Name: gravitational binding and potential energy
Uploaded: 2023-04-30T11:06:30.000Z
Duration: 1 min 27 s
Channel: JoVE

K12 Standards

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

9-12

Physics

Energy

Gravitational potential energy

Potential Energy due to Gravitation

最初に地球の中心から距離r1にある質量mの物体は、r2に変位します。物体は地球の重力場を移動するため、その位置エネルギーはU1からU2に変化します。位置エネルギーの変化は、物体を変位させる重力によって行われる仕事に等しくなります。
ここで、行われた作業は、重力と変位の大きさの負の積分に等しくなります。重力を代入し、r1 と r2 の限界内で積分すると、U2 から U1 を引いた値は、重力定数に 2 つの質量の積に r1 に 1 を掛けたものから r2 に 1 を引いたものに等しくなります。
距離 r2 を無限大とみなし、地球の重力場は無視できるほど小さいと、位置エネルギー U2 はゼロに等しくなります。
したがって、距離 r で隔てられた任意の 2 つの質量の重力ポテンシャルエネルギーは、2 つの質量の積を r で割った G のマイナス G 倍として表されます。
通常、質量が離れるほど位置エネルギーは増加します。その最大値は、互いに無限に離れている質量に対してゼロです。

重力による位置エネルギー

重力は保存的な力であるため、物体が存在する重力場の 2 点間で物体を移動させるために行われる仕事量は、たどる経路には依存しません。 したがって、重力場と同様に、空間座標のみに依存する重力位置エネルギー関数を定義できます。

別の物体に重力で結合している質量を考えてみましょう。 たとえば、地球は重力によって太陽の重力場に結合しています。 太陽の重力場における地球の位置エネルギーは、その値が太陽に近づくと負になり、太陽から遠く離れるとゼロに増加するように定義されます。

地球と太陽は特殊なケースではないため、結果は任意の 2 つの天体に対して一般化できます。 したがって、重力の影響下では、すべての質量は、運動エネルギーが増加する一方で、より高い位置エネルギーからより低い位置エネルギーに落下します。 したがって、この定義はエネルギー保存則と一致しています。 系の総エネルギーが正の場合、その系は重力の束縛を受けません。

位置エネルギーの大きさは、2 つの物体間の距離に応じて減少します。 重力は距離に逆二乗するため、距離に反比例します。



このテキストは <a href="https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy"> から編集されています。 Openstax, University Physics Volume 1, Section 13.3: Gravitational Potential Energy and Total Energy.</a>

重力は保存力であるため、物体が磁場内の2点間で物体を移動するために行われる作業量は...

Education

JoVE Core

物理学

Unit 1

ユニット1

Gravitation

万有引力

KEY TERMS AND CONCEPTS

主要用語と概念

gravitational binding and potential energy

Gravitational potential energy describes how gravity stores energy between two masses. Because gravity is a conservative force, the work needed to move an object between two points depends on the starting and ending positions, not on the path taken. That is why a gravitational potential energy function can be written in terms of spatial coordinates.

For a mass bound by gravity to another object, the potential energy is negative near the attracting body and rises toward zero as the distance becomes very large. A clear example is the Earth in the Sun’s gravitational field. The Earth and the Sun are not special cases, so this idea applies to any two objects.

As an object falls under gravity, it moves from higher potential energy to lower potential energy while its kinetic energy increases. This fits the conservation of energy principle. If the total energy of a system is positive, the object is not gravitationally bound.

The size of the gravitational potential energy also gets smaller as the distance between the two objects increases. This happens because the force of gravity follows an inverse-square dependence on distance. As a result, the potential energy is inversely proportional to distance as well.

Gravitational Binding and Potential Energy

gravitational-binding-and-potential-energy

gravitation

Newton’s Law Behind Planet Motion

Newton’s law of gravitation links falling objects, the Moon’s orbit, and planetary motion. Before Newton, many people thought the sky and Earth followed different laws. Kepler later described the three laws of planetary motion, but their connection to a deeper force was still unclear. Isaac Newton made that connection in 1665 to 1666. He showed that the same phenomenon explains why objects fall to Earth and why the Moon moves around Earth. He also found that the Moon’s centripetal acceleration...

