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

Considérons un système ressort-masse sans frottement dont la position est définie par une variable indépendante. Ici, l’énergie potentielle est la somme des énergies potentielles gravitationnelles et élastiques, et le négatif de sa variation est égal au travail effectué sur le système.
Si le système est en équilibre et subit un déplacement virtuel, le travail effectué peut être remplacé par un travail virtuel.
D’après le principe du travail virtuel, le travail est nul pour tous les déplacements virtuels, et donc la variation de l’énergie potentielle est également nulle. Pour les petits déplacements virtuels, l’énergie potentielle est exprimée en fonction de sa dérivée première avec la coordonnée de position.
Comme le déplacement virtuel n’est pas nul, la dérivée première de l’énergie potentielle doit être nulle. Ainsi, un système est en équilibre lorsque la dérivée première de son énergie potentielle est nulle.
L’application de ce critère au système ressort-masse donne sa position d’équilibre.
Si l’énergie potentielle dépend de plusieurs variables indépendantes, alors la dérivée partielle de l’énergie potentielle pour chaque coordonnée doit être nulle à l’équilibre.

<ol>
	<li>Hibbeler, R.C. (2016). Mécanique du génie : Statique. Quatorzième édition, New Jersey : Pearson. P. 600 à 602.</li>
	<li>Meriam, J.L., Kraige, L.G. et Bolten, J.N. (2016). Mécanique de l’ingénierie : statique et dynamique. Huitième édition, Singapour : John Wiley & Sons. Pages 415 et 416.</li>
</ol>

critère d'énergie potentielle pour l'équilibre

L'énergie potentielle ou la fonction potentielle joue un rôle essentiel dans la détermination de la stabilité d'un système mécanique. Si un système est soumis à la fois à des forces gravitationnelles et élastiques, la fonction potentielle du système peut être exprimée comme la somme algébrique de l'énergie potentielle gravitationnelle et élastique. Si le système est en équilibre et est déplacé d'une petite quantité, alors le travail effectué sur le système est égal au négatif du changement d'énergie potentielle du système de la position initiale à la position finale. Si le système subit un déplacement virtuel plutôt qu'un déplacement réel, alors la relation de travail virtuel suggère que le changement d'énergie potentielle du système devrait être nul pour tous les déplacements virtuels. Cela signifie que la configuration d'équilibre d'un système mécanique est celle pour laquelle l'énergie potentielle totale du système a une valeur stationnaire. Pour un système d'un degré de liberté où l'énergie potentielle et ses dérivées sont une fonction continue d'une variable unique x, qui décrit la configuration, la condition d'équilibre peut être écrite comme suit : <img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;"> La formule ci–dessus indique qu'un système mécanique est en équilibre lorsque la dérivée de son énergie potentielle totale est nulle. Pour les systèmes avec plusieurs degrés de liberté, la dérivée partielle de l'énergie potentielle par rapport à chaque coordonnée doit être nulle pour l'équilibre. La stabilité de la configuration d'équilibre reste tant que son énergie potentielle est minimale à cette configuration. Pour garantir une énergie potentielle minimale, la deuxième dérivée de la fonction potentielle par rapport aux coordonnées de déplacement doit être positive. C'est la deuxième condition essentielle pour un équilibre stable d'un système.

Critère d'énergie potentielle pour l'équilibre

L’énergie potentielle ou la fonction potentielle joue un rôle essentiel dans la détermination de la stabilité d’un système mécanique. Si un système est soumis à la fois ...

Education

Enseignement

JoVE Core

Mechanical Engineering

Génie mécanique

Unit 1

Unité 1

Virtual Work

Travail virtuel

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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

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 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-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.

Stable Equilibrium Through Potential Energy | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

The universe is composed of matter in different forms, and all forms of matter contain energy.&#160; The different forms of energy on Earth originate from ...

Types of Potential Energy

Potential energy is also known as energy at rest or stored energy. Common types of potential energy include the gravitational potential energy stored in ...

Types of Kinetic Energy

The amount of kinetic energy of an object depends on its mass and speed. Consider two balls of different masses rolling down an inclined plane at the same ...

Kinetic Energy - I

It&#8217;s plausible to suppose that the greater the velocity of a body, the greater effect it could have on other bodies. This does not depend on the ...

Kinetic Energy - II

The kinetic energy of a particle is one-half of the product of the particle&#8217;s mass and the square of its speed. Note that just as Newton&#8217;s ...

Potential Energy due to Gravitation

Since gravitational force is a conservative force, the amount of work done to move an object between two points in the gravitational field in which it ...

