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Mechanical equilibrium
Mechanical equilibrium
Mechanical equilibrium
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An object resting on a surface and the corresponding free body diagram showing the forces acting on the object. The normal force N is equal, opposite, and collinear to the gravitational force mg so the net force and 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.[1]: 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.[1]: 45–46 [2]

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.[2]

More generally in conservative systems, equilibrium is established at a point in 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.[3][4] Since all particles in equilibrium have constant velocity, it is always possible to find an inertial reference frame in which the particle is stationary with respect to the frame.

Stability

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An important property of systems at mechanical equilibrium is their stability.

Potential energy stability test

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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 points of the function describing the system's potential energy. These points can be located using the fact that the 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 denoting the static equation of motion of a system with a single degree of freedom the following calculations can be performed:

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.
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, 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 (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.
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

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Main article: Statically indeterminate

Sometimes the equilibrium equations – force and moment equilibrium conditions – are insufficient to determine the forces and 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.

Ship stability illustration explaining the stable and unstable dynamics of buoyancy (B), center of buoyancy (CB), center of gravity (CG), and weight (W)

Examples

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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 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 collapsing.

Objects in motion can also be in equilibrium. A child sliding down a 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.[5] Such an object is called a gömböc.

See also

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Notes and references

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Mechanical equilibrium

View on Grokipedia
from Grokipedia
Mechanical equilibrium is a fundamental concept in physics describing a state in which a body experiences no net external or , resulting in zero linear and , such that the body either remains at rest or continues moving with constant velocity in a straight line. This condition arises directly from Newton's first law of motion, which states that an object will maintain its state of motion unless acted upon by an unbalanced . In mechanical equilibrium, the system satisfies two primary conditions: translational equilibrium, where the vector sum of all external forces equals zero (ΣF = 0), ensuring no change in linear momentum; and rotational equilibrium, where the sum of all external torques about any axis equals zero (Στ = 0), preventing angular acceleration. These conditions must hold simultaneously for rigid bodies, and they apply in both static equilibrium, where the body is stationary (velocity = 0), and dynamic equilibrium, where the body moves with uniform velocity without rotation. The principle underpins the analysis of structures, machines, and particle systems in , enabling engineers and physicists to predict stability and balance under various loads. For instance, in problems, free-body diagrams are used to resolve forces into components, verifying that horizontal and vertical sums are zero, while torque calculations often involve choosing a pivot point to simplify equations. Mechanical equilibrium extends beyond isolated particles to complex systems like bridges or satellites, where internal forces may adjust to maintain overall balance.

Definition and Conditions

Definition of Mechanical Equilibrium

Mechanical equilibrium refers to a state of a physical system, typically a rigid body, in which there is no net change in its linear or rotational motion, resulting from the balance of all applied forces and torques. In this condition, the velocity of the center of mass remains constant, and the angular velocity remains constant (possibly zero), ensuring the body neither accelerates translationally nor rotationally. This concept assumes basic familiarity with forces—vector quantities that can cause linear motion—and torques, which are the rotational equivalents of forces, producing twisting effects around an axis. The resultant force, defined as the vector sum of all individual forces acting on the body, must be zero for translational balance, while the resultant torque, the sum of all moments about a point, must similarly be zero to prevent rotation. A key distinction exists between static and dynamic mechanical equilibrium. Static equilibrium describes a system at rest, where the center of mass is zero and no motion occurs, such as a lying motionless on a table under balanced gravitational and normal forces. In contrast, dynamic equilibrium involves constant non-zero of the center of mass with zero and constant angular , like a sliding at uniform speed on frictionless ice; however, in most and contexts, the term emphasizes the static case due to its prevalence in structural and design applications. The foundational idea of mechanical equilibrium is rooted in Isaac Newton's first law of motion, also known as the law of inertia, which states that an object remains at rest or in uniform motion unless acted upon by a net external force. This principle was formally articulated by Newton in his 1687 work, , marking a pivotal 17th-century advancement in that shifted understanding from Aristotelian notions of natural rest to inertial persistence. Specific conditions for achieving equilibrium, such as the precise balance of forces and torques, are explored further in dedicated analyses.

Conditions for Equilibrium

A system is in mechanical equilibrium when the net external force acting on it is zero, ensuring no acceleration of its center of mass. This condition, derived from Newton's second law, requires that the vector sum of all forces ∑F⃗=0\sum \vec{F} = 0
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