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

K12 Standards

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Physics

PHYS.1

IPC.1.G

stability-of-equilibrium-configuration-problem-solving

Stability of Equilibrium Configuration: Problem Solving

Weźmy pod uwagę dysk połączony z systemem sprężynowego koła pasowego z kulą przymocowaną do jego obręczy.
Dysk jest obracany i rysowany jest wykres swobodnego ciała tylko tych sił, które działają. Ugięcia przy obrotach są pokazane w odniesieniu do osi współrzędnych.
W celu wyznaczenia położeń równowagowych układu i ich stabilności uzyskuje się sumę sprężystych i grawitacyjnych funkcji energii potencjalnej.
Pionowe ugięcie sprężyny i kuli jest określane i podstawiane do funkcji energii potencjalnej.
W stanie równowagi pierwsza pochodna funkcji energii potencjalnej wynosi zero. Rozwiązanie i podstawienie wartości liczbowych daje pozycje równowagi pod względem kąta obrotu.
Stateczność kątów równowagi wyznacza się na podstawie drugiej pochodnej energii potencjalnej.
Jeśli druga pochodna jest ujemna, równowaga jest niestabilna. Pod tym kątem niewielki ruch wprawia system w ruch obrotowy.
Jeśli jest dodatnia, równowaga jest stabilna, co oznacza, że pod wpływem obrotu układ zatrzymuje się pod tym kątem.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska: Statyka. Czternasta edycja, New Jersey: Pearson. Sekcja 11.7, s. 603-606.</li>
	<li>Beer, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. i Self, B.P. (2016). Mechanika wektorowa dla inżynierów. Jedenasta edycja, Nowy Jork: McGraw-Hill Education. Strony 601.</li>
</ol>

stabilność konfiguracji równowagi: rozwiązywanie problemów

Stabilność konfiguracji równowagi jest ważnym pojęciem w fizyce, inżynierii i innych powiązanych dziedzinach. W prostych słowach oznacza to tendencję obiektu lub układu do powrotu do swojego punktu równowagi po zaburzeniu. Stabilność konfiguracji równowagi można analizować, biorąc pod uwagę funkcję energii potencjalnej układu i badając jej zachowanie w pobliżu punktu równowagi.

Rozwiązywanie problemów w kontekście stabilności konfiguracji równowagi obejmuje kilka kroków. Po pierwsze, należy zidentyfikować punkty równowagi układu. Można to osiągnąć, ustawiając pochodną funkcji energii potencjalnej na zero i rozwiązując równanie dla wartości zmiennych niezależnych, które spełniają równanie.

Po ustaleniu punktów równowagi, kolejnym krokiem jest sprawdzenie ich stabilności. Aby to zrobić, ocenia się drugą pochodną funkcji energii potencjalnej w każdym punkcie równowagi. Jeśli druga pochodna jest dodatnia, równowaga jest stabilna, co oznacza, że układ powróci do pozycji równowagi po zaburzeniu. W przeciwnym przypadku, jeśli druga pochodna jest ujemna, równowaga jest niestabilna, co oznacza, że układ oddali się od pozycji równowagi po zaburzeniu.

Warto zauważyć, że układ może mieć wiele punktów równowagi, z których każdy musi być analizowany oddzielnie, aby określić jego stabilność. Ponadto, stabilność konfiguracji równowagi może być wpływana przez różne czynniki, takie jak obecność tłumienia lub sił zewnętrznych.

Stabilność konfiguracji równowagi: Rozwiązywanie problemów

Stabilność konfiguracji równowagowych jest ważnym pojęciem w fizyce, inżynierii i innych pokrewnych dziedzinach. Mówiąc prościej, odnosi się do ...

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JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

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Virtual Work

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Kluczowe terminy i pojęcia

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

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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