Editing Mechanical equilibrium

{{Short description|When the net force on a particle is zero}}
{{redirect-distinguish|Point of equilibrium|Equilibrium point (mathematics)}}

[[File:WeightNormal.svg|thumb|An object resting on a surface and the corresponding [[free body diagram]] showing the [[force]]s acting on the object. The [[normal force]] ''N'' is equal, opposite, and collinear to the [[Standard gravitational acceleration|gravitational force]] ''mg'' so the net force and [[moment (physics)|moment]] is zero. Consequently, the object is in a state of static mechanical equilibrium.]]

In [[classical mechanics]], a [[particle]] is in '''mechanical equilibrium''' if the [[net force]] on that particle is zero.<ref name="Synge"/>{{rp|39}} By extension, a [[physical system]] made up of many parts is in mechanical equilibrium if the [[net force]] on each of its individual parts is zero.<ref name="Synge">{{cite book |title=Principles of Mechanics |author1=John L Synge |author2=Byron A Griffith |name-list-style=amp |edition= 2nd |publisher=McGraw-Hill |year=1949 |url=https://archive.org/details/principlesofmech031468mbp}}</ref>{{rp|45–46}}<ref name="beer">{{cite book |author=Beer FP, Johnston ER, Mazurek DF, Cornell PJ, and Eisenberg, ER |year=2009 |title=Vector Mechanics for Engineers: Statics and Dynamics |edition=9th |publisher=McGraw-Hill |page=158 }}</ref>

In addition to defining mechanical equilibrium in terms of force, there are many alternative definitions for mechanical equilibrium which are all mathematically equivalent. 
* In terms of momentum, a system is in equilibrium if the momentum of its parts is all constant.
* In terms of velocity, the system is in equilibrium if velocity is constant. * In a rotational mechanical equilibrium the angular momentum of the object is conserved and the net [[torque]] is zero.<ref name="beer"/> 

More generally in [[conservative system]]s, equilibrium is established at a point in [[Configuration space (physics)|configuration space]] where the [[gradient]] of the [[potential energy]] with respect to the [[generalized coordinates]] is zero.

If a particle in equilibrium has zero velocity, that particle is in '''static equilibrium'''{{anchor|Static}}.<ref name="Stehle">{{cite book |title=Classical Mechanics |page=113 |author1=Herbert Charles Corben |author2=Philip Stehle |name-list-style=amp |url=https://books.google.com/books?id=1gxk4oq9trYC&dq=%22static+equilibrium%22&pg=PA113 |isbn=0-486-68063-0 |publisher=Courier Dover Publications |edition=Reprint of 1960 second |year=1994}}</ref><ref name=Rao>{{cite book |title=Engineering Mechanics |author1=Lakshmana C. Rao |author2=J. Lakshminarasimhan |author3=Raju Sethuraman |author4=Srinivasan M. Sivakumar |isbn=81-203-2189-8 |publisher=PHI Learning Pvt. Ltd. |url=https://books.google.com/books?id=F7gaa1ShPKIC&dq=%22static+equilibrium%22&pg=PA90 |page=6 |year=2004}}</ref> Since all particles in equilibrium have constant velocity, it is always possible to find an [[inertial reference frame]] in which the particle is [[Rest frame|stationary]] with respect to the frame.

== Stability ==
An important property of systems at mechanical equilibrium is their [[Stability theory|stability]].

===Potential energy stability test===
In a function which describes the system's potential energy, the system's equilibria can be determined using [[calculus]]. A system is in mechanical equilibrium at the [[critical point (mathematics)|critical point]]s of the function describing the system's [[potential energy]]. These points can be located using the fact that the [[First derivative test|derivative]] of the function is zero at these points. To determine whether or not the system is stable or unstable, the [[second derivative test]] is applied. With <math>V </math> denoting the static [[Equations of motion|equation of motion]] of a system with a single [[Degrees of freedom (mechanics)|degree of freedom]] the following calculations can be performed: 

[[File:Diagram of a ball placed in an unstable equilibrium.svg|thumb|Diagram of a ball placed in an unstable equilibrium.]]

