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

Stellen Sie sich eine Scheibe vor, die mit einem Federrollensystem verbunden ist und an dessen Felge eine Kugel befestigt ist.
Die Scheibe wird gedreht und das Freikörperdiagramm nur der Kräfte gezeichnet, die wirken. Die Auslenkungen bei Rotationen werden in Bezug auf die Koordinatenachse angezeigt.
Um die Gleichgewichtspositionen des Systems und deren Stabilität zu bestimmen, wird die Summe der elastischen und der potentiellen Gravitationsenergiefunktionen erhalten.
Die vertikale Auslenkung der Feder und der Kugel wird ermittelt und in die potentielle Energiefunktion eingesetzt.
Im Gleichgewicht ist die erste Ableitung der potentiellen Energiefunktion Null. Das Lösen und Ersetzen der numerischen Werte ergibt die Gleichgewichtspositionen in Bezug auf den Drehwinkel.
Die Stabilität der Gleichgewichtswinkel wird aus der zweiten Ableitung der potentiellen Energie bestimmt.
Ist die zweite Ableitung negativ, ist das Gleichgewicht instabil. Bei diesem Winkel wird das System durch eine leichte Bewegung in Rotation versetzt.
Ist es positiv, ist das Gleichgewicht stabil, was bedeutet, dass das System unter Rotation in diesem Winkel zur Ruhe kommt.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik: Statik. Vierzehnte Auflage, New Jersey: Pearson. Abschnitt 11.7, S. 603-606.</li>
	<li>Bier, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. und Self, B.P. (2016). Vektormechanik für Ingenieure. Elfte Auflage, New York: McGraw-Hill Education. S. 601.</li>
</ol>

stabilität der gleichgewichtskonfiguration: problemlösung

Die Stabilität von Gleichgewichtskonfigurationen ist ein wichtiger Begriff in der Physik, Ingenieurwissenschaft und anderen verwandten Bereichen. Im einfachsten Sinne bezieht er sich auf die Tendenz eines Objekts oder Systems, nach einer Störung in seine Gleichgewichtsposition zurückzukehren. Die Stabilität einer Gleichgewichtskonfiguration kann analysiert werden, indem die potenzielle Energie-Funktion des Systems betrachtet und ihr Verhalten nahe dem Gleichgewichtspunkt untersucht wird. Das Problemlösen im Kontext der Stabilität der Gleichgewichtskonfiguration umfasst mehrere Schritte. Zunächst müssen die Gleichgewichtspunkte des Systems identifiziert werden. Dies kann erreicht werden, indem man die Ableitung der potenziellen Energie-Funktion auf null setzt und die Werte der unabhängigen Variablen löst, die die Gleichung erfüllen. Sobald die Gleichgewichtspunkte bestimmt wurden, ist der nächste Schritt, ihre Stabilität zu untersuchen. Hierfür wird die zweite Ableitung der potenziellen Energie-Funktion an jedem Gleichgewichtspunkt ausgewertet. Ist die zweite Ableitung positiv, ist das Gleichgewicht stabil und zeigt an, dass das System in die Gleichgewichtsposition zurückkehrt, wenn es gestört wird. Ist die zweite Ableitung hingegen negativ, ist das Gleichgewicht instabil, was bedeutet, dass das System sich von der Gleichgewichtsposition wegbewegen wird, wenn es gestört wird. Es ist wichtig zu beachten, dass ein System mehrere Gleichgewichtspunkte haben kann und jeder separat analysiert werden muss, um seine Stabilität zu bestimmen. Zudem kann die Stabilität einer Gleichgewichtskonfiguration von verschiedenen Faktoren beeinflusst werden, wie beispielsweise dem Vorhandensein von Dämpfung oder externen Kräften.

Stabilität der Gleichgewichtskonfiguration: Problemlösung

Die Stabilität von Gleichgewichtskonfigurationen ist ein wichtiges Konzept in der Physik, im Ingenieurwesen und in anderen verwandten Bereichen. Einfach ausgedrückt bezieht es sich auf die ...

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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...

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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

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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

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mechanical efficiency of real machines

Friction’s Role in Machine Efficiency

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conservative forces

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potential energy

Gravitational and Elastic Energy Stored

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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...

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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 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

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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Equilibrium Stability Through Potential Energy

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