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

Рассмотрим систему пружина-масса без трения, положение которой определяется одной независимой переменной. Здесь потенциальная энергия является суммой гравитационной и упругой потенциальной энергий, а отрицательная величина ее изменения равна работе, проделанной над системой.
Если система находится в равновесии и претерпевает виртуальное смещение, то проделанная работа может быть заменена виртуальной работой.
С точки зрения принципа виртуальной работы, работа равна нулю для всех виртуальных перемещений, и поэтому изменение потенциальной энергии также равно нулю. Для малых виртуальных перемещений потенциальная энергия выражается в терминах ее первой производной с координатой положения.
Поскольку виртуальное смещение не равно нулю, то первая производная потенциальной энергии должна быть равна нулю. Таким образом, система находится в равновесии, когда первая производная ее потенциальной энергии равна нулю.
Применение этого критерия к системе пружина-масса дает ее равновесное положение.
Если потенциальная энергия зависит от нескольких независимых переменных, то частная производная потенциальной энергии для каждой координаты должна быть равна нулю для равновесия.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика: Статика. Четырнадцатое издание, Нью-Джерси: Пирсон. С. 600-602.</li>
	<li>Мериам, Д.Л., Крайге, Л.Г. и Болтен, Д.Н. (2016). Инженерная механика: статика и динамика. Восьмое издание, Сингапур: John Wiley & Sons. С. 415-416.</li>
</ol>

Критерий потенциальной энергии для равновесия

Потенциальная энергия или потенциальная функция играют важную роль в определении стабильности механической системы. Если система подвержена как гравитационным, так и упругим силам, потенциальная функция системы может быть выражена как алгебраическая сумма гравитационной и упругой потенциальной энергии. Если система находится в равновесии и смещается на небольшое расстояние, то работа, совершаемая над системой, равна отрицательной изменению потенциальной энергии системы от начального до конечного положения. Если система подвергается виртуальному смещению вместо фактического, то виртуальное соотношение работы подразумевает, что изменение потенциальной энергии системы должно быть нулевым для всех виртуальных смещений. Это означает, что равновесная конфигурация механической системы является конфигурацией, для которой общая потенциальная энергия системы имеет стационарное значение. Для системы с одной степенью свободы, где потенциальная энергия и ее производные являются непрерывной функцией единственной переменной x, описывающей конфигурацию, условие равновесия может быть записано следующим образом:

<img alt="Equation 1" src="/files/ftp_upload/14359/PT_14359_Equation_1.png" style="text-align: center; height: 30px;" />

Вышеприведенное уравнение гласит, что механическая система находится в равновесии, когда производная от ее общей потенциальной энергии равна нулю. Для систем с несколькими степенями свободы частная производная потенциальной энергии по каждой координате должна быть равна нулю для равновесия.

Состоятельность равновесной конфигурации сохраняется, пока ее потенциальная энергия минимальна в этой конфигурации. Чтобы обеспечить минимальную потенциальную энергию, вторая производная потенциальной функции по координатам смещения должна быть положительной. Это второе необходимое условие для стабильного равновесия системы.

Потенциальная энергия или потенциальная функция играет существенную роль в определении стабильности механической системы. Если система подвергается обоим ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Virtual Work

Виртуальная работа

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

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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

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Finding Forces with Virtual Work in Linked Bodies

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principle of virtual work: problem solving

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Virtual Work for Finding Unknown Forces

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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 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-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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