Name: gradient force from 3d potential energy
Uploaded: 2023-04-30T10:57:30.000Z
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K12 Standards

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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

L'espressione della forza e dell'energia potenziale può essere estesa a tre dimensioni.
Si consideri una particella che si muove nello spazio tridimensionale a causa di una forza conservativa che è una funzione di tutte e tre le coordinate spaziali.
La forza lungo una coordinata spaziale può essere scritta come il rapporto negativo tra la variazione di energia potenziale e lo spostamento lungo quella coordinata. Quando lo spostamento è minimo, i rapporti diventano derivati.
Sommando la forza lungo le tre coordinate spaziali, la forza netta può essere scritta come la somma vettoriale delle derivate parziali dell'energia potenziale rispetto a ciascuna coordinata, moltiplicata per il corrispondente vettore unitario. 
Quindi, la forza in tre dimensioni può essere espressa come il negativo del gradiente della funzione di energia potenziale.
Per la verifica della relazione tra forza ed energia potenziale, si consideri l'energia potenziale gravitazionale. Quando viene sostituita nella relazione tra forza ed energia, si ottiene l'espressione della forza gravitazionale.

forza ed energia potenziale in tre dimensioni

Consideriamo una particella che si muove grazie all'azione di una forza conservativa avente componenti lungo ogni asse di coordinate. Ogni componente della forza è una funzione delle coordinate. La funzione dell'energia potenziale U è anche la funzione di tutte e tre le coordinate spaziali. La forza in una dimensione può essere scritta come il rapporto negativo del cambiamento dell'energia potenziale rispetto allo spostamento lungo le coordinate. Per uno spostamento minimo, il rapporti diventano derivati. Se una funzione ha molte variabili, la derivata considera solo la variabile specificata dalla derivata parziale; le altre variabili vengono mantenute costanti. Pertanto, aggiungendo le forze lungo le tre coordinate spaziali, la forza netta può essere scritta come la somma vettoriale delle derivate spaziali dell'energia potenziale rispetto ad ogni coordinata, moltiplicata per il corrispondente vettore unitario. In altre parole, la forza in tre dimensioni può essere espressa come il negativo del gradiente della funzione dell'energia potenziale.

Cerchiamo di capirlo con l'aiuto di un esempio. L'energia potenziale di una forza tridimensionale è data da:

<img alt="Figure1" src="/files/ftp_upload/12678/12678_Equation_1.png" style="text-align: center; height: 40px;" />

Dove k è una costante. Ricavata da Fx, Fy e Fz, e successivamente descrivere la forza totale in ogni punto delle sue coordinate. Qui, la quantità conosciuta è l'energia potenziale. Le quantità sconosciute Fx, Fy e Fz, e il vettore della forza totale devono essere calcolati. La forza è collegata all'energia potenziale dalla seguente equazione:

<img alt="Figure2" src="/files/ftp_upload/12678/12678_Equation_2.png" style="text-align: center; height: 37px;" />

Sostituendo la U nell'equazione precedente ci da l'espressione per Fx, Fy e Fz in termini di coordinate x, y e z. La forza del vettore è la somma della forza lungo ogni coordinata. In altre parole:

<img alt="Figure3" src="/files/ftp_upload/12678/12678_Equation_3.png" style="text-align: center; height: 25px;" />

Sostituendo i valori di di Fx, Fy e Fz nell'equazione precedente, otteniamo il vettore della forza netta:

<img alt="Figure4" src="/files/ftp_upload/12678/12678_Equation_4.png" style="text-align: center; height: 37px;" />

Questo testo è adattato da <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>

Forza ed energia potenziale in tre dimensioni

Si consideri una particella che si muove sotto l'azione di una forza conservativa che ha componenti lungo ogni asse di coordinate. Ogni componente della forza è una funzione ...

Education

Didattica

JoVE Core

Fisica

Unit 1

Unità 1

Potential Energy and Energy Conservation

Energia Potenziale e Conservazione dell'Energia

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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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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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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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HS-PS3-2.

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Gradient Force from 3D Potential Energy

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