Energy in Simple Harmonic Motion

To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to ...

/files/media/science-education/science-education-thumbs/12765.jpg

problem-solving-energy-in-simple-harmonic-motion

Problem Solving: Energy in Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion in time and position, in which an object oscillates back and forth around an equilibrium ...

/files/media/science-education/science-education-thumbs/14743.jpg

Kinetic and Potential Energy of a Wave

All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge ...

Energy and Power of a Wave

The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy. The average rate of energy transfer associated ...

Molecular Kinetic Energy

The word &#34;gas&#34;&#160;comes from the Flemish word meaning &#34;chaos,&#34; first used to describe vapors by the chemist J. B. van Helmont. Consider ...

Electric Potential Energy

When an electric field accelerates a free positive charge q, it is given kinetic energy. The process is analogous to an object accelerated by a ...

Energy Basics

Chemical reactions, such as those that occur when you light a match, involve changes in energy as well as matter.

Chemical changes and their accompanying ...

Internal Energy

The total of all possible kinds of energy present in a substance is called the internal energy (U), sometimes symbolized as E. Suppose a system with ...

Enthalpy

Chemists ordinarily use a property known as enthalpy (H) to describe the thermodynamics of chemical and physical processes. Enthalpy is defined as the sum ...

Energy In A Magnetic Field

If a magnetic field is sustained, there must be a current in a closed circuit or loop, implying some energy has been spent in creating the field. If this ...

using-differential-scanning-calorimetry-to-measure-changes-in-enthalpy

Using Differential Scanning Calorimetry to Measure Changes in Enthalpy

Source: Laboratory of Dr. Terry Tritt &#8212; Clemson University
Differential Scanning Calorimetry (DSC) is a method of thermodynamic analysis based on ...

Hess's Law

There are two ways to determine the amount of heat involved in a chemical change: measure it experimentally, or calculate it from other experimentally ...

Entropy

The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the ...

Work-energy Theorem

According to Newton&#x2019;s second law of motion, the sum of all the forces acting on a particle (net force) determines the rate of change in the ...

Conservation of Mechanical Energy

The mechanical energy E of a system is the sum of its potential energy U and the kinetic energy K of the objects within it. What happens to this ...

/files/media/science-education/science-education-thumbs/14702.jpg

Conservation of Energy

The terms &#39;conserved quantity&#39;&#160;and &#39;conservation law&#39;&#160;have specific scientific meanings in physics, which differ from the ...

Conservation of Energy: Application

When solving problems using the energy conservation law, the object (system) to be studied should first be identified. Often, in applications of energy ...

Energy and Work

Source: Ketron Mitchell-Wynne, PhD,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of ...

conservation-of-energy-approach-to-system-analysis

Conservation of Energy Approach to System Analysis

Source: Ricardo Mejia-Alvarez and Hussam Hikmat Jabbar, Department of Mechanical Engineering, Michigan State University, East Lansing, MI
The purpose of ...

Hooke's Law and Simple Harmonic Motion

energy-associated-with-a-charge-distribution

Energy Associated With a Charge Distribution

The work done to bring a charge through a distance r is given by the potential difference between the initial and the final position. To assemble a ...

Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known. [Clarification Statement: Emphasis is on explaining the meaning of mathematical expressions used in the model.] [Assessment Boundary: Assessment is limited to basic algebraic expressions or computations; to systems of two or three components; and to thermal energy, kinetic energy, and/or the energies in gravitational, magnetic, or electric fields.]

HS-PS3-1

force-and-potential-energy-in-one-dimension

Force and Potential Energy in One Dimension

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular ...

force-and-potential-energy-in-three-dimensions

Force and Potential Energy in Three Dimensions

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function ...

Energy Diagrams - I

The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the ...

Energy Diagrams - II

Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the ...

Sound Waves

Sound waves can be thought of as fluctuations in the pressure of a medium through which they propagate. Since the pressure also makes the medium&#39;s ...

Energy Carried By Electromagnetic Waves

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes, this energy is obvious, such as in the summer sun&#39;s ...

electric-potential-energy-in-a-uniform-electric-field

Electric Potential Energy in a Uniform Electric Field

When an electric field accelerates a free positive charge, it acquires kinetic energy. This process is analogous to an object being accelerated by a ...

Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy associated with the relative position of particles (objects). [Clarification Statement: Examples of phenomena at the macroscopic scale could include the conversion of kinetic energy to thermal energy, the energy stored due to position of an object above the earth, and the energy stored between two electrically-charged plates. Examples of models could include diagrams, drawings, descriptions, and computer simulations.]

