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

마찰이 없는 스프링-질량 시스템이 그 위치가 하나의 독립 변수로 정의되는 것을 생각해 보십시오. 여기서 위치 에너지는 중력 위치 에너지와 탄성 위치 에너지의 합이며, 그 변화의 음수는 시스템에서 수행된 작업과 같습니다.
시스템이 평형 상태에 있고 가상 변위를 겪는 경우 수행된 작업은 가상 작업으로 대체될 수 있습니다.
가상 작업의 원리에 따라 모든 가상 변위에 대해 작업은 0이므로 위치 에너지의 변화도 0입니다. 작은 가상 변위의 경우, 위치 에너지는 위치 좌표와의 1차 도함수로 표현됩니다.
가상 변위가 0이 아니기 때문에 위치 에너지의 1차 도함수는 0이어야 합니다. 따라서 시스템은 위치 에너지의 1차 도함수가 0일 때 평형 상태에 있습니다.
이 기준을 스프링-질량 시스템에 적용하면 평형 위치가 제공됩니다.
위치 에너지가 여러 독립 변수에 의존하는 경우 각 좌표에 대한 위치 에너지의 편도함수는 평형을 위해 0이어야 합니다.

<ol>
	<li>히벨러, R.C. (2016). 엔지니어링 역학: 정적. 14판, 뉴저지: 피어슨. 600-602쪽.</li>
	<li>Meriam, J.L., Kraige, L.G. 및 Bolten, J.N.(2016). Engineering Mechanics: Statics and Dynamics(정역학 및 동역학). 여덟 번째 판, 싱가포르 : John Wiley & Sons. 415-416쪽.</li>
</ol>

평형에 대한 퍼텐셜에너지 기준

퍼텐셜에너지 또는 퍼텐셜함수는 역학적 시스템의 안정성을 결정하는 데 중요한 역할을 합니다. 시스템이 중력과 탄성력을 모두 받으면 시스템의의 퍼텐셜함수는 중력 퍼텐셜에너지와 탄성 퍼텐셜에너지의 대수적 합으로 표현될 수 있습니다. 시스템이 평형상태에 있고 적은 양만큼 변위가 있다면 시스템에 한 일은 초기 위치에서 최종 위치까지 시스템의 퍼텐셜에너지 변화의 음수와 같습니다. 시스템이 실제 변위가 아닌 가상 변위를 받으면 가상 일 관계는 시스템의 퍼텐셜에너지 변화가 모든 가상 변위에 대해 0이어야 함을 시사합니다. 이것은 역학적 시스템의 평형구성이 시스템의 총 퍼텐셜에너지가 정지된 값을 갖는 것임을 의미합니다. 퍼텐셜에너지와 그 도함수가 단일 변수의 연속 함수인 한 자유도 시스템의 경우, 평형조건은 다음과 같이 나타낼 수 있습니다.
<img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;" />
위의 식은 전체 퍼텐셜에너지의 도함수가 0일 때 역학 시스템이 평형상태에 있다는 것을 나타냅니다. 여러 자유도를 가진 시스템의 경우 각 좌표에 대한 퍼텐셜에너지의 편미분은 평형상태에 대해 0이어야 합니다.
평형상태에서의 안정성은 위치에너지가 최소인 한 유지됩니다. 최소 위치에너지를 보장하기 위해서는 변위좌표에 대한 위치함수의 이계도함수가 양수이어야 합니다. 이것은 시스템의 안정한 평형을 위한 두 번째 필수조건입니다.

위치 에너지 또는 위치 함수는 기계 시스템의 안정성을 결정하는 데 필수적인 역할을 합니다. 시스템이 두 가지 모두에 노출되면 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Virtual Work

가상일

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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...

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 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-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

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Stable Equilibrium Through Potential Energy

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