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newton's law of gravitation

Newton's Gravity and the Motion of Bodies

Newton's gravity links the fall of objects on Earth with the motion of the Moon and planets. People once thought the downward pull seen near Earth was unique to our planet. Later, Nicolaus Copernicus proposed the heliocentric theory, which placed Earth and the other planets around the Sun, with the Moon orbiting Earth. Isaac Newton connected these motions in the 17th century. He reasoned that the same attraction that keeps the Moon on its path around Earth also makes an apple fall from a tree.

gravitation between spherically symmetric masses

gravitation-between-spherically-symmetric-masses

Gravitational Potential Inside a Shell

Gravitational potential energy for spherically symmetric bodies depends on their masses and the distance between them. The idea is to treat each body as if its mass were concentrated at its center of mass. This makes the calculation easier for high school physics. A spherically symmetric mass can be thought of as many concentric spherical shells. Now place a point mass at a distance r from the center of one shell. Every particle on a given spherical ring is the same distance from that point...

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gravity between spherical bodies

Point-Mass Approximation in Spherical Gravity

Newton’s law of gravitation gives the force between two point masses. For large spherical objects, such as the Earth, the Moon, and other planets, the same law can be used by treating each body as if its mass were concentrated at its center. This idea can be proven by comparing gravitational potential energy. The transcript shows that a hollow sphere of mass M and a point mass m give the same energy result as two extended solid spherical objects. To set up the proof, a tiny ring on the shell...

reduced mass coordinates: isolated two-body problem

reduced-mass-coordinates-isolated-two-body-problem

Center of Mass in the Two-Body Problem

Reduced mass coordinates help simplify the classical two-body problem. In classical mechanics, this problem describes the motion of two interacting bodies under gravity or another central force. The key idea is that the system can be treated like a single-body problem when reduced mass is used. For this description, one body is imagined to move with the reduced mass while the other body is fixed in place with infinite mass. The reduced mass is a property of the two-body system and is defined...

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acceleration due to gravity on earth

Gravity on Earth: Mass, Direction, and Value

Gravity near Earth gives objects a downward acceleration called acceleration due to gravity. It comes from Earth’s gravitational pull and points toward the center of the Earth. Newton’s law of gravitation shows that the gravitational force depends on mass, while Newton’s second law shows that the acceleration produced by a force depends on how that force acts on the object. Because of this relationship, the acceleration due to gravity is independent of an object’s mass. An object and the Earth...

acceleration due to gravity on other planets

acceleration-due-to-gravity-on-other-planets

Measuring Gravity on Distant Planets

Gravity near Earth’s surface is measured as acceleration due to gravity. On Earth, students can test it with simple experiments. On other planets, that kind of direct experiment is not possible. Instead, astronomers use observations to find a planet’s gravity. They study how the planet affects nearby objects, especially its satellite. The key idea is that the acceleration depends on the planet’s gravity, not on the object’s mass. To make the calculation, scientists track a satellite’s path...

apparent weight and the earth's rotation

Earth's Spin and Apparent Weight

Earth’s spin changes how weight is measured on the surface. Because the Earth turns once every 24 hours, objects on the surface move in a circle. That motion requires a net centripetal force, which points toward the center of the circle. The north and south poles are the only exception. At the equator, the force that provides this inward pull comes from Earth’s gravity. It is the difference between an object’s true weight and the normal force from the ground. The normal force is the support...

variation in acceleration due to gravity near the earth's surface

variation-in-acceleration-due-to-gravity-near-the-earth-s-surface

How Earth’s Shape Affects Apparent Weight

Apparent weight changes slightly on Earth because of rotation and the planet’s shape. Apparent weight is the weight measured by a spring balance at a location. It is different from true weight, which is the force Earth pulls on an object with. Earth’s rotation reduces apparent weight by adding a centripetal force. Centripetal force is the inward force that keeps an object moving in a circle. The apparent weight equals the true weight minus this force. The difference depends on the square of...

potential energy due to gravitation

Gravitational potential energy describes how gravity stores energy between two masses. Because gravity is a conservative force, the work needed to move an object between two points depends on the starting and ending positions, not on the path taken. That is why a gravitational potential energy function can be written in terms of spatial coordinates. For a mass bound by gravity to another object, the potential energy is negative near the attracting body and rises toward zero as the distance...

the principle of superposition and the gravitational field

the-principle-of-superposition-and-the-gravitational-field

Adding Gravitational Forces with Superposition

Gravitational forces can be added using the principle of superposition. For objects that are sufficiently far apart, the net gravitational force on a point object is the vector sum of the forces from each mass. This means each force is found first, and then the forces are added as vectors. Superposition is useful because gravity from more than one object can combine in ways that are not always easy to see. One force may be increased or reduced by another force, or the final effect may be a...

