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Statics
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Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment.

If is the total of the forces acting on the system, is the mass of the system and is the acceleration of the system, Newton's second law states that (the bold font indicates a vector quantity, i.e. one with both magnitude and direction). If , then . As for a system in static equilibrium, the acceleration equals zero, the system is either at rest, or its center of mass moves at constant velocity.

The application of the assumption of zero acceleration to the summation of moments acting on the system leads to , where is the summation of all moments acting on the system, is the moment of inertia of the mass and is the angular acceleration of the system. For a system where , it is also true that

Together, the equations (the 'first condition for equilibrium') and (the 'second condition for equilibrium') can be used to solve for unknown quantities acting on the system.

History

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Archimedes (c. 287–c. 212 BC) did pioneering work in statics.[1][2] Later developments in the field of statics are found in works of Thebit.[3]

Background

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Force

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Force is the action of one body on another. A force is either a push or a pull, and it tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application (or point of contact). Thus, force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.[4]

Forces are classified as either contact or body forces. A contact force is produced by direct physical contact; an example is the force exerted on a body by a supporting surface. A body force is generated by virtue of the position of a body within a force field such as a gravitational, electric, or magnetic field and is independent of contact with any other body; an example of a body force is the weight of a body in the Earth's gravitational field.[5]

Moment of a force

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In addition to the tendency to move a body in the direction of its application, a force can also tend to rotate a body about an axis. The axis may be any line which neither intersects nor is parallel to the line of action of the force. This rotational tendency is known as moment of force (M). Moment is also referred to as torque.

Moment about a point

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Diagram of the moment arm of a force F.

The magnitude of the moment of a force at a point O, is equal to the perpendicular distance from O to the line of action of F, multiplied by the magnitude of the force: M = F · d, where

F = the force applied
d = the perpendicular distance from the axis to the line of action of the force. This perpendicular distance is called the moment arm.

The direction of the moment is given by the right hand rule, where counter clockwise (CCW) is out of the page, and clockwise (CW) is into the page. The moment direction may be accounted for by using a stated sign convention, such as a plus sign (+) for counterclockwise moments and a minus sign (−) for clockwise moments, or vice versa. Moments can be added together as vectors.

In vector format, the moment can be defined as the cross product between the radius vector, r (the vector from point O to the line of action), and the force vector, F:[6]

Varignon's theorem

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Varignon's theorem states that the moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.

Equilibrium equations

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The static equilibrium of a particle is an important concept in statics. A particle is in equilibrium only if the resultant of all forces acting on the particle is equal to zero. In a rectangular coordinate system the equilibrium equations can be represented by three scalar equations, where the sums of forces in all three directions are equal to zero. An engineering application of this concept is determining the tensions of up to three cables under load, for example the forces exerted on each cable of a hoist lifting an object or of guy wires restraining a hot air balloon to the ground.[7]

Moment of inertia

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In classical mechanics, moment of inertia, also called mass moment, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI units kg·m²) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbols I and J are usually used to refer to the moment of inertia or polar moment of inertia.

While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.

The concept was introduced by Leonhard Euler in his 1765 book Theoria motus corporum solidorum seu rigidorum; he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.

Applications

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Solids

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Statics is used in the analysis of structures, for instance in architectural and structural engineering. Strength of materials is a related field of mechanics that relies heavily on the application of static equilibrium. A key concept is the center of gravity of a body at rest: it represents an imaginary point at which all the mass of a body resides. The position of the point relative to the foundations on which a body lies determines its stability in response to external forces. If the center of gravity exists outside the foundations, then the body is unstable because there is a torque acting: any small disturbance will cause the body to fall or topple. If the center of gravity exists within the foundations, the body is stable since no net torque acts on the body. If the center of gravity coincides with the foundations, then the body is said to be metastable.

Fluids

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Hydrostatics, also known as fluid statics, is the study of fluids at rest (i.e. in static equilibrium). The characteristic of any fluid at rest is that the force exerted on any particle of the fluid is the same at all points at the same depth (or altitude) within the fluid. If the net force is greater than zero the fluid will move in the direction of the resulting force. This concept was first formulated in a slightly extended form by French mathematician and philosopher Blaise Pascal in 1647 and became known as Pascal's law. It has many important applications in hydraulics. Archimedes, Abū Rayhān al-Bīrūnī, Al-Khazini[8] and Galileo Galilei were also major figures in the development of hydrostatics.

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Statics

View on Grokipedia
from Grokipedia
Statics is the branch of classical mechanics that deals with the analysis of forces and torques on physical systems in a state of equilibrium, where there is no net acceleration or rotation. This field focuses on rigid bodies or particles that are either at rest or moving with constant velocity, ensuring that the sum of all forces and moments acting on them is zero. Key principles include translational equilibrium, where the vector sum of forces equals zero (∑F = 0), and rotational equilibrium, where the sum of torques about any point is zero (∑τ = 0). These concepts stem directly from Newton's first law of motion, which states that an object remains at rest or in uniform motion unless acted upon by a net external force. The historical foundations of statics trace back to ancient Greece, with early contributions from Archimedes in the 3rd century BCE. The field was further developed during the Scientific Revolution by Galileo Galilei and Isaac Newton, who applied and formalized principles of mechanics relevant to equilibrium. In modern engineering and physics, statics serves as a foundational tool for designing stable structures and mechanisms across disciplines such as civil, mechanical, and aerospace engineering. It is applied to calculate internal forces in beams, trusses, and frames, ensuring safety in bridges, buildings, and aircraft components. Additionally, statics principles extend to biomechanics, where they model forces on the human body during posture or lifting, and to geotechnical engineering for analyzing soil stability. The field's emphasis on vector mechanics and free-body diagrams enables precise predictions of load distribution, preventing failures in real-world applications.

Historical Development

Ancient and Classical Contributions

The ancient Egyptians and Mesopotamians made practical contributions to statics through empirical applications in construction and measurement around 2500 BCE. In Egypt, pyramid construction required an intuitive understanding of balance and weight distribution, employing ramps, levers, and counterweights to position massive stone blocks stably, as evidenced by archaeological analyses of building techniques. Mesopotamians similarly advanced balance concepts via pan scales for trade, using standardized weights to achieve equilibrium, with artifacts from sites like Nippur demonstrating precise mensuration systems that relied on static principles. These developments emphasized observational stability without formal theory, laying groundwork for later mechanical insights. Archimedes, in the 3rd century BCE, formalized key statics principles in his work On the Equilibrium of Planes. He established the law of the lever, stating that for balance on a fulcrum, the distances from the fulcrum are proportional to the inverse of the weights, enabling predictions of equilibrium in lever systems. Archimedes also introduced the concept of the center of gravity as the point where a body's weight acts, applying it to plane figures and deriving locations for uniform shapes like triangles and parabolas. His investigations extended to buoyancy, linking hydrostatic equilibrium to displaced fluid weights, though primarily through static balances. Aristotle, in the 4th century BCE, offered qualitative ideas on balance and rest in his Physics, viewing equilibrium as objects returning to their natural places—earth downward, fire upward—where motion ceases without external influence. These concepts influenced medieval thought despite inaccuracies in dynamics. In the Hellenistic period, Hero of Alexandria advanced statics in the 1st century CE through his Mechanica, detailing pulley systems for lifting weights by distributing forces across multiple ropes and wheels to achieve equilibrium. He analyzed balances, levers, and compound pulleys, showing how equal weights at equal distances maintain stability and how mechanical advantage reduces effort for heavy loads. These empirical treatments built on Archimedes, focusing on practical weight manipulation in engineering.

17th to 19th Century Advances

The foundations of modern statics began to take shape in the late 16th and 17th centuries with contributions that emphasized mathematical rigor over empirical intuition. Simon Stevin, a Flemish engineer, published influential work in 1586 on the equilibrium of forces along inclined planes and introduced the triangle of forces theorem, which demonstrated that forces in equilibrium form a closed triangle proportional to their magnitudes. This approach, building on lever principles, provided a graphical method for resolving forces and was widely adopted in the 17th century to analyze static systems like hydrostatic pressure and mechanical advantage on inclines. Stevin's innovations marked an early shift toward systematic force composition, influencing subsequent European mechanists during the scientific revolution. Galileo Galilei advanced the application of statics in his 1638 work Two New Sciences, where he analyzed the strength of materials under load, examining how beams and structures resist bending and fracture to maintain equilibrium. This integrated static principles with practical engineering concerns about stability and failure. A pivotal advancement occurred in 1687 with Isaac Newton's Philosophiæ Naturalis Principia Mathematica, which formalized the laws of motion and directly addressed static equilibrium. Newton's first law states that a body remains at rest or in uniform motion unless acted upon by an external force, establishing that equilibrium requires zero net force on a body. This principle, derived from observations of pendulums and balanced systems, unified statics with broader mechanics by treating rest as a special case of inertial motion. In that same year, French mathematician Pierre Varignon developed his theorem on the equivalence of force systems, showing that the moment of a force system about any point equals the moment of its resultant; it was published posthumously in 1725. These works by Newton and Varignon integrated statics into a mathematical framework, distinguishing it from qualitative ancient treatments. The 19th century saw the maturation of statics through vector-based methods, separating it more clearly from dynamics as a field focused on equilibrium without time-dependent motion. Jean-Victor Poncelet, a French engineer, advanced force resolution in the early 1800s by developing graphical vector techniques for static calculations, such as resolving concurrent forces in engineering structures like bridges and machines. His methods, outlined in works on projective geometry, enabled precise decomposition of force polygons without algebraic computation, influencing practical statics in civil engineering. Concurrently, August Ferdinand Möbius contributed to vector statics in 1827 with his introduction of barycentric coordinates, a coordinate-free system for expressing force equilibria and mass distributions in geometric terms. Möbius's barycentric calculus treated forces as weighted points in space, facilitating the analysis of rigid body equilibrium and establishing statics as an independent mathematical discipline by mid-century. These developments, alongside Poncelet's, emphasized statics' autonomy, prioritizing balance conditions over dynamic trajectories.

Fundamental Concepts

Forces

In statics, a force is defined as an action that tends to change the state of motion or shape of a body upon which it acts, and it is fundamentally a vector quantity characterized by three essential attributes: magnitude, direction (including sense), and point of application. The magnitude quantifies the strength of the force, typically measured along its line of action; the direction specifies the line along which the force acts, while the sense indicates the specific way it points along that line (e.g., toward or away from the body); and the point of application denotes the precise location on the body where the force is exerted, which is crucial for analyzing its effects on rigid bodies. This vector nature allows forces to be represented graphically as arrows, where the length corresponds to magnitude and the arrowhead to sense, providing a complete description only when all attributes are specified. Forces in statics are classified into two primary categories based on their origin and mode of action: contact forces and body forces. Contact forces, also known as surface forces, arise from direct physical interaction between bodies at their interface, such as friction, which opposes relative motion between surfaces, or the normal force, which acts perpendicular to the contact surface to prevent penetration. Body forces, conversely, act throughout the volume of a body without requiring physical contact, distributed uniformly or variably over its mass, with examples including gravitational forces, which pull objects toward Earth's center proportional to mass, and electromagnetic forces, which arise from charged particle interactions across space. Additionally, forces are distinguished as sliding vectors in statics, which are bound to a specific line of action and can be translated along that line without altering their effect on a rigid body (useful for equivalent systems), while their point of application along the line may matter for detailed analysis. Free vectors, translatable anywhere, apply to quantities like net force at a point where position is irrelevant, and bound vectors are fixed to a specific point. For analytical purposes in statics, forces are represented in vector form, most commonly resolved into components using Cartesian coordinates for three-dimensional problems or polar coordinates for two-dimensional cases. In Cartesian coordinates, a force F⃗\vec{F}
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