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

Wyrażenie siły i energii potencjalnej można rozszerzyć do trzech wymiarów.
Rozważmy cząstkę poruszającą się w przestrzeni trójwymiarowej z powodu siły zachowawczej, która jest funkcją wszystkich trzech współrzędnych przestrzennych.
Siła wzdłuż współrzędnej przestrzennej może być zapisana jako ujemny stosunek zmiany energii potencjalnej do przemieszczenia wzdłuż tej współrzędnej. Gdy przemieszczenie jest minimalne, współczynniki stają się pochodnymi.
Dodając siłę wzdłuż trzech współrzędnych przestrzennych, siłę wypadkową można zapisać jako sumę wektorów pochodnych cząstkowych energii potencjalnej względem każdej współrzędnej, pomnożoną przez odpowiedni wektor jednostkowy. 
Zatem siła w trzech wymiarach może być wyrażona jako ujemna wartość gradientu funkcji energii potencjalnej.
W celu sprawdzenia zależności między siłą a energią potencjalną należy wziąć pod uwagę grawitacyjną energię potencjalną. Po podstawieniu do relacji siły i energii, otrzymuje się wyrażenie dla siły grawitacji.

siła i energia potencjalna w trzech wymiarach

Rozważmy cząstkę poruszającą się pod działaniem siły zachowawczej, której składowe znajdują się wzdłuż każdej osi współrzędnych. Każda składowa siły jest funkcją współrzędnych. Funkcja energii potencjalnej U jest także funkcją wszystkich trzech współrzędnych przestrzennych. Siłę w jednym wymiarze można zapisać jako ujemny stosunek zmiany energii potencjalnej do przemieszczenia wzdłuż tej współrzędnej. W przypadku minimalnego przemieszczenia stosunki stają się pochodnymi. Jeśli funkcja ma wiele zmiennych, pochodna przyjmuje tylko tę zmienną, którą określa pochodna cząstkowa; pozostałe zmienne pozostają stałe. Zatem, dodając siłę wzdłuż trzech współrzędnych przestrzennych, siłę wypadkową można zapisać jako sumę wektorową pochodnych cząstkowych energii potencjalnej w odniesieniu do każdej współrzędnej, pomnożoną przez odpowiedni wektor jednostkowy. Innymi słowy, siłę w trzech wymiarach można wyrazić jako ujemną wartość gradientu funkcji energii potencjalnej.

Zrozumiemy to na przykładzie. Energię potencjalną siły trójwymiarowej wyraża się wzorem

<img alt="Rysunek1" src="/files/ftp_upload/12678/12678_Equation_1.png" style="text-align: center; height: 40px;" />

gdzie k jest stałą. Wyprowadź Fx,Fy, i Fz, a następnie opisz całkowitą siłę w każdym punkcie za pomocą współrzędnych. Tutaj znana wielkość to energia potencjalna. Nieznane wielkości Fx, Fy,&Fz i należy obliczyć całkowity wektor siły. Siła jest powiązana z energią potencjalną poprzez &; następujące równanie:

<img alt="Figure2" src="/files/ftp_upload/12678/12678_Equation_2.png" style="text-align: center; height: 37px;" />

Podstawienie U w powyższym równaniu daje wyrażenie dla Fx, Fy i&Fz w postaci współrzędnych x, y i z. Siła wektorowa jest sumą sił wzdłuż każdej współrzędnej. Innymi słowy,

<img alt="Figure3" src="/files/ftp_upload/12678/12678_Equation_3.png" style="text-align: center; height: 25px;" />

Zastępowanie wartości& Fx, Fy i&&Fz w powyższym równaniu otrzymujemy wektor siły wypadkowej jako

<img alt="Figure4" src="/files/ftp_upload/12678/12678_Equation_4.png" style="text-align: center; height: 37px;" />

Ten tekst jest adaptacją <a href="https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams- i-stabilność">Openstax, University Physics, tom 1, sekcja 8.4: Diagramy energii potencjalnej i stabilność</a>.

Siła i energia potencjalna w trzech wymiarach

Rozważmy cząstkę poruszającą się pod wpływem siły zachowawczej, która ma składowe wzdłuż każdej osi współrzędnych. Każda składowa siły jest funkcją ...

Education

JoVE Core

Fizyka

Unit 1

Jednostka 1

Potential Energy and Energy Conservation

Energia potencjalna i zasada zachowania energii

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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

physics

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

Next Generation Science Standards

HS-PS3-2

south-carolina-college-and-career-ready-science-standards

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.

The universe is composed of matter in different forms, and all forms of matter contain energy.&#160; The different forms of energy on Earth originate from ...

Types of Potential Energy

Potential energy is also known as energy at rest or stored energy. Common types of potential energy include the gravitational potential energy stored in ...

Types of Kinetic Energy

The amount of kinetic energy of an object depends on its mass and speed. Consider two balls of different masses rolling down an inclined plane at the same ...

Gravitational Potential Energy

Potential energy is not just a property of each object, but also a property of the interactions between objects in a chosen system. For each type of ...

Elastic Potential Energy

Elastic potential energy is the energy stored as a result of the deformation of an elastic object, such as the stretching of a spring. An object is ...

Conservation of Mechanical Energy

The mechanical energy E of a system is the sum of its potential energy U and the kinetic energy K of the objects within it. What happens to this ...

Conservative Forces

According to the law of conservation of energy, any transition between kinetic and potential energy conserves the total energy of the system. Hence, the ...

Non-conservative Forces

Non-conservative forces are dissipative forces such as friction or air resistance. These forces take energy away from a system as it progresses. Unlike ...

Conservation of Energy

The terms &#39;conserved quantity&#39;&#160;and &#39;conservation law&#39;&#160;have specific scientific meanings in physics, which differ from the ...

Conservation of Energy: Application

When solving problems using the energy conservation law, the object (system) to be studied should first be identified. Often, in applications of energy ...

Use models to qualitatively and quantitatively analyze the kinetic and potential energy in a system.

PS.Phy.6.1

Work

Work is done when energy is transferred from one object to another. In other words, work is when a force acts on something that undergoes a displacement ...

Positive, Negative, and Zero Work

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Work-energy Theorem

According to Newton&#x2019;s second law of motion, the sum of all the forces acting on a particle (net force) determines the rate of change in the ...

Work and Energy for Variable Forces

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work-energy-theorem-for-motion-along-a-curve

Work-Energy Theorem for Motion Along a Curve

The work-energy theorem can be generalized to the motion of a particle along any curved path. The simple argument here is that the curved path can be ...

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Work Done Over an Inclined Plane

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Work Done by Many Forces

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Work Done by Gravity

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Like other forces, ...

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Power

The concept of work involves force and displacement; meanwhile, the work-energy theorem relates the net work done on a body to the difference in its ...

Power Expended by a Constant Force

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Force and Potential Energy in One Dimension

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Energy Diagrams - I

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Energy Diagrams - II

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Analyze and interpret data to qualitatively and quantitatively explain the relationship among work, power, and energy.

Understand the relationship among work, energy, and power.

Gradient Force from 3D Potential Energy

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