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escape velocity

Launching a Satellite Beyond Gravity

Launching a satellite beyond gravity requires enough escape velocity to leave a planet or other body and never return. Escape velocity is the minimum initial speed needed to break free from the gravitational pull of an object that is holding the satellite in orbit or near its surface. For Earth, the idea is often shown with a satellite launched from the ground. The calculation assumes that no energy is lost to friction. In real life, Earth’s atmosphere also slows the satellite down, so the...

circular orbits and critical velocity for satellites

circular-orbits-and-critical-velocity-for-satellites

Orbital Speed and Period in Circular Paths

Orbital speed and period in circular paths explain how the Moon, planets, and satellites stay in motion around larger bodies. The Moon orbits Earth. Earth and the other planets orbit the Sun. The space above our atmosphere also contains many artificial satellites. A circular orbit is the simplest orbit to study. It helps show how an object’s speed and its period, or the time for one full orbit, depend on where it is and what it circles. Nicolaus Copernicus first suggested that Earth and the...

energy of a satellite in a circular orbit

energy-of-a-satellite-in-a-circular-orbit

Satellite Orbits and Energy in Space

Satellite orbits and energy in space depend on altitude, orbit shape, and gravity. Thousands of artificial satellites circle Earth every day at many different distances. Satellites below 1,600 km are in low-Earth orbit, or LEO. Research satellites and Earth observation satellites are usually placed in LEO. They mostly move in elliptical orbits, which are oval-shaped paths. Navigation satellites are placed in medium-Earth orbit, or MEO, which ranges from 2,000 km to 36,000 km above Earth’s...

kepler's first law of planetary motion

Planet Orbits as Ellipses

Planet orbits are not perfect circles. Johannes Kepler used planet data from Tycho Brahe to show that each planet follows an ellipse, with the Sun at one focus. This idea came from studying Mars, which did not fit the circular path expected in the Copernican heliocentric model. Copernicus had proposed that the solar system is heliocentric, meaning the Sun is at the center and the planets, including Earth, move around it. His model treated those paths as circles. Brahe later recorded the...

kepler's second law of planetary motion

Kepler's Equal-Area Planet Motion

Kepler's second law explains how a planet moves faster and slower in its orbit around the Sun. Johannes Kepler, a German astronomer and mathematician in the early 17th century, developed three laws for planetary motion. His first law says that planets travel in elliptical orbits, with the Sun at one focus of the ellipse. Because the planet-Sun distance changes during the orbit, the planet does not move at the same speed all the time. In an elliptical orbit, the total energy of the planet is...

kepler's third law of planetary motion

Kepler’s Planet Distance Law

Kepler’s third law links a planet’s average distance from the Sun to the time it takes to orbit the Sun. This rule gives a clear mathematical relationship between orbital distance and orbital period, which is the time for one full revolution. Johannes Kepler developed this law in the early 17th century after building on the observations of Nicolaus Copernicus and Tycho Brahe. Kepler first formed his first two laws from those earlier observations. He later published the third law of planetary...

tidal forces

How the Moon Shapes Earth’s Tides

Earth’s ocean tides are driven by gravitational tidal forces. Scientists have studied the origin of tides for more than 2000 years, and Newton’s work marked the start of a proper understanding. These same tidal forces also act on other astronomical bodies. They can produce internal heat that leads to volcanic activity on Io, one of Jupiter’s moons, and they can tear apart stars that pass too close to black holes. Newton’s law of gravitation says that gravitational force is inversely...

schwarzschild radius and event horizon

Escape Velocity Limits and Black Holes

Escape velocity helps explain how black holes and event horizons form. It is the minimum speed a projectile needs to leave the surface of an object and eventually escape its gravitational field. In this context, escape velocity does not depend on the mass of the object being launched. The key idea is that no object with finite mass can move faster than the speed of light in a vacuum. When escape velocity is compared with that speed limit, the Schwarzschild radius becomes important. If an...

detection of black holes

How Astronomers Find Black Holes

Black holes are found through indirect astronomical evidence. They were first proposed in the 1920s, but they stayed outside observational astronomy until the 1970s. Their closest known cousins are neutron stars, which are made mostly of packed neutrons and are extremely dense. A neutron star can have the mass of the Sun while its diameter is only a few kilometers. Because of that, the escape velocity from its surface is close to the speed of light. Interest in black holes rose again in the...