HS-PS3-2

What is Energy?

The universe is composed of matter in different forms, and all forms of matter contain energy. &#xA0;The different forms of energy on Earth originate from ...

Chemical Reactions

A chemical reaction is a process by which the bonds in the atoms of substances are rearranged to generate new substances. Matter cannot be created or ...

Energy Transfer in Chemical Reactions

Chemical reactions require sufficient energy to cause the matter to collide with enough precision and force that old chemical bonds can be broken and new ...

Standard Entropy Change for a Reaction

Entropy is a state function, so the standard entropy change for a chemical reaction (&#x394;S&#xB0;rxn) can be calculated from the difference in standard ...

Gibbs Free Energy

One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy ...

Effects of Temperature on Free Energy

The spontaneity of a process depends upon the temperature of the system. Phase transitions, for example, will proceed spontaneously in one direction or ...

Calculating Standard Free Energy Changes

The free energy change for a reaction that occurs under the standard conditions of 1 bar pressure and at 298 K is called the standard free energy change. ...

free-energy-changes-for-nonstandard-states

Free Energy Changes for Nonstandard States

The free energy change for a process taking place with reactants and products present under nonstandard conditions (pressures other than 1 bar; ...

Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for &#x394;G represents a driving force for the ...

gibbs-free-energy-and-thermodynamic-favorability

Gibbs Free Energy and Thermodynamic Favorability

Design, build, and refine a device that works within given constraints to convert one form of energy into another form of energy. [Clarification Statement: Emphasis is on both qualitative and quantitative evaluations of devices. Examples of devices could include Rube Goldberg devices, wind turbines, solar cells, solar ovens, and generators. Examples of constraints could include use of renewable energy forms and efficiency.] [Assessment Boundary: Assessment for quantitative evaluations is limited to total output for a given input. Assessment is limited to devices constructed with materials provided to students.]

HS-PS3-3

Mechanisms of Heat Transfer I

Just as interesting as the effects of heat transfer on a system are the methods by which the heat transfer occur. Whenever there is a temperature ...

Mechanisms of Heat Transfer II

In convection, thermal energy is carried by the large-scale flow of matter. Ocean currents and large-scale atmospheric circulation, which result from the ...

conduction-convection-and-radiation-problem-solving

Conduction, Convection and Radiation: Problem Solving

There are three methods by which heat transfer can take place: conduction, convection, and radiation. Each method has unique and interesting ...

Thermodynamic Systems

A thermodynamic system is a set of objects whose thermodynamic properties are of interest. The system is considered to be embedded in its surroundings or ...

first-law-of-thermodynamics-problem-solving

First Law Of Thermodynamics: Problem-Solving

The first law of thermodynamics states that the change in internal energy of the system is equal to the net heat transfer into the system minus the net ...

The First Law of Thermodynamics

The first law of thermodynamics deals with the total amount of energy in the universe. It states that this total amount of energy is constant. In other ...

statements-of-the-second-law-of-thermodynamics

Statements of the Second Law of Thermodynamics

The second law of thermodynamics can be stated in several different ways, and all of them can be shown to imply the others. The Clausius&#8217; statement ...

Heat Engines

A heat engine is a device used to extract heat from a source and then convert it into mechanical work used for various applications. For example, a steam ...

Refrigerators and Heat Pumps

Refrigerators or heat pumps are heat engines operating in a reverse direction. For a refrigerator, the focus is on removing heat from a specific area, ...

Plan and conduct an investigation to provide evidence that the transfer of thermal energy when two components of different temperature are combined within a closed system results in a more uniform energy distribution among the components in the system (second law of thermodynamics). [Clarification Statement: Emphasis is on analyzing data from student investigations and using mathematical thinking to describe the energy changes both quantitatively and conceptually. Examples of investigations could include mixing liquids at different initial temperatures or adding objects at different temperatures to water.] [Assessment Boundary: Assessment is limited to investigations based on materials and tools provided to students.]

HS-PS3-4

Gravitational Potential Energy

Potential energy is not just a property of each object, but also a property of the interactions between objects in a chosen system. For each type of ...

Magnetic Fields

A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on ...

Develop and use a model of two objects interacting through electric or magnetic fields to illustrate the forces between objects and the changes in energy of the objects due to the interaction. [Clarification Statement: Examples of models could include drawings, diagrams, and texts, such as drawings of what happens when two charges of opposite polarity are near each other.] [Assessment Boundary: Assessment is limited to systems containing two objects.]

Stable Equilibrium Through Potential Energy

Description

Transcript

Tags

More videos by Standards