Name: equilibrium stability through potential energy
Uploaded: 2023-09-22T11:49:38.000Z
Duration: 1 min 13 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Physics

PHYS.1

IPC.1.G

stability-of-equilibrium-configuration-problem-solving

Stability of Equilibrium Configuration: Problem Solving

Considere um disco conectado a um sistema de polia de mola com uma esfera presa ao seu aro.
O disco é girado e o diagrama de corpo livre apenas das forças que fazem o trabalho é desenhado. As deflexões nas rotações são mostradas em relação ao eixo de coordenadas.
Para determinar as posições de equilíbrio do sistema e sua estabilidade, obtém-se a soma das funções de energia potencial elástica e gravitacional.
A deflexão vertical da mola e da esfera é determinada e substituída na função de energia potencial.
No equilíbrio, a primeira derivada da função de energia potencial é zero. Resolvendo e substituindo os valores numéricos, obtém-se as posições de equilíbrio em termos de ângulo de rotação.
A estabilidade dos ângulos de equilíbrio é determinada a partir da segunda derivada da energia potencial.
Se a segunda derivada for negativa, o equilíbrio é instável. Nesse ângulo, um leve movimento coloca o sistema em rotação.
Se for positivo, o equilíbrio é estável, o que implica que, sob rotação, o sistema fica em repouso neste ângulo.

<ol>
	<li>Hibbeler, R.C. (2016). Engenharia Mecânica: Estática. Décima quarta edição, Nova Jersey: Pearson. Seção 11.7, pp. 603-606.</li>
	<li>Cerveja, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. e Self, B.P. (2016). Mecânica vetorial para engenheiros. Décima primeira edição, Nova York: McGraw-Hill Education. Pág. 601.</li>
</ol>

estabilidade da configuração de equilíbrio: resolução de problemas

A estabilidade das configurações de equilíbrio é um conceito importante na física, engenharia e outras áreas relacionadas. Em termos simples, refere-se à tendência de um objeto ou sistema de voltar à sua posição de equilíbrio após ser perturbado. A estabilidade de uma configuração de equilíbrio pode ser analisada considerando a função de energia potencial do sistema e examinando seu comportamento próximo ao ponto de equilíbrio.

A resolução de problemas no contexto da estabilidade da configuração de equilíbrio envolve várias etapas. Em primeiro lugar, é necessário identificar os pontos de equilíbrio do sistema. Isso pode ser feito configurando a derivada da função de energia potencial como zero e resolvendo os valores das variáveis independentes que satisfazem a equação.

Uma vez que os pontos de equilíbrio tenham sido determinados, o próximo passo é examinar sua estabilidade. Para fazer isso, a segunda derivada da função de energia potencial é avaliada em cada ponto de equilíbrio. Se a segunda derivada for positiva, o equilíbrio é estável, indicando que o sistema retornará à posição de equilíbrio quando perturbado. Ao contrário, se a segunda derivada for negativa, o equilíbrio é instável, o que significa que o sistema se afastará da posição de equilíbrio quando perturbado.

É importante observar que um sistema pode ter vários pontos de equilíbrio e cada um deve ser analisado separadamente para determinar sua estabilidade. Além disso, a estabilidade de uma configuração de equilíbrio pode ser afetada por vários fatores, como a presença de amortecimento ou forças externas.

Estabilidade da Configuração de Equilíbrio: Resolução de Problemas

A estabilidade das configurações de equilíbrio é um conceito importante na física, engenharia e outros campos relacionados. Em termos simples, refere-se ao ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Virtual Work

Trabalho Virtual

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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 the equation. A system may have more than one equilibrium point, so each point must be checked on its own.

The next step is to test stability at each equilibrium point. For this, the second derivative of the potential energy function is evaluated at the equilibrium point. If the second derivative is positive, the equilibrium is stable. If it is negative, the equilibrium is unstable and the system moves away after being disturbed.

Other factors can also affect stability. Damping and external forces may change how a system behaves near equilibrium. Because of this, equilibrium analysis is an important problem-solving tool in physics, engineering, and related fields.

Equilibrium Stability Through Potential Energy

equilibrium-stability-through-potential-energy

work

Work in Force and Motion

Work connects force, motion, and energy in mechanics. It happens when a force acts on an object and the object moves along the line of action of that force. Work can also be described as energy transfer caused by applying a force, which leads to a change in the object’s motion or position. Work is calculated as the force multiplied by the part of the displacement in the same direction as the force. It can also be written as the dot product of the applied force and the object’s displacement. In...

work of a couple moment

Couple Moment Work in Rotation

Mechanical engineering studies motion, energy, and force, and it uses the idea of a couple moment to describe how two equal and opposite forces act on a rigid body. A couple moment is created when those forces act at two points on the body that are separated by a distance. When a rigid body moves because of a couple, its displacement can be split into two parts. First, the two points move by the same amount to their final positions. Then the body rotates about one of those points. During the...

virtual work

Virtual Work in Force Equilibrium

Virtual work is a mechanics principle used to check equilibrium with imaginary motion. If a body is in static or dynamic equilibrium, the total virtual work from external forces and couple moments must be zero for any virtual displacement. A virtual displacement is an imagined small movement. It can be a shift or a rotation. The virtual work done by a force equals the force dotted with the virtual displacement in the force’s direction. For a couple moment, virtual rotational work is found by...

virtual work for a system of connected rigid bodies

virtual-work-for-a-system-of-connected-rigid-bodies

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,...

principle of virtual work: problem solving

principle-of-virtual-work-problem-solving

Virtual Work for Finding Unknown Forces

The principle of virtual work helps find unknown forces and moments in equilibrium problems. It is an important idea in mechanics and engineering. The method uses a small, hypothetical motion called a virtual displacement. If the system is in equilibrium, the work done by the forces during that displacement is zero. To apply this principle, the positions of the internal and external forces must be written in terms of one common coordinate parameter. That parameter can be an angle, a distance,...

mechanical efficiency of real machines

Friction’s Role in Machine Efficiency

Mechanical efficiency shows how well a machine turns input work into output work. It is calculated as the ratio of output work to input work. An ideal machine has an efficiency of one, which means the input work equals the output work. Real machines are never perfectly ideal because some energy is lost. Friction is one of the main causes of that loss. In a toggle vise used to compress wooden blocks, the sliding block rubs against the horizontal plane and creates friction. To find the...

conservative forces

Conservative Forces and Path-Independent Work

Conservative forces are forces whose work depends only on an object’s starting and ending positions. The path between those positions does not change the amount of work done. This idea is important in mechanical engineering because it helps describe how energy moves through a system. A conservative force also conserves energy in the sense that its work is path independent. Gravity is one example. Electrostatic forces and springs modeled by Hooke’s law are also examples. A ball rolling down a...

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-criterion-for-equilibrium

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...

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...

one-degree-of-freedom system

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...

stability of equilibrium configuration: problem solving

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 to find equilibrium points and test stability with potential energy, first and second derivatives, and signs of stable or unstable motion.

Equilibrium Stability Through Potential Energy | JoVE K12

O JoVE Core Mechanical Engineering oferece vídeos animados concisos e de fácil compreensão que abrangem tópicos-chave como equilíbrio, forças, análise de treliças e mecânica estrutural. Explore suas aplicações no mundo real e observe esses conceitos em ação por meio de experimentos conduzidos por cientistas.

mechanical engineering

Chemistry and Physics

IPC.1

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PHYS.1.C

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PHYS.1.D

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PHYS.1.E

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Equilibrium Stability Through Potential Energy

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