Mechanical equilibrium

Encyclopedia

A standard definition of

This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next.

A standard definition of

The necessary conditions for

As applied to a rigid body, the necessary and sufficient conditions become:

A rigid body in mechanical equilibrium is undergoing neither 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 applies to conservative systems and often proves more useful is:

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

. The consequences to the stability of the equilibrium state are as follows:

In more than one dimension, it is possible to get different results in different directions, for example stability with respect to displacements in the

. 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.

An example of mechanical equilibrium is a person trying to press a spring. He or she can push it up to a point after which it reaches a state where the force trying to compress it and the resistive force from the spring are equal, so the person cannot further press it. At this state the system will be in mechanical equilibrium. When the pressing force is removed the spring attains its original state.

**static equilibrium**is:- A system of particles is in static equilibrium when all the particles of the system are at rest and the total force on each particle is permanently zero.

This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next.

A standard definition of

**mechanical equilibrium**for a particle is:- The necessary and sufficient conditions for a particle to be in mechanical equilibrium is that the net forceNet forceIn physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...

acting upon the particle is zero.

The necessary conditions for

**mechanical equilibrium**for a system of particles are:-
- (i)The vector sum of all
*external forces*is zero; - (ii) The sum of the moments of all
*external forces*about any line is zero.

- (i)The vector sum of all

As applied to a rigid body, the necessary and sufficient conditions become:

- A rigid bodyRigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

is in mechanical equilibrium when the sum of all forces on all particles of the system is zero, and also the sum of all torqueTorqueTorque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

s on all particles of the system is zero.

A rigid body in mechanical equilibrium is undergoing neither 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

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

, 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

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

.

An alternative definition of equilibrium that applies to conservative systems and often proves more useful is:

- A system is in mechanical equilibrium if its position in configuration spaceConfiguration space- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...

is a point at which the gradientGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

with respect to the generalized coordinatesGeneralized coordinatesIn the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....

of the potential energyPotential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

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

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, 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 testSecond derivative test

In calculus, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point....

. The consequences to the stability of the equilibrium state are as follows:

- Second derivativeSecond derivativeIn calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...

< 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 testFirst derivative testIn calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.-Intuitive explanation:...

. 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 pointSaddle point

In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...

. 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

Gömböc

A gömböc is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter...

. A child sliding down a slide

Playground

A playground or play area is a place with a specific design for children be able to play there. It may be indoors but is typically outdoors...

at constant speed would be in mechanical equilibrium, but not in static equilibrium.

An example of mechanical equilibrium is a person trying to press a spring. He or she can push it up to a point after which it reaches a state where the force trying to compress it and the resistive force from the spring are equal, so the person cannot further press it. At this state the system will be in mechanical equilibrium. When the pressing force is removed the spring attains its original state.

## See also

- Dynamic equilibriumDynamic equilibriumA dynamic equilibrium exists once a reversible reaction ceases to change its ratio of reactants/products, but substances move between the chemicals at an equal rate, meaning there is no net change. It is a particular example of a system in a steady state...
- Engineering mechanics
- MetastabilityMetastabilityMetastability describes the extended duration of certain equilibria acquired by complex systems when leaving their most stable state after an external action....
- Statically indeterminateStatically indeterminateIn statics, a structure is statically indeterminate when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure....
- StaticsStaticsStatics is the branch of mechanics concerned with the analysis of loads on 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 a constant velocity...
- WaterWaterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

## Further reading

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