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

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Force and Potential Energy in Three Dimensions

The force and potential energy expression can be extended to three dimensions.
Consider a particle moving in three-dimensional space due to a conservative force that is a function of all the three spatial coordinates.
Force along a spatial coordinate can be written as the negative ratio of potential energy change to the displacement along that coordinate. When the displacement is minimal, the ratios become derivatives.
Adding the force along the three spatial coordinates, the net force can be written as the vector sum of partial derivatives of potential energy with respect to each coordinate, multiplied by the corresponding unit vector. 
Thus, force in three dimensions can be expressed as the negative of the gradient of the potential energy function.
For verification of the force and potential energy relationship, consider gravitational potential energy. When substituted into the force and energy relationship, expression for gravitational force is obtained.

force and potential energy in three dimensions

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function of the coordinates. The potential energy function U is also a function of all three spatial coordinates. Force in one dimension can be written as the negative ratio of potential energy change to the displacement along that coordinate. For minimal displacement, the ratios become derivatives. If a function has many variables, the derivative only takes the variable that the partial derivative specifies; the other variables are held constant. Thus, adding the force along the three spatial coordinates, the net force can be written as the vector sum of partial derivatives of potential energy with respect to each coordinate, multiplied by the corresponding unit vector. In other words, force in three dimensions can be expressed as the negative of the gradient of the potential energy function.
Let us understand this with the help of an example. The potential energy of a 3-dimensional force is given by
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where k is a constant. Derive Fx, Fy, and Fz, and then describe the total force at each point in terms of its coordinates. Here, the known quantity is potential energy. The unknown quantities Fx, Fy, Fz, and the total force vector need to be calculated. Force is related to potential energy by the following equation:
<img alt="Euler's formula: ei&#952;=cos&#952;+isin&#952; diagram; complex analysis, polar representation, angle &#952;." src="/files/ftp_upload/12678/12678_Equation_2.png" style="text-align: center; height: 37px;" data-altupdatedat="1761830439000" data-alt="Figure2" />
Substitution of U in the above equation gives the expression for Fx, Fy, and Fz in terms of the coordinates x, y, and z. The vector force is the sum of the force along each coordinate. In other words,
<img alt="Static equilibrium beams, &#931;Fx=0, forces diagram, illustrating mechanical stability analysis." src="/files/ftp_upload/12678/12678_Equation_3.png" style="text-align: center; height: 25px;" data-altupdatedat="1761830439000" data-alt="Figure3" />
Substituting the values of Fx, Fy, and Fz in the above equation, we get the net force vector as
<img alt="Lewis structure diagram of carbon dioxide (CO&#8322;), showing molecule bonding with linear geometry." src="/files/ftp_upload/12678/12678_Equation_4.png" style="text-align: center; height: 37px;" data-altupdatedat="1761830439000" data-alt="Figure4" />
This text is adapted from <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>.

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function ...

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Unit 1

Potential Energy and Energy Conservation

KEY TERMS AND CONCEPTS

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...

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Mechanical Energy in Isolated Systems

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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...

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,...

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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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Gradient Force from 3D Potential Energy

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