principle of equivalence

Weightlessness and Einstein’s Equivalence

Weightlessness and Einstein’s equivalence principle are closely connected in gravity. A person in free fall under gravity, such as someone diving toward Earth from an airplane, would feel completely weightless. A person descending in a lift may feel partially weightless. The principle of equivalence says that an object in a uniform gravitational field and an object accelerating at a constant rate with no gravity are under the same experimental conditions. In other words, no experiment can tell...

space-time curvature and the general theory of relativity

space-time-curvature-and-the-general-theory-of-relativity

Einstein’s Gravity and Mercury’s Orbit

Einstein’s theory of gravity explains how space-time affects motion. In 1905, Einstein published his special theory of relativity. It says that no object can travel faster than the speed of light in a vacuum, which is the speed limit of the universe. Special relativity also changed the way scientists think about space and time. Two observers moving relative to each other do not agree on object length or on how time passes. Newton’s laws still work well for many cases, even for objects moving...

key terms and concepts

Learn how gravitational potential energy depends on distance, why it can be negative near massive objects, and how it connects to binding energy and motion.

Gravitational Binding and 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。

Virginia  Standards of Learning

HIgh School

PH.3

PH.3.b

Arizona Science Standards

High School

Physical Science

HS.P2

HS.P2U1.5

HS+Phy.P3

HS+Phy.P3U1.2

Next Generation Science Standards

HS-PS3-1

HS-PS3-2

HS-PS3-3

Florida Department of Education

Earth and Space Science

SC.912.E.5

SC.912.E.5.10

SC.912.P.10

SC.912.P.10.6

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.H

new-york-state-science-learning-standards

New York State Science Learning Standards

HS. Forces and Interactions

HS-PS2-4.

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

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 ...

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Since gravitational force is a conservative force, the amount of work done to move an object between two points in the gravitational field in which it ...

The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

Gravitational potential energy

Content elaboration: When two attracting masses interact, the kinetic energies of both objects change but neither is acting as the energy source or the receiver. Instead, the energy is transferred into or out of the gravitational field around the system as gravitational potential energy. A single mass does not have gravitational potential energy. Only the system of attracting masses can have gravitational potential energy. When two masses are moved farther apart, energy is transferred into the field as gravitational potential energy. When two masses are moved closer together, gravitational potential energy is transferred out of the field.

Hooke's Law

Hooke&#39;s law states that stress applied to a material is directly proportional to the strain it experiences within the elastic limit. For ductile ...

Types of Potential Energy

Potential energy is also known as energy at rest or stored energy. Common types of potential energy include the gravitational potential energy stored in ...

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 ...

Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and ...

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Energy in springs

Content elaboration: The approximation for the change in the potential elastic energy of an elastic object (e.g., a spring) is $\Delta E_{elastic}=\frac{1}{2}k\Delta x^{2}$ where $\Delta x$ is the distance the elastic object is stretched or compressed from its relaxed length.

Energy in springs

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 ...

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

The relationship between work done and the time taken to do it can be explained using the concept of power. For example, several sprinters in a race may ...

Work and power

Content elaboration: Work can be calculated for situations in which the force and the displacement are at angles to one another using the equation $W=F\Delta x(cos~\theta)$ where W is the work, F is the force, $\Delta x$ is the displacement, and is the angle between the force and the displacement. This means when the force and the displacement are at right angles, no work is done and no energy is transferred between the objects. Such is the case for circular motion. The rate of energy change or transfer is called power (P) and can be mathematically represented by $P=\Delta E/\Delta t$ or $P=W/\Delta t$. Power is a scalar property. The unit of power is the watt (W), which is equivalent to one joule of energy transferred in one second $(J/s)$.

Work and power

The universe is composed of matter in different forms, and all forms of matter contain energy.&#160; The different forms of energy on Earth originate from ...

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 ...

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 ...

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Conservation of Energy

The terms &#39;conserved quantity&#39;&#160;and &#39;conservation law&#39;&#160;have specific scientific meanings in physics, which differ from the ...

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 ...

Conservation of energy

Content elaboration: The total initial energy of the system and the energy entering the system are equal to the total final energy of the system and the energy leaving the system. Although the various forms of energy appear very different, each can be measured in a way that makes it possible to keep track of how much of one form is converted into another. Situations involving energy transformations can be represented with verbