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 ${\displaystyle {\textbf {F}}}$ is the total of the forces acting on the system, ${\displaystyle m}$ is the mass of the system and ${\displaystyle {\textbf {a}}}$ is the acceleration of the system,

states that ${\displaystyle {\textbf {F}}=m{\textbf {a}}\,}$ (the bold font indicates a ). If ${\displaystyle {\textbf {a}}=0}$, then ${\displaystyle {\textbf {F}}=0}$. 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 ${\displaystyle {\textbf {M}}=I\alpha =0}$, where ${\displaystyle {\textbf {M}}}$ is the summation of all moments acting on the system, ${\displaystyle I}$ is the moment of inertia of the mass and ${\displaystyle \alpha }$ is the angular acceleration of the system. For a system where ${\displaystyle \alpha =0}$, it is also true that ${\displaystyle {\textbf {M}}=0.}$

Together, the equations ${\displaystyle {\textbf {F}}=m{\textbf {a}}=0}$ (the 'first condition for equilibrium') and ${\displaystyle {\textbf {M}}=I\alpha =0}$ (the 'second condition for equilibrium') can be used to solve for unknown quantities acting on the system.

History

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

Force

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.^[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

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

(M). Moment is also referred to as torque.

Moment about a point

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]

${\displaystyle {\textbf {M}}_{O}={\textbf {r}}\times {\textbf {F}}}$

${\displaystyle r=\left({\begin{array}{cc}x_{00}&...&x_{0j}\\x_{01}&...&x_{1j}\\...&...&...\\x_{i0}&...&x_{ij}\\\end{array}}\right)}$

${\displaystyle F=\left({\begin{array}{cc}f_{00}&...&f_{0j}\\f_{01}&...&f_{1j}\\...&...&...\\f_{i0}&...&f_{ij}\\\end{array}}\right)}$

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.

Equilibrium equations

The

Moment of inertia

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

Solids

Statics is used in the analysis of structures, for instance in

.

Fluids

were also major figures in the development of hydrostatics.

See also

• Cremona diagram
• Dynamics
• Solid mechanics

Notes

1. ISBN 9780226482316
   .
2. ^ Grant, Edward (2007). A History of Natural Philosophy. New York: Cambridge University Press. p. 309-10.
3. ISBN 978-3-642-14440-0
   .
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. ISBN 978-0-13-607790-9
   .
7. ISBN 0-07-121830-0
   .
8. medieval science
   ."

References

• 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
  .
• Morelon, Régis; Rashed, Roshdi, eds. (1996),
  ISBN 978-0415124102

External links

Wikimedia Commons has media related to Particle equilibrium.

Wikibooks has a book on the topic of: Statics