Name: stable equilibrium through potential energy
Uploaded: 2023-09-22T11:49:37.000Z
Duration: 1 min 16 s
Channel: JoVE

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potential-energy-criterion-for-equilibrium

Potential-Energy Criterion for Equilibrium

Consider a frictionless spring-mass system whose position is defined by one independent variable. Here, the potential energy is the sum of the gravitational and elastic potential energies, and the negative of its change equals the work done on the system.
If the system is in equilibrium and undergoes a virtual displacement, the work done can be replaced by virtual work.
From the principle of virtual work, the work is zero for all virtual displacements, and so the change in potential energy is also zero. For small virtual displacements, the potential energy is expressed in terms of its first derivative with the position coordinate.
Since the virtual displacement is not zero, the first derivative of the potential energy must be zero. So, a system is in equilibrium when the first derivative of its potential energy is zero.
Applying this criterion to the spring-mass system gives its equilibrium position.
If the potential energy depends on several independent variables, then the partial derivative of the potential energy for each each coordinate must be zero for equilibrium.

potential-energy criterion for equilibrium

Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the final position. If the system undergoes a virtual displacement rather than an actual displacement, then the virtual work relation suggests that the change in potential energy of the system should be zero for all virtual displacements. This means that the equilibrium configuration of a mechanical system is one for which the total potential energy of the system has a stationary value. For a system of one degree of freedom where the potential energy and its derivatives are a continuous function of a single variable x, which describes the configuration, the equilibrium condition can be written as
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The above equation states that a mechanical system is in equilibrium when the derivative of its total potential energy is zero. For systems with several degrees of freedom, the partial derivative of potential energy with respect to each coordinate must be zero for equilibrium.
The stability at the equilibrium configuration remains as long as its potential energy is minimum at that configuration. To ensure minimum potential energy, the second derivative of the potential function with respect to the displacement coordinates must be positive. This is the second essential condition for the stable equilibrium of a system.

Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both ...

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stable equilibrium through potential energy

Potential energy helps determine when a mechanical system is stable at equilibrium. When a system is acted on by gravitational and elastic forces, its total potential function is the algebraic sum of gravitational potential energy and elastic potential energy.

If the system is already in equilibrium and is given a small displacement, the work done on the system equals the negative of the change in potential energy between the starting and ending positions. For a virtual displacement, which is an imagined small shift rather than an actual motion, the virtual work relation shows that the change in potential energy must be zero for every virtual displacement.

This means the equilibrium configuration is a state where the total potential energy has a stationary value. For a system with one degree of freedom, where the configuration is described by a single variable x and the potential energy is a continuous function of x, equilibrium occurs when the derivative of total potential energy with respect to x is zero. For systems with several degrees of freedom, the partial derivative of potential energy with respect to each coordinate must also be zero.

Stability depends on whether the potential energy is at a minimum at the equilibrium position. To make sure the equilibrium is stable, the second derivative of the potential function with respect to the displacement coordinates must be positive. This positive second-derivative condition is the second requirement for stable equilibrium.

Stable Equilibrium Through Potential Energy

stable-equilibrium-through-potential-energy

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Gravitational and elastic potential energy are forms of stored energy from conservative forces. A conservative force, such as gravity or an elastic force, gives a body the capacity to do work. This capacity is measured as potential energy and depends on the body’s position relative to a fixed reference point, or datum. For gravitational potential energy, the datum is the level where the energy is taken as zero. If a body is above this reference plane, its weight has positive gravitational...

Potential energy helps determine when a mechanical system is stable at equilibrium. When a system is acted on by gravitational and elastic forces, its total potential function is the algebraic sum of gravitational potential energy and elastic potential energy. If the system is already in equilibrium and is given a small displacement, the work done on the system equals the negative of the change in potential energy between the starting and ending positions. For a virtual displacement, which is...

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stability of equilibrium configuration: problem solving

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Equilibrium Stability Through Potential Energy

Stability of equilibrium in physics and engineering can be studied by using a system’s potential energy. An equilibrium configuration is a state where the system can remain at rest. Stability describes whether the system tends to return to that state after a small disturbance. Problem solving begins by finding the equilibrium points of the system. This is done by setting the derivative of the potential energy function equal to zero and solving for the independent variable values that satisfy...

key terms and concepts

Learn how gravitational and elastic potential energy determine equilibrium, why the first derivative is zero, and how a positive second derivative shows stability.

Stable Equilibrium Through Potential Energy | JoVE K12

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Stable Equilibrium Through Potential Energy

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