Name: finding force from potential energy
Uploaded: 2023-04-30T10:57:22.000Z
Duration: 1 min 13 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-one-dimension

Force and Potential Energy in One Dimension

保守的な力は、位置エネルギーで表すことができます。
位置の関数である保存力のために直線に沿って移動する物体を考えてみましょう。
さて、保存力によって行われる仕事は、その位置エネルギーの変化のマイナスに等しくなります。
行われる仕事は、力と力によって引き起こされる変位の積にも等しくなります。
したがって、身体にかかる力は、位置エネルギーの変化を変位で割った負の値に等しくなります。 
負の符号は、保存力がシステムを低い位置エネルギーに引きずり込むことを示します。
非常に小さな変位の場合、1次元での力と位置エネルギーの表現を確立できます。
力とポテンシャルエネルギーの関係を検証するために、弾性ポテンシャルエネルギーの例を考えてみましょう。力とエネルギーの関係に置き換えられると、フックの法則の公式が確立されます。

一次元の力と位置エネルギー

力は、位置の関数である位置エネルギーの式から計算できます。 特定の方向における保存力の成分は、その方向の変位に関する対応する位置エネルギーの導関数の負の値に等しくなります。 位置エネルギーが変位とともに急速に変化する領域では、行われる仕事と力は最大になります。 また、正の座標軸に沿って力が加えられると、変位の増加とともに位置エネルギーが減少します。 したがって、力と変位に関する位置エネルギーの導関数は反対の符号を持ちます。

たとえば、x 軸に沿って 1 次元運動する粒子の位置エネルギーが次の場合、

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_1.png" style="height: 32px;" />

ここで、c = 8 N/m3 の場合、与えられた位置エネルギーに対する力はいくらになりますか? ここで、既知の量とは位置エネルギーです。 未知の量の力を計算する必要があります。

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_2.png" style="height: 32px;" />

上の式に位置エネルギーの値を組み込むと、x 方向に沿った力は −cx3。 c の値を代入すると、力は −8x3 と計算されます。

二原子分子内の 2 つの原子間の力の位置エネルギー関数は、近似的に次のように与えられます。

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_3.png" style="height: 32px;" />

a と b は定数、 x は原子間の距離です。 分子の解離エネルギーが

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_4.png" style="height: 14px;" />

D の値は何ですか? ここで、既知の量は位置エネルギーです。 未知の量は分子の解離エネルギーです。 ポテンシャルエネルギー関数を代入すると、力の式が得られます。 平衡状態では正味の力はゼロなので、次のようになります。

<img alt="Equation1" src="/files/ftp_upload/12677/12677_Equation_5.png" style="height: 32px;" />

ここで、x6 の値が位置エネルギーの式に置き換えられます。 無限大での位置エネルギーはゼロであるため、分子の解離エネルギーは b2/4a と計算されます。

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>.

力は、位置の関数である位置エネルギーの式から計算できます。保守的な力の構成要素は、特定の...

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

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 decreases as displacement increases. In one-dimensional motion, this relationship is often used to move from a known potential energy function to the force it produces.

For example, if a particle moves along the x-axis and its potential energy is given with c = 8 N/m^3, the force can be calculated from that expression. The potential energy is the known quantity, and the force is the unknown quantity. Differentiating the energy function gives the x-direction force as -cx^3, which becomes -8x^3 after substituting the value of c.

The same idea also applies to the force between two atoms in a diatomic molecule. Their potential energy can be written approximately as a function of distance x, with a and b as constants. If the dissociation energy is needed, the force is first found from the potential energy, and the equilibrium point is used because the net force is zero there. Then the value of x^6 is substituted into the energy expression, and since the potential energy at infinity is zero, the dissociation energy is b^2/4a.

Finding Force from 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 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

force-and-potential-energy-in-three-dimensions

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,...

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 to calculate force from potential energy in one dimension, including the sign rule, a particle example, and dissociation energy for atoms.

Finding Force from 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

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

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Non-conservative Forces

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Conservation of Energy

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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

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Positive, Negative, and Zero Work

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Work-energy Theorem

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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

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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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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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Finding Force from Potential Energy

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