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Statics

For static analysis in economics, see Comparative statics. For the technique of static correction used in exploration geophysics, see Reflection seismology. For other uses, see Static analysis.

Statics is the branch of mechanics that is concerned with the analysis of (force and torque, or "moment") acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. The application of Newton's second law to a system gives:

F = m a . {\displaystyle {\textbf {F}}=m{\textbf {a}}\,.}

Where bold font indicates a vector that has magnitude and direction. F {\displaystyle {\textbf {F}}} is the total of the forces acting on the system, m {\displaystyle m} is the mass of the system and a {\displaystyle {\textbf {a}}} is the acceleration of the system. The summation of forces will give the direction and the magnitude of the acceleration and will be inversely proportional to the mass. The assumption of static equilibrium of a {\displaystyle {\textbf {a}}} = 0 leads to:

F = 0 . {\displaystyle {\textbf {F}}=0\,.}

The summation of forces, one of which might be unknown, allows that unknown to be found. So when in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mass moves at constant velocity. Likewise the application of the assumption of zero acceleration to the summation of moments acting on the system leads to:

M = I α = 0 . {\displaystyle {\textbf {M}}=I\alpha =0\,.}

Here, M {\displaystyle {\textbf {M}}} is the summation of all moments acting on the system, I {\displaystyle I} is the moment of inertia of the mass and α {\displaystyle \alpha } = 0 the angular acceleration of the system, which when assumed to be zero leads to:

M = 0 . {\displaystyle {\textbf {M}}=0\,.}

The summation of moments, one of which might be unknown, allows that unknown to be found. These two equations together, can be applied to solve for as many as two loads (forces and moments) acting on the system.

From Newton's first law, this implies that the net force and net torque on every part of the system is zero. The net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium. See statically indeterminate.

Contents

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

Example of a beam in static equilibrium. The sum of force and moment is zero.

A scalar is a quantity which only has a magnitude, such as mass or temperature. A vector has a magnitude and a direction. There are several notations to identify a vector, including:

  • A bold faced character V
  • An underlined character V
  • A character with an arrow over it V {\displaystyle {\overrightarrow {V}}} .

Vectors are added using the parallelogram law or the triangle law. Vectors contain components in orthogonal bases. Unit vectors i, j, and k are, by convention, along the x, y, and z axes, respectively.

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. Thus, force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.

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.

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 the moment (M) of the force. Moment is also referred to as torque.

Moment about a point

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:

M O = r × F {\displaystyle {\textbf {M}}_{O}={\textbf {r}}\times {\textbf {F}}}
r = ( x 00 . . . x 0 j x 01 . . . x 1 j . . . . . . . . . x i 0 . . . x i j ) {\displaystyle r=\left({\begin{array}{cc}x_{00}&...&x_{0j}\\x_{01}&...&x_{1j}\\...&...&...\\x_{i0}&...&x_{ij}\\\end{array}}\right)}
F = ( f 00 . . . f 0 j f 01 . . . f 1 j . . . . . . . . . f i 0 . . . f i j ) {\displaystyle F=\left({\begin{array}{cc}f_{00}&...&f_{0j}\\f_{01}&...&f_{1j}\\...&...&...\\f_{i0}&...&f_{ij}\\\end{array}}\right)}

Varignon's theorem

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.

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.

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.

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.

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 and Galileo Galilei were also major figures in the development of hydrostatics.

  1. Lindberg, David C. (1992).The Beginnings of Western Science. Chicago: The University of Chicago Press. p. 108-110.
  2. Grant, Edward (2007).A History of Natural Philosophy. New York: Cambridge University Press. p. 309-10.
  3. Holme, Audun (2010).Geometry : our cultural heritage (2nd ed.). Heidelberg: Springer. p. 188. ISBN 3-642-14440-3.
  4. Meriam, James L., and L. Glenn Kraige. Engineering Mechanics (6th ed.) Hoboken, N.J.: John Wiley & Sons, 2007; p. 23.
  5. Engineering Mechanics, p. 24
  6. Hibbeler, R. C. (2010).Engineering Mechanics: Statics, 12th Ed. New Jersey: Pearson Prentice Hall. ISBN 0-13-607790-0.
  7. Beer, Ferdinand (2004). Vector Statics For Engineers. McGraw Hill. ISBN 0-07-121830-0.
  8. Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in (Morelon & Rashed 1996, pp. 614–642) harv error: no target: CITEREFMorelonRashed1996 (help):

    "Using a whole body of mathematical methods (not only those inherited from the antique theory of ratios and infinitesimal techniques, but also the methods of the contemporary algebra and fine calculation techniques), Arabic scientists raised statics to a new, higher level. The classical results of Archimedes in the theory of the centre of gravity were generalized and applied to three-dimensional bodies, the theory of ponderable lever was founded and the 'science of gravity' was created and later further developed in medieval Europe. The phenomena of statics were studied by using the dynamic approach so that two trends - statics and dynamics - turned out to be inter-related within a single science, mechanics. The combination of the dynamic approach with Archimedean hydrostatics gave birth to a direction in science which may be called medieval hydrodynamics. [...] Numerous experimental methods were developed for determining the specific weight, which were based, in particular, on the theory of balances and weighing. The classical works of al-Biruni and al-Khazini may be considered the beginning of the application of experimental methods in medieval science."

  • Beer, F.P. & Johnston Jr, E.R. (1992). Statics and Mechanics of Materials. McGraw-Hill, Inc.
  • Beer, F.P.; Johnston Jr, E.R.; Eisenberg (2009). Vector Mechanics for Engineers: Statics, 9th Ed. McGraw Hill. ISBN 978-0-07-352923-3.
Wikimedia Commons has media related toStatics.
Look up statics in Wiktionary, the free dictionary.
Wikibooks has a book on the topic of: Statics

Statics
Statics Language Watch Edit For static analysis in economics see Comparative statics For the technique of static correction used in exploration geophysics see Reflection seismology For other uses see Static analysis Statics is the branch of mechanics that is concerned with the analysis of force and torque or moment acting on physical systems that do not experience an acceleration a 0 but rather are in static equilibrium with their environment The application of Newton s second law to a system gives F m a displaystyle textbf F m textbf a Where bold font indicates a vector that has magnitude and direction F displaystyle textbf F is the total of the forces acting on the system m displaystyle m is the mass of the system and a displaystyle textbf a is the acceleration of the system The summation of forces will give the direction and the magnitude of the acceleration and will be inversely proportional to the mass The assumption of static equilibrium of a displaystyle textbf a 0 leads to F 0 displaystyle textbf F 0 The summation of forces one of which might be unknown allows that unknown to be found So when in static equilibrium the acceleration of the system is zero and the system is either at rest or its center of mass moves at constant velocity Likewise the application of the assumption of zero acceleration to the summation of moments acting on the system leads to M I a 0 displaystyle textbf M I alpha 0 Here M displaystyle textbf M is the summation of all moments acting on the system I displaystyle I is the moment of inertia of the mass and a displaystyle alpha 0 the angular acceleration of the system which when assumed to be zero leads to M 0 displaystyle textbf M 0 The summation of moments one of which might be unknown allows that unknown to be found These two equations together can be applied to solve for as many as two loads forces and moments acting on the system From Newton s first law this implies that the net force and net torque on every part of the system is zero The net forces equaling zero is known as the first condition for equilibrium and the net torque equaling zero is known as the second condition for equilibrium See statically indeterminate Contents 1 History 2 Vectors 3 Force 4 Moment of a force 4 1 Moment about a point 4 2 Varignon s theorem 5 Equilibrium equations 6 Moment of inertia 7 Solids 8 Fluids 9 See also 10 Notes 11 References 12 External linksHistory EditArchimedes 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 Vectors Edit Example of a beam in static equilibrium The sum of force and moment is zero A scalar is a quantity which only has a magnitude such as mass or temperature A vector has a magnitude and a direction There are several notations to identify a vector including A bold faced character V An underlined character V A character with an arrow over it V displaystyle overrightarrow V Vectors are added using the parallelogram law or the triangle law Vectors contain components in orthogonal bases Unit vectors i j and k are by convention along the x y and z axes respectively Force EditForce 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 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 EditIn 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 the moment M of the force Moment is also referred to as torque Moment about a point Edit 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 M O r F displaystyle textbf M O textbf r times textbf F r x 00 x 0 j x 01 x 1 j x i 0 x i j displaystyle r left begin array cc x 00 amp amp x 0j x 01 amp amp x 1j amp amp x i0 amp amp x ij end array right F f 00 f 0 j f 01 f 1 j f i 0 f i j displaystyle F left begin array cc f 00 amp amp f 0j f 01 amp amp f 1j amp amp f i0 amp amp f ij end array right Varignon s theorem Edit 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 EditThe 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 EditIn classical mechanics moment of inertia also called mass moment rotational inertia polar moment of inertia of mass or the angular mass SI units kg m 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 Solids EditStatics 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 EditHydrostatics 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 Abu Rayhan al Biruni Al Khazini 8 and Galileo Galilei were also major figures in the development of hydrostatics See also Edit Physics portal Cremona diagram Dynamics Mechanical equilibrium Solid mechanicsNotes Edit Lindberg David C 1992 The Beginnings of Western Science Chicago The University of Chicago Press p 108 110 Grant Edward 2007 A History of Natural Philosophy New York Cambridge University Press p 309 10 Holme Audun 2010 Geometry our cultural heritage 2nd ed Heidelberg Springer p 188 ISBN 3 642 14440 3 Meriam James L and L Glenn Kraige Engineering Mechanics 6th ed Hoboken N J John Wiley amp Sons 2007 p 23 Engineering Mechanics p 24 Hibbeler R C 2010 Engineering Mechanics Statics 12th Ed New Jersey Pearson Prentice Hall ISBN 0 13 607790 0 Beer Ferdinand 2004 Vector Statics For Engineers McGraw Hill ISBN 0 07 121830 0 Mariam Rozhanskaya and I S Levinova 1996 Statics p 642 in Morelon amp Rashed 1996 pp 614 642 harv error no target CITEREFMorelonRashed1996 help Using a whole body of mathematical methods not only those inherited from the antique theory of ratios and infinitesimal techniques but also the methods of the contemporary algebra and fine calculation techniques Arabic scientists raised statics to a new higher level The classical results of Archimedes in the theory of the centre of gravity were generalized and applied to three dimensional bodies the theory of ponderable lever was founded and the science of gravity was created and later further developed in medieval Europe The phenomena of statics were studied by using the dynamic approach so that two trends statics and dynamics turned out to be inter related within a single science mechanics The combination of the dynamic approach with Archimedean hydrostatics gave birth to a direction in science which may be called medieval hydrodynamics Numerous experimental methods were developed for determining the specific weight which were based in particular on the theory of balances and weighing The classical works of al Biruni and al Khazini may be considered the beginning of the application of experimental methods in medieval science References EditBeer F P amp Johnston Jr E R 1992 Statics and Mechanics of Materials McGraw Hill Inc Beer F P Johnston Jr E R Eisenberg 2009 Vector Mechanics for Engineers Statics 9th Ed McGraw Hill ISBN 978 0 07 352923 3 External links EditWikimedia Commons has media related to Statics Look up statics in Wiktionary the free dictionary Online test of statics conceptual knowledge meant for teachers Free engineering Statics courseware with about 300 interactive exercises with hints and feedback Carnegie Mellon Open Learning Initiative Statics for RoboticsWikibooks has a book on the topic of Statics 1 Engineering statics A course at the University of Nebraska Lincoln Retrieved from https en wikipedia org w index php title Statics amp oldid 1041618302, wikipedia, wiki, book,

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