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

La expresión de la fuerza y la energía potencial se puede extender a tres dimensiones.
Considere una partícula que se mueve en un espacio tridimensional debido a una fuerza conservadora que es una función de las tres coordenadas espaciales.
La fuerza a lo largo de una coordenada espacial se puede escribir como la relación negativa entre el cambio de energía potencial y el desplazamiento a lo largo de esa coordenada. Cuando el desplazamiento es mínimo, las proporciones se convierten en derivadas.
Sumando la fuerza a lo largo de las tres coordenadas espaciales, la fuerza neta se puede escribir como la suma vectorial de derivadas parciales de energía potencial con respecto a cada coordenada, multiplicada por el vector unitario correspondiente. 
Por lo tanto, la fuerza en tres dimensiones se puede expresar como el negativo del gradiente de la función de energía potencial.
Para verificar la relación entre la fuerza y la energía potencial, considere la energía potencial gravitatoria. Cuando se sustituye en la relación de fuerza y energía, se obtiene la expresión de la fuerza gravitatoria.

fuerza y energía potencial en tres dimensiones

Considere una partícula que se mueve bajo la acción de una fuerza conservativa que tiene componentes a lo largo de cada eje de coordenadas. Cada componente de la fuerza es una función de las coordenadas. La función de energía potencial U también es una función de las tres coordenadas espaciales. La fuerza en una dimensión se puede escribir como la razón negativa del cambio de energía potencial a la desplazamiento a lo largo de esa coordenada. Para un desplazamiento mínimo, las ratios se convierten en derivadas. Si una función tiene muchas variables, la derivada solo toma la variable que especifica la derivada parcial; las otras variables se mantienen constantes. Así, al sumar la fuerza a lo largo de las tres coordenadas espaciales, la fuerza neta se puede escribir como la suma vectorial de las derivadas parciales de la energía potencial con respecto a cada coordenada, multiplicada por el correspondiente vector unitario. En otras palabras, la fuerza en tres dimensiones se puede expresar como el negativo del gradiente de la función de energía potencial. 

 Entendamos esto con la ayuda de un ejemplo. La energía potencial de una fuerza tridimensional se da por <img alt="Figure1" src="/files/ftp_upload/12678/12678_Equation_1.png" style="text-align: center; height: 40px;"> 

 Donde k es una constante. Derive Fx, Fy, y Fz, y luego describa la fuerza total en cada punto en términos de sus coordenadas. Aquí, la magnitud conocida es la energía potencial. Las magnitudes desconocidas Fx, Fy y Fz, y el vector de fuerza total deben ser calculados. La fuerza está relacionada con la energía potencial por la siguiente ecuación: 
 <img alt="Figure2" src="/files/ftp_upload/12678/12678_Equation_2.png" style="text-align: center; height: 37px;">

 La sustitución de U en la ecuación anterior da la expresión para Fx, Fy, y Fz en términos de las coordenadas x, y y z. La fuerza vectorial es la suma de la fuerza a lo largo de cada coordenada. En otras palabras, 
 <img alt="Figure3" src="/files/ftp_upload/12678/12678_Equation_3.png" style="text-align: center; height: 25px;"> 

 Sustituyendo los valores de Fx, Fy y Fz en la ecuación anterior, obtenemos el vector de fuerza neta como
 <img alt="Figure4" src="/files/ftp_upload/12678/12678_Equation_4.png" style="text-align: center; height: 37px;"> 

 Este texto es adaptado de <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>.

Fuerza y energía potencial en tres dimensiones

Considere una partícula que se mueve bajo la acción de una fuerza conservadora que tiene componentes a lo largo de cada eje de coordenadas. Cada componente de la fuerza es una función ...

Education

Educación

JoVE Core

Física

Unit 1

Unidad 1

Potential Energy and Energy Conservation

Energía Potencial y Conservación de la Energía

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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 Física presenta la teoría y la aplicación de conceptos físicos fundamentales, como la cinemática, dinámica, mecánica de fluidos, estática, ondas y acústica, mediante animaciones audiovisuales concisas y fáciles de entender. Además, los vídeos de científicos en acción muestran la aplicación de los conceptos relacionados en experimentos de investigación clásicos y originales, realizados en laboratorios de todo el mundo.

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.

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Types of Kinetic Energy

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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

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Non-conservative Forces

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Conservation of Energy

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Conservation of Energy: Application

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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

Work is done on an object when energy is transferred to the object. In other words, work is done when a force acts on a body that undergoes a displacement ...

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

When an object is acted upon by a variable force, the amount of work done and the change in energy of the object can be more complex to calculate compared ...

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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

The total work done on an object acted upon by multiple forces can be computed using two methods that give the same result. In one method, the work done ...

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Work Done by Gravity

Gravitation is one of the four fundamental forces in nature. The force between objects on Earth and Earth itself is called gravity.
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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Understand the relationship among work, energy, and power.

Gradient Force from 3D Potential Energy

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