;Second derivative < 0: The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away.

[[File:Diagram of a ball placed in a stable equilibrium.svg|thumb|Diagram of a ball placed in a stable equilibrium.]]

;Second derivative > 0: The potential energy is at a local minimum. This is a stable equilibrium. The response to a small perturbation is forces that tend to restore the equilibrium. If more than one stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states.

;Second derivative = 0: The state is neutral to the lowest order and nearly remains in equilibrium if displaced a small amount. To investigate the precise stability of the system, [[Taylor expansion|higher order derivatives]] can be examined. The state is unstable if the lowest nonzero derivative is of odd order or has a negative value, stable if the lowest nonzero derivative is both of even order and has a positive value. If all derivatives are zero then it is impossible to derive any conclusions from the derivatives alone. For example, the function <math>e^{-1/x^2}</math> (defined as 0 in x=0) has all derivatives equal to zero. At the same time, this function has a local minimum in x=0, so it is a stable equilibrium. If this function is multiplied by the [[Sign function]], all derivatives will still be zero but it will become an unstable equilibrium.

[[File:Diagram of a ball placed in a neutral equilibrium.svg|thumb|Diagram of a ball placed in a neutral equilibrium.]]

;Function is locally constant: In a truly neutral state the energy does not vary and the state of equilibrium has a finite width. This is sometimes referred to as a state that is marginally stable, or in a state of indifference, or astable equilibrium.

When considering more than one dimension, it is possible to get different results in different directions, for example stability with respect to displacements in the ''x''-direction but instability in the ''y''-direction, a case known as a [[saddle point]]. Generally an equilibrium is only referred to as stable if it is stable in all directions.

===Statically indeterminate system===
{{main|Statically indeterminate}}
Sometimes the [[Mechanical equilibrium|equilibrium]] equations{{snd}} force and moment equilibrium conditions{{snd}} are insufficient to determine the forces and [[Reaction (physics)|reactions]]. Such a situation is described as ''statically indeterminate''.

Statically indeterminate situations can often be solved by using information from outside the standard equilibrium equations.

[[File:Ship stability.svg|thumb|[[Ship stability]] illustration explaining the stable and unstable dynamics of buoyancy (B), center of buoyancy (CB), center of gravity (CG), and weight (W)]]

== Examples ==
A stationary object (or set of objects) is in "static equilibrium," which is a special case of mechanical equilibrium. A paperweight on a desk is an example of static equilibrium. Other examples include a [[rock balancing|rock balance]] sculpture, or a stack of blocks in the game of [[Jenga]], so long as the sculpture or stack of blocks is not in the state of [[Structural integrity and failure|collapsing]].

Objects in motion can also be in equilibrium. A child sliding down a [[Playground|slide]] at constant speed would be in mechanical equilibrium, but not in static equilibrium (in the reference frame of the earth or slide).

Another example of mechanical equilibrium is a person pressing a spring to a defined point. He or she can push it to an arbitrary point and hold it there, at which point the compressive load and the spring reaction are equal. In this state the system is in mechanical equilibrium. When the compressive force is removed the spring returns to its original state.

The minimal number of static equilibria of homogeneous, convex bodies (when resting under gravity on a horizontal surface) is of special interest. In the planar case, the minimal number is 4, while in three dimensions one can build an object with just one stable and one unstable balance point.<ref>{{cite web |url=https://gomboc.eu/en/mathematics/ |title= Mathematics|author=<!--Not stated--> |date=2021 |website=Gömböc |publisher= |access-date=12 November 2023 |quote=}}</ref> Such an object is called a [[gömböc]].

== See also ==
* [[Dynamic equilibrium (mechanics)]]
* [[Engineering mechanics]]
* [[Metastability]]
* [[Statically indeterminate]]
* [[Statics]]
* [[Hydrostatic equilibrium]]

==Notes and references==
{{reflist}}

==Further reading==
* Marion JB and Thornton ST. (1995) ''Classical Dynamics of Particles and Systems.'' Fourth Edition, Harcourt Brace & Company.

[[Category:Statics]]