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Encyclopedia > Mechanical equilibrium

A standard definition of mechanical equilibrium is:

A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero.
Στext = 0

A particle in mechanical equilibrium is neither undergoing linear nor rotational acceleration; however it could be translating or rotating at a constant velocity. In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ...

However, this definition is of little use in continuum mechanics, for which the idea of a particle is foreign. In addition, this definition gives no information as to one of the most important and interesting aspects of equilibrium states – their stability. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Look up stability in Wiktionary, the free dictionary. ...

An alternative definition of equilibrium that is more general and often more useful is

A system is in mechanical equilibrium if its position in configuration space is a point at which the gradient of the potential energy is zero.

Because of the fundamental relationship between force and energy, this definition is equivalent to the first definition. However, the definition involving energy can be readily extended to yield information about the stability of the equilibrium state. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... For other uses, see Gradient (disambiguation). ... Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ...

For example, from elementary calculus, we know that a necessary condition for a local minimum or a maximum of a differentiable function is a vanishing first derivative (that is, the first derivative is becoming zero). To determine whether a point is a minimum or maximum, one may be able to use the second derivative test. The consequences to the stability of the equilibrium state are as follows: Calculus (from Latin, counting stone) is a major area in mathematics with further widespread applications in science and engineering used to solve more complex and expansive problems which cannot be solved by algebra alone. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum... In calculus, a branch of mathematics, the second derivative test determines whether a given stationary point of a function (where its first derivative is zero) is a maximum, a minimum, or neither. ...

  • 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.
  • 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 or does not exist: The second derivative test fails, and one must typically resort to using the first derivative test. Both of the previous results are still possible, as is a third: this could be a region in which the energy does not vary, in which case the equilibrium is called neutral or indifferent or marginally stable. To lowest order, if the system is displaced a small amount, it will stay in the new state.

In 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. Without further qualification, an equilibrium is stable only if it is stable in all directions. In calculus, a branch of mathematics, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither. ... Plot of y = x3 with a saddle-point at (0,0). ...

The special case of mechanical equilibrium of a stationary object is static equilibrium. A paperweight on a desk would be in static equilibrium. 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, this is called Gomboc. A child sliding down a slide at constant speed would be in mechanical equilibrium, but not in static equilibrium. The mono-monostatic Gomboc shape A Gomboc (Hungarian: Gömböc) is an artificial three-dimensional shape with one stable and one unstable point of equilibrium, enabling it to mimic the self-righting abilities of shelled animals such as turtles and beetles. ... Combination playground structure for small children; slides, climbers (stairs in this case), playhouse A playground is an area designed for children to play freely. ...

See also

A dynamic equilibrium occurs when two reversible processes occur at the same rate. ... Engineering mechanics is a branch of the physical sciences which looks to understand the actions and reactions of bodies at rest or in motion. ... A metastable system with a weakly stable state (1), an unstable transition state (2) and a strongly stable state (3) Metastability is the ability of a non-equilibrium state to persist for some period of time. ... Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ...

Further reading

  • Marion & Thornton, Classical Dynamics of Particles and Systems. Fourth Edition, Harcourt Brace & Company (1995).

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