Name: gradient force from 3d potential energy
Uploaded: 2023-04-30T10:57:30.000Z
Duration: 1 min 4 s
Channel: JoVE

K12 Standards

north-carolina-k-12-science-standard-course-of-study

North Carolina K-12 Science Standard Course of Study

9-12

Physics

Energy

PS.Phy.6.2

force-and-potential-energy-in-three-dimensions

Force and Potential Energy in Three Dimensions

Выражение силы и потенциальной энергии может быть расширено до трех измерений.
Рассмотрим частицу, движущуюся в трехмерном пространстве благодаря консервативной силе, которая является функцией всех трех пространственных координат.
Сила вдоль пространственной координаты может быть записана как отрицательное отношение изменения потенциальной энергии к смещению вдоль этой координаты. Когда смещение минимально, соотношения становятся производными.
Сложив силу по трем пространственным координатам, результирующую силу можно записать как векторную сумму частных производных потенциальной энергии относительно каждой координаты, умноженную на соответствующий единичный вектор. 
Таким образом, сила в трех измерениях может быть выражена как отрицание градиента функции потенциальной энергии.
Для проверки взаимосвязи силы и потенциальной энергии рассмотрим гравитационную потенциальную энергию. При подстановке в соотношение силы и энергии получается выражение для гравитационной силы.

Сила и потенциальная энергия в трёх измерениях

Рассмотрим частицу, движущуюся под действием консервативной силы, компоненты которой направлены вдоль каждой координатной оси. Каждая компонента силы является функцией координат. Функция потенциальной энергии U также является функцией всех трёх пространственных координат. Сила в одном измерении может быть записана как отрицательное отношение изменения потенциальной энергии к перемещению вдоль соответствующей координаты. Для минимального смещения соответствующие отношения становятся производными. Если функция имеет несколько переменных, производная принимает только переменную, которую указывает частная производная; остальные переменные остаются постоянными. Таким образом, сложение силы вдоль трёх пространственных координат позволяет записать полную силу как векторную сумму частных производных потенциальной энергии по каждой координате, умноженных на соответствующий единичный вектор. Другими словами, сила в трёх измерениях может быть выражена как отрицательный градиент функции потенциальной энергии.

Давайте разберём это на примере. Потенциальная энергия трёхмерной силы задается выражением

<img alt="Figure1" src="/files/ftp_upload/12678/12678_Equation_1.png" style="text-align: center; height: 40px;" />

где k - постоянная. Производные Fx, Fy и Fz могут быть выведены, а затем описана общая сила в каждой точке в терминах её координат. Здесь известная величина - потенциальная энергия. Неизвестные величины Fx,Fy, Fz и вектор полной силы должны быть вычислены. Сила связана с потенциальной энергией следующим  уравнением:

<img alt="Figure2" src="/files/ftp_upload/12678/12678_Equation_2.png" style="text-align: center; height: 37px;" />

Подстановка U в вышеприведённое уравнение даёт выражение для Fx, Fy и Fz в терминах координат x, y и z. Векторная сила является суммой силы вдоль каждой координаты. Другими словами,

<img alt="Figure3" src="/files/ftp_upload/12678/12678_Equation_3.png" style="text-align: center; height: 25px;" />

Подстановка значений Fx, Fy и Fz в вышеприведённое уравнение даёт вектор силы как

<img alt="Figure4" src="/files/ftp_upload/12678/12678_Equation_4.png" style="text-align: center; height: 37px;" />

Этот текст адаптирован из <a href="https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams-and-stability">Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability.</a>

Рассмотрим частицу, движущуюся под действием консервативной силы, компоненты которой расположены вдоль каждой оси координат. Каждый компонент силы является функцией...

Education

JoVE Core

Физика

Unit 1

Раздел 1

Potential Energy and Energy Conservation

Потенциальная энергия и сохранение энергии

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

gradient force from 3d potential energy

Gradient force from 3D potential energy shows how a conservative force changes along the x, y, and z axes. The potential energy function U depends on all three spatial coordinates. Each force component is linked to how U changes with position along one coordinate.

In one dimension, force can be written as the negative ratio of the change in potential energy to the displacement along that axis. For very small displacements, these ratios become derivatives. When a function has several variables, a partial derivative changes only the named variable and keeps the others constant.

Adding the force components along the three spatial coordinates gives the net force as a vector sum of partial derivatives of potential energy with respect to x, y, and z. This means force in three dimensions can be written as the negative gradient of the potential energy function. The gradient points in the direction of the steepest increase in U, so the force points in the opposite direction.

The example in the transcript uses a 3D potential energy function with k as a constant. The goal is to find F x, F y, and F z, and then describe the total force at each point using its coordinates. Since U is known, the force components are the unknowns that must be calculated.

Substituting the given U into the force equation gives expressions for F x, F y, and F z in terms of x, y, and z. The total force vector is then the sum of the three coordinate forces. After substituting the component values, the net force is written as a coordinate-based vector field.

Gradient Force from 3D Potential Energy

gravitational potential energy

Gravitational Energy from Height and Weight

Gravitational potential energy depends on an object’s weight and its height above the ground. It is one part of the total potential energy of a system, which comes from the interactions between objects. In a system, the total potential energy is the sum of the potential energies for all the objects. Potential energy is usually grouped into two main categories: gravitational potential energy and elastic potential energy. Gravitational potential energy describes the energy tied to a body’s...

elastic potential energy

Spring Force and Stored Energy

Elastic potential energy is stored when an elastic object is deformed, such as when a spring is stretched. An object is elastic if it returns to its original shape and size after the force is removed. This stored energy is part of a system’s potential energy. An ideal spring also exerts an elastic force. The work done by this force can be described as a change in the spring’s elastic potential energy. For a perfectly elastic spring in one dimension, that work depends only on the spring...

conservation of mechanical energy

Mechanical Energy in Isolated Systems

Mechanical energy in an isolated system is the total of kinetic energy and potential energy. Kinetic energy is the energy of motion, and potential energy is stored energy. The transcript focuses on what happens when energy shifts within a system without outside influence. In this case, only conservative forces act on the objects in the system. Conservative forces can transfer energy between kinetic energy and potential energy. Friction and drag do not act here, and no external force from...

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work done on a system by external force

External Work, Energy, and Thermal Loss

External work on a system can change its energy in more than one way. For a single particle, work done by an external force changes kinetic energy. For a system of interacting particles, internal forces also matter, and potential energy helps describe their effect. The net work done by an external force can be written as the total change in mechanical energy. Mechanical energy includes both kinetic energy and potential energy. When the system gains or loses this energy, the work expression...

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conservative forces

Conservative Forces and Path Independence

Conservative forces are forces for which the work done depends only on the starting and ending points. In this case, the work is path independent. That means the total energy of the system, including kinetic energy and potential energy, stays conserved during the change. This idea follows the law of conservation of energy. When energy shifts between kinetic and potential forms, the total amount remains the same. Because the work is completely reversible, an object can start and stop at any two...

non-conservative forces

Friction and Energy Loss in Motion

Non-conservative forces, such as friction and air resistance, remove energy from a system as it moves. The energy is not stored as potential energy. Instead, it is lost from the system and cannot be used later for useful work. These forces are path-dependent, which means the route matters. Where an object starts and stops can change the work done. A grinding wheel is one example, because the work depends on how many rotations the wheel makes. Non-conservative forces also turn large-scale...

conservation of energy

Energy Changes in a Closed System

Energy changes in a closed system follow a conservation law in physics. A conserved quantity is not the same as the everyday use of the word "conserved." In science, a conserved quantity stays constant, or it changes by a definite amount that is transferred to another system and converted into another form of the same quantity. This means a conserved quantity can change form, but it cannot be created or destroyed in a strict physical sense. For that reason, physics does not have a law of...

conservation of energy: application

Energy Conservation Problem-Solving Steps

Energy conservation problem-solving uses a clear set of steps. First, identify the system you want to study. In many cases, that system includes more than one body at the same time. Next, identify the forces acting on the object and decide whether each force is conservative. A conservative force is one whose work can be described with potential energy. If a non-conservative force such as friction does work, then mechanical energy is not conserved, and the system must be analyzed with...

force and potential energy in one dimension

force-and-potential-energy-in-one-dimension

Finding Force from Potential Energy

Force can be found from potential energy when the energy is written as a function of position. For a conservative force, the force component in a given direction equals the negative derivative of the matching potential energy with respect to displacement. This means force and the slope of potential energy have opposite signs. Where potential energy changes quickly with displacement, the force and the work done are largest. When a force acts along the positive coordinate axis, potential energy...

force and potential energy in three dimensions

Gradient force from 3D potential energy shows how a conservative force changes along the x, y, and z axes. The potential energy function U depends on all three spatial coordinates. Each force component is linked to how U changes with position along one coordinate. In one dimension, force can be written as the negative ratio of the change in potential energy to the displacement along that axis. For very small displacements, these ratios become derivatives. When a function has several variables,...

energy diagrams - i

Potential Energy Diagrams and Stability

Potential energy diagrams help describe how a mechanical system moves. Because energy is a scalar quantity, the system’s behavior is easier to interpret from the diagram. A skater on a parabolic ramp is a clear example. The ramp height changes with horizontal position, so the potential energy changes as well. As the skater moves down from the highest point, potential energy changes into kinetic energy. Kinetic energy is the energy of motion. The net force on the skater is gravitational force,...

energy diagrams - ii

Stability in Potential Energy Diagrams

Potential energy diagrams show how stable a system is at different positions. Their shape, or topology, helps identify equilibrium points. An equilibrium point is a place where the system can stay in balance because the net force is zero. At a local minimum, the potential energy is lowest. The system tends to remain there unless a net force acts on it. The slope at the minimum is zero. The slope is positive on both sides, so the force acts as a restoring force and pushes the system back. A...

key terms and concepts

Learn how force in three dimensions comes from potential energy. Find force components with partial derivatives and write the net force as the negative gradient.

Gradient Force from 3D Potential Energy | JoVE K12

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JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Pennsylvania State Academic Standards

Physical Science

3.2.9-12.S

Louisiana Student Standards for Science

High School

HS-PS3-5

Tennessee Academic Standards

PHYS.PS3.3

PHYS.PS3.8

Virginia  Standards of Learning

HIgh School

PH.4

PH.4.b

Alabama Course of Study Science

9-12.P.2.3.b

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HS-PS3-2

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South Carolina College and Career Ready Science Standards

P-PS3-5

new-york-state-science-learning-standards

New York State Science Learning Standards

HS. Energy

HS-PS3-2.

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Gravitational Potential Energy

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Elastic Potential Energy

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Conservation of Mechanical Energy

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Conservative Forces

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Conservation of Energy

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PS.Phy.6.1

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Work-energy Theorem

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Work and Energy for Variable Forces

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Work-Energy Theorem for Motion Along a Curve

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Gradient Force from 3D Potential Energy

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