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,
:$Sigma\; au\_\{ext\}\; =\; 0.,$
A particle in mechanical equilibrium is neither undergoing linear nor rotational acceleration; however it could be translating or rotating at a constant velocity.
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.
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.
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:

★ 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.
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.

Contents

See also

Further reading

See also

★ Dynamic equilibrium

★ Engineering mechanics

★ Metastability

★ Statically indeterminate

★ Statics

★ Water

Further reading

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