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

림에 볼이 부착된 스프링 풀리 시스템에 연결된 디스크를 생각해 보십시오.
디스크가 회전하고 작동하는 힘만 나타내는 자유물체 다이어그램이 그려집니다. 회전의 처짐은 좌표 축을 기준으로 표시됩니다.
시스템의 평형 위치와 안정성을 결정하기 위해 탄성 및 중력 위치 에너지 함수의 합을 구합니다.
스프링과 볼의 수직 처짐이 결정되고 위치 에너지 함수로 대체됩니다.
평형에서 위치 에너지 함수의 1차 도함수는 0입니다. 숫자 값을 풀고 대체하면 회전 각도 측면에서 평형 위치가 제공됩니다.
평형 각도의 안정성은 위치 에너지의 2차 도함수에서 결정됩니다.
2차 도함수가 음수이면 평형이 불안정합니다. 이 각도에서 약간의 움직임은 시스템을 회전으로 설정합니다.
양수이면 평형이 안정적이며, 이는 회전 하에서 시스템이 이 각도에서 정지한다는 것을 의미합니다.

<ol>
	<li>히벨러, R.C. (2016). 엔지니어링 역학: 정적. 14판, 뉴저지: 피어슨. 섹션 11.7, pp. 603-606.</li>
	<li>Beer, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. 및 Self, B.P. (2016). 엔지니어를 위한 벡터 역학. 제 11 판, 뉴욕 : McGraw-Hill Education. 601쪽.</li>
</ol>

평형구성의 안정성: 문제해결

평형 구성의 안정성은 물리학, 공학 및 기타 관련 분야에서 중요한 개념입니다. 간단한 용어로 물체나 시스템이 교란된 후 평형 위치로 되돌아오는 경향을 말합니다. 평형 구성의 안정성은 시스템의 퍼텐셜 에너지 함수를 고려하고 평형점 근처에서의 동작을 조사하여 분석할 수 있습니다.
평형 구성의 안정성의 맥락에서 문제 해결은 여러 단계를 포함합니다. 먼저 시스템의 평형점을 식별해야 합니다. 이는 퍼텐셜 에너지 함수의 도함수를 0으로 설정하고 방정식을 만족하는 독립 변수 값을 해결함으로써 달성할 수 있습니다.
평형점이 결정되면 다음 단계는 평형점의 안정성을 조사하는 것입니다. 이를 위해 각 평형점에서 퍼텐셜에너지함수의 이계도함수를 평가합니다. 이계도함수가 양수이면 평형이 안정적인데, 이는 시스템이 교란될 때 평형위치로 되돌아간다는 것을 의미합니다. 반대로 이계도함수가 음수이면 평형이 불안정하므로 교란될 때 시스템이 평형위치에서 멀어짐을 의미합니다.
한 시스템은 여러 개의 평형점을 가질 수 있으며, 각 시스템의 안정성을 결정하기 위해 개별적으로 분석해야 합니다. 또한 평형 구성의 안정성은 감쇠 또는 외부 힘의 존재와 같은 다양한 요인에 의해 영향을 받을 수 있습니다.

평형 구성의 안정성은 물리학, 공학 및 기타 관련 분야에서 중요한 개념입니다. 간단히 말해서, 그것은 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Virtual Work

가상일

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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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

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

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conservative forces

Conservative Forces and Path-Independent Work

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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

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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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IPC.1

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Equilibrium Stability Through Potential Energy

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