions. *See* Buoyancy, Hydrostatics, Mechanics

In statics the bodies being studied are in equilibrium. The equilibrium conditions are very similar in the planar, or two-dimensional, and the three-dimensional rigid-body statics. These are that the vector sum of all forces acting upon the body must be zero; and the resultant of all torques about any point must be zero. Thus it is necessary to understand the vector sums of forces and torques.

In studying statics problems, two principles, superposition and transmissibility, are used repeatedly on force vectors. They are applicable to all vectors, but specifically to forces and torques (first moments of forces). The principle of superposition of vectors is that the sum of any two vectors is another vector. The principle of transmissibility of a force applied to a rigid body is that the same mechanical effect is produced by any shift of the application of the force along its line of action. To use the superposition principle to add two vectors, the principle of transmissibility is used to move some vectors along their line of action in order to add to their components.

The moment of a force about a directed line is a signed number whose value can be obtained by applying these two rules: (1) The moment of a force about a line parallel to the force is zero. (2) The moment of a force about a line normal to a plane containing the force is the product of the magnitude of the force and the least distance from the line to the line of the force. *See* Equilibrium of forces, Force, Torque

## Statics

the branch of mechanics that studies the equilibrium condition of bodies acted on by forces.

Statics is divided into geometric and analytic statics. The basis of analytic statics is the virtual work principle, which gives the general condition of equilibrium of any mechanical system. Geometric statics is based on the axioms of statics, which express the properties of the forces acting on a material particle and on an ideal rigid body, that is, on a body in which the distances between points always remain constant.

The fundamental axioms of statics establish that (1) two forces acting on a material particle have a resultant determined by the law of parallelogram of forces, (2) two forces acting on a material particle or ideal rigid body are balanced only when they are identical in magnitude and act in opposite directions along a straight line, and (3) the addition or subtraction of balanced forces does not change the action of the given system on a rigid body. Here, balanced forces are those forces under the action of which a free rigid body may be at rest with respect to an inertial frame of reference.

The statics of rigid bodies is studied by means of the methods of geometric statics. Here, the solutions of the following two kinds of problems are considered: (1) reduction of the system of forces acting on a rigid body to its simplest form and (2) determination of the equilibrium condition of the forces acting on a rigid body.

The necessary and sufficient conditions of equilibrium of elasti-cally deformable bodies, of liquids, and of gases are considered in elasticity theory, hydrostatics, and aerostatics, respectively.

The basic concepts of statics include the concepts of force, the moment of force relative to the center and relative to the axis, and the couple. The addition of forces and their moments relative to a center is accomplished according to the rule of addition of vectors. The quantity **R**, equal to the geometric sum of all forces **F _{k}** acting on a given body, is called the resultant vector of this system of forces, and the quantity

**M**, equal to the geometric sum of the moments

_{0}**m**(

_{0}**F**) of these forces relative to the center

_{k}*O*, is called the resultant moment of the system of forces relative to the given center:

**R = ∑F _{k} M_{o} = ∑m_{o} (F_{k})**

The solution of the problem of the reduction of forces gives the following basic result: any system of forces acting on an ideal rigid body is equivalent to a single force, equal to the resultant vector R of the system and applied to an arbitrarily selected center *O*, and to a single couple, with a moment equal to the resultant moment M_{0} of the system relative to this center. It therefore follows that any system of forces acting on a rigid body can be defined by the system’s resultant vector and resultant moment. This principle is widely used in practice in determining, for example, the desired aerodynamic forces acting on an airplane or rocket or the forces in a cross section of a beam.

The simplest form to which a given system of forces can be reduced depends on the values of **R** and **M**_{0}. If **R** = 0 and **M**_{0} ≠ 0, then the given system of forces can be replaced by a single couple with moment M_{0}. If R ≠ 0 and M_{0} = 0 or M_{0} ≠ 0 but the vectors R and M_{0} are mutually perpendicular (which, for example, always holds for parallel forces or forces lying in a single plane), then the system of forces reduces to a resultant equal to R. Finally, when R ≠ 0, M_{0} ≠ 0, and these vectors are not mutually perpendicular, the system of forces may be replaced by the joint action of the force and the couple (or by two concurrent forces), and the system has no resultant.

For equilibrium to exist in any system of forces acting on a rigid body, it is necessary and sufficient for the quantities R and M_{0} to vanish (*see*EQUILIBRIUM OF A MECHANICAL SYSTEM for a discussion of the equations that follow from this condition and that must be satisfied by the forces acting on the body at equilibrium). The equilibrium of a system of bodies is studied by constructing the equations of equilibrium for each body separately and by taking into account the law of action and reaction. If the total number of constraints is greater than the number of equations describing these constraints, then the corresponding system of bodies is statically indeterminate. To study the equilibrium of such a system, allowance must be made for the deformation of the bodies.

Graphic methods of solving problems of statics are based on the construction of a polygon of forces and a funicular polygon.

### REFERENCES

Poinsot, L.*Nachala statiki*. Paris, 1920.

Zhukovskii, N. E.

*Teoreticheskaia mekhanika*, 2nd ed. Moscow-Leningrad, 1952.

Voronkov, I. M.

*Kurs teoreticheskoi mekhaniki*, 9th ed. Moscow, 1961.

Targ, S. M.

*Kratkii kurs teoreticheskoi mekhaniki*, 9th ed. Moscow, 1974.

See also references under MECHANICS.

S. M. TARG