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

力と位置エネルギーの表現は、3次元に拡張できます。
3 つの空間座標すべての関数である保存力によって 3 次元空間を移動する粒子について考えてみましょう。
空間座標に沿った力は、その座標に沿った変位に対する位置エネルギー変化の負の比率として記述できます。変位が最小になると、比率は微分になります。
3つの空間座標に沿って力を加えると、正味の力は、各座標に対する位置エネルギーの偏導関数のベクトル和に、対応する単位ベクトルを掛けたものとして書くことができます。 
したがって、3次元の力は、ポテンシャルエネルギー関数の勾配の負として表すことができます。
力とポテンシャルエネルギーの関係を検証するために、重力ポテンシャルエネルギーを考えてみましょう。力とエネルギーの関係に代入すると、重力の表現が得られます。

三次元の力と位置エネルギー

各座標軸に沿った成分を持つ保存力の作用下で移動する粒子を考えてみましょう。 力の各成分は座標の関数です。 ポテンシャル エネルギー関数Uは、3 つの空間座標すべての関数でもあります。 1 次元の力は、その座標に沿った変位に対する位置エネルギー変化の負の比率として記述できます。 最小の変位の場合、比率は導関数になります。 関数に多くの変数がある場合、導関数は偏導関数で指定された変数のみを受け取ります。 他の変数は一定に保たれます。 したがって、3 つの空間座標に沿って力を加算すると、正味の力は、各座標に関する位置エネルギーの偏導関数の合力に、対応する単位ベクトルを乗じたものとして書くことができます。 言い換えれば、3 次元の力は、位置エネルギー関数の勾配の負の値として表現できます。

例を使ってこれを理解しましょう。 3 次元の力の位置エネルギーは次の式で与えられます。

<img alt="Figure1" src="/files/ftp_upload/12678/12678_Equation_1.png" style="text-align: center; height: 40px;" />

k は定数です。 Fx、Fy、、および Fz を使用して、各点における合計の力を座標で記述します。 ここで、既知の量とは位置エネルギーです。 未知の量 Fx、Fy、 Fz と合計の力のベクトルを計算する必要があります。 力は位置エネルギーと関係しています。 次の方程式:

<img alt="Figure2" src="/files/ftp_upload/12678/12678_Equation_2.png" style="text-align: center; height: 37px;" />

上記の方程式に U を代入すると、Fx、Fy の式が得られます。 、Fz (座標 x、y、および <) em>z。 ベクトル力は、各座標に沿った力の合計です。 つまり、

<img alt="Figure3" src="/files/ftp_upload/12678/12678_Equation_3.png" style="text-align: center; height: 25px;" />

  の値を代入します。 Fx、Fy、Fz 上記の方程式では、正味の力ベクトルは次のように得られます。

<img alt="Figure4" src="/files/ftp_upload/12678/12678_Equation_4.png" style="text-align: center; height: 37px;" />

このテキストは <a href="https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams-and-stability"> から編集されたものです。 Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability</a>.

各座標軸に沿ってコンポーネントを持つ保存力の作用下で移動する粒子について考えてみましょう。力の各構成要素は関数です...

Education

JoVE Core

物理学

Unit 1

ユニット1

Potential Energy and Energy Conservation

ポテンシャルエネルギーとエネルギー保存の法則

KEY TERMS AND CONCEPTS

主要用語と概念

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...

/files/media/science-education/science-education-thumbs/14702.jpg

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 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

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

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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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PS.Phy.6.1

Work

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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

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Power

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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.

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Gradient Force from 3D Potential Energy

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