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

Stellen Sie sich ein reibungsfreies Feder-Masse-System vor, dessen Position durch eine unabhängige Variable definiert wird. Hier ist die potentielle Energie die Summe der potentiellen Gravitations- und elastischen Energien, und das Negativ ihrer Änderung entspricht der Arbeit, die am System geleistet wurde.
Befindet sich das System im Gleichgewicht und erfährt eine virtuelle Verschiebung, kann die geleistete Arbeit durch virtuelle Arbeit ersetzt werden.
Nach dem Prinzip der virtuellen Arbeit ist die Arbeit für alle virtuellen Verschiebungen Null, und damit ist auch die Änderung der potentiellen Energie Null. Bei kleinen virtuellen Verschiebungen wird die potentielle Energie in Form ihrer ersten Ableitung mit der Ortskoordinate ausgedrückt.
Da die virtuelle Verschiebung nicht Null ist, muss die erste Ableitung der potentiellen Energie Null sein. Ein System befindet sich also im Gleichgewicht, wenn die erste Ableitung seiner potentiellen Energie Null ist.
Wendet man dieses Kriterium auf das Feder-Masse-System an, erhält man dessen Gleichgewichtsposition.
Wenn die potentielle Energie von mehreren unabhängigen Variablen abhängt, dann muss die partielle Ableitung der potentiellen Energie für jede Koordinate für das Gleichgewicht Null sein.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik: Statik. Vierzehnte Auflage, New Jersey: Pearson. S. 600-602.</li>
	<li>Meriam, J.L., Kraige, L.G. und Bolten, J.N. (2016). Technische Mechanik: Statik und Dynamik. Achte Auflage, Singapur: John Wiley & Sons. S. 415-416.</li>
</ol>

potentialenergie-kriterium für gleichgewicht

Die Potentialenergie oder das Potentialfunktions spielt eine wichtige Rolle bei der Bestimmung der Stabilität eines mechanischen Systems. Wenn ein System sowohl schweren als auch elastischen Kräften ausgesetzt ist, kann die Potentialfunktion des Systems als algebraische Summe aus Gravititations- und elastischer potentieller Energie ausgedrückt werden. Wenn das System im Gleichgewicht ist und um einen kleinen Betrag verschoben wird, dann ist die auf das System ausgeübte Arbeit gleich dem negativen Änderung der potentiellen Energie des Systems von der Anfangs- zur Endposition. Wenn das System eine virtuelle Verschiebung anstatt einer tatsächlichen Verschiebung erfährt, dann legt die virtuelle Arbeitsrelation nahe, dass die Änderung der potentiellen Energie des Systems für alle virtuellen Verschiebungen null sein sollte. Das bedeutet, dass die Gleichgewichtskonfiguration eines mechanischen Systems eine solche ist, für die die Gesamtpotentialenergie des Systems einen stationären Wert aufweist. Für ein System mit einem Freiheitsgrad, bei dem die potentielle Energie und deren Ableitungen eine stetige Funktion einer einzigen Variablen x darstellen, die die Konfiguration beschreibt, kann die Gleichgewichtsbedingung wie folgt geschrieben werden <img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;"> Die obige Gleichung besagt, dass ein mechanisches System im Gleichgewicht ist, wenn die Ableitung seiner Gesamtpotentialenergie null ist. Bei Systemen mit mehreren Freiheitsgraden muss die partielle Ableitung der potentiellen Energie nach jeder Koordinate für das Gleichgewicht null sein. Die Stabilität der Gleichgewichtskonfiguration bleibt erhalten, solange seine Potentialenergie in dieser Konfiguration minimal ist. Um eine minimale potentielle Energie zu gewährleisten, muss die zweite Ableitung der Potentialfunktion in Bezug auf die Verschiebungskoordinaten positiv sein. Dies ist die zweite wesentliche Bedingung für das stabile Gleichgewicht eines Systems.

Potentialenergie-Kriterium für Gleichgewicht

Die potentielle Energie bzw. potentielle Funktion spielt eine wesentliche Rolle bei der Bestimmung der Stabilität eines mechanischen Systems. Ist eine Anlage sowohl ...

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KEY TERMS AND CONCEPTS

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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

work

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Finding Forces with Virtual Work in Linked Bodies

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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 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...

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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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