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

K12 Standards

Next Generation Science Standards

High School

Physical Science

Energy

HS-PS3-5

potential-energy-criterion-for-equilibrium

Potential-Energy Criterion for Equilibrium

Considere un sistema de masa de resorte sin fricción cuya posición está definida por una variable independiente. Aquí, la energía potencial es la suma de las energías potenciales gravitacional y elástica, y el negativo de su cambio es igual al trabajo realizado en el sistema.
Si el sistema está en equilibrio y experimenta un desplazamiento virtual, el trabajo realizado puede ser reemplazado por el trabajo virtual.
Desde el principio del trabajo virtual, el trabajo es cero para todos los desplazamientos virtuales, por lo que el cambio en la energía potencial también es cero. Para desplazamientos virtuales pequeños, la energía potencial se expresa en términos de su primera derivada con la coordenada de posición.
Dado que el desplazamiento virtual no es cero, la primera derivada de la energía potencial debe ser cero. Por lo tanto, un sistema está en equilibrio cuando la primera derivada de su energía potencial es cero.
La aplicación de este criterio al sistema resorte-masa da su posición de equilibrio.
Si la energía potencial depende de varias variables independientes, entonces la derivada parcial de la energía potencial para cada coordenada debe ser cero para el equilibrio.

<ol>
	<li>Hibbeler, R.C. (2016). Ingeniería Mecánica: Estática. Decimocuarta edición, Nueva Jersey: Pearson. Págs. 600-602.</li>
	<li>Meriam, J.L., Kraige, L.G. y Bolten, J.N. (2016). Ingeniería Mecánica: Estática y Dinámica. Octava edición, Singapur: John Wiley & Sons. Págs. 415-416.</li>
</ol>

criterio de energía potencial para el equilibrio

La energía potencial o función potencial juega un papel esencial en determinar la estabilidad de un sistema mecánico. Si un sistema está sujeto a fuerzas gravitacionales y elásticas, la función potencial del sistema puede expresarse como la suma algebraica de la energía potencial gravitacional y elástica. Si el sistema está en equilibrio y se desplaza una pequeña cantidad, el trabajo realizado sobre el sistema es igual al negativo del cambio en la energía potencial del sistema desde la posición inicial hasta la final. Si el sistema sufre un desplazamiento virtual en lugar de un desplazamiento real, entonces la relación del trabajo virtual sugiere que el cambio en la energía potencial del sistema debe ser cero para todos los desplazamientos virtuales. Esto significa que la configuración de equilibrio de un sistema mecánico es aquella para la cual la energía potencial total del sistema tiene un valor estacionario. Para un sistema de un grado de libertad donde la energía potencial y sus derivadas son una función continua de una sola variable x que describe la configuración, la condición de equilibrio puede escribirse como <img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;"> La ecuación anterior establece que un sistema mecánico está en equilibrio cuando la derivada de su energía potencial total es cero. Para sistemas con varios grados de libertad, la derivada parcial de la energía potencial con respecto a cada coordenada debe ser cero para el equilibrio. La estabilidad en la configuración de equilibrio se mantiene siempre y cuando su energía potencial sea mínima en esa configuración. Para asegurar una energía potencial mínima, la segunda derivada de la función potencial con respecto a las coordenadas de desplazamiento debe ser positiva. Esta es la segunda condición esencial para el equilibrio estable de un sistema.

Criterio de energía potencial para el equilibrio

La energía potencial o función potencial juega un papel esencial en la determinación de la estabilidad de un sistema mecánico. Si un sistema está sujeto a ambos ...

Education

Educación

JoVE Core

Mechanical Engineering

Ingeniería Mecánica

Unit 1

Unidad 1

Virtual Work

Trabajo virtual

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

work

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Finding Forces with Virtual Work in Linked Bodies

Virtual work helps find forces in a system of connected rigid bodies. It is a useful method when several links move together and the system is in equilibrium. In that case, the total virtual work is zero. To analyze the system, start with a free-body diagram. Use one independent coordinate to describe the link configuration. Then mark the deflected position that comes from a positive virtual displacement. This makes the small change in position clear before any calculations begin. Next,...

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potential energy

Gravitational and Elastic Energy Stored

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 criterion for equilibrium

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...

stability of equilibrium configuration

Types of Equilibrium in Mechanical Systems

Types of equilibrium in mechanical systems are stable, neutral, and unstable. These three cases help explain how a structure or object responds when it is pushed slightly away from balance. They are important for predicting whether a system will return to its original position, stay where it is, or move farther away. Stable equilibrium occurs when a system returns toward its original position after a small displacement. In this case, the potential energy is at a minimum. A cantilever beam...

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Spring-Mass Motion and Energy Stability

A one-degree-of-freedom system uses a single independent variable to describe its state and behavior. In mechanical engineering, this kind of model helps explain how parts in a larger system respond to force. It also gives engineers a way to predict motion before a design is built in real life. A simple harmonic oscillator is one example of a one-degree-of-freedom system. A mass attached to a spring moves along one axis, and that position determines the system’s state. The spring force makes...

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stability-of-equilibrium-configuration-problem-solving

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.

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Stable Equilibrium Through Potential Energy

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