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

Rozważmy beztarciowy układ sprężyna-masa, którego położenie jest zdefiniowane przez jedną zmienną niezależną. Tutaj energia potencjalna jest sumą grawitacyjnej i sprężystej energii potencjalnej, a ujemna jej zmiana równa się pracy wykonanej w układzie.
Jeśli system jest w równowadze i ulega wirtualnemu przemieszczeniu, wykonana praca może zostać zastąpiona pracą wirtualną.
Z zasady pracy wirtualnej, praca wynosi zero dla wszystkich wirtualnych przemieszczeń, a więc zmiana energii potencjalnej również wynosi zero. W przypadku małych przemieszczeń wirtualnych energia potencjalna jest wyrażona w postaci jej pierwszej pochodnej ze współrzędną położenia.
Ponieważ przemieszczenie wirtualne nie jest równe zeru, pierwsza pochodna energii potencjalnej musi wynosić zero. Tak więc układ jest w równowadze, gdy pierwsza pochodna jego energii potencjalnej wynosi zero.
Zastosowanie tego kryterium do układu sprężyna-masa daje jego położenie równowagi.
Jeśli energia potencjalna zależy od kilku zmiennych niezależnych, to pochodna cząstkowa energii potencjalnej dla każdej współrzędnej musi wynosić zero dla równowagi.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska: Statyka. Czternasta edycja, New Jersey: Pearson. Strony 600-602.</li>
	<li>Meriam, J.L., Kraige, L.G. i Bolten, J.N. (2016). Mechanika inżynierska: statyka i dynamika. Ósma edycja, Singapur: John Wiley & Sons. Strony 415-416.</li>
</ol>

kryterium energii potencjalnej dla równowagi

Energia potencjalna lub funkcja potencjalna odgrywa istotną rolę w określaniu równowagi układu mechanicznego. Jeżeli układ jest poddany siłom grawitacyjnym i sprężystym, funkcja potencjalna układu może być wyrażona jako suma algebraiczna energii potencjalnej grawitacyjnej i sprężystej. Jeżeli układ znajduje się w równowadze i zostanie przesunięty o niewielką wartość, praca wykonana na układ równa się zmianie energii potencjalnej układu od pozycji początkowej do końcowej, z przeciwnym znakiem. Jeżeli układ wykonuje wirtualne przemieszczenie zamiast rzeczywistego, zasada pracy wirtualnej sugeruje, że zmiana energii potencjalnej układu powinna być równa zero dla wszystkich wirtualnych przemieszczeń. Oznacza to, że równowagowa konfiguracja układu mechanicznego jest taką, dla której całkowita energia potencjalna układu ma stałą wartość. Dla układu o jednym stopniu swobody, w którym energia potencjalna i jej pochodne są ciągłymi funkcjami jednej zmiennej x, opisującej konfigurację, warunek równowagi może być zapisany jako

<img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;" />

Powyższe równanie mówi, że układ mechaniczny jest w równowadze, gdy pochodna jego całkowitej energii potencjalnej wynosi zero. Dla układów o wielu stopniach swobody, cząstkowe pochodne energii potencjalnej względem każdej współrzędnej muszą być równe zero dla równowagi.

Ciało jest w równowadze tak długo, jak długo energia potencjalna jest minimalna w tej konfiguracji. Aby zagwarantować minimalną energię potencjalną, druga pochodna funkcji potencjalnej względem współrzędnych przemieszczenia musi być dodatnia. Jest to drugi warunek niezbędny dla równowagi układu.

Kryterium energii potencjalnej dla równowagi

Energia potencjalna lub funkcja potencjału odgrywa zasadniczą rolę w określaniu stabilności układu mechanicznego. Jeśli system jest poddany zarówno ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Virtual Work

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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...

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Couple Moment Work in Rotation

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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,...

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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...

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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...

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.

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Stable Equilibrium Through Potential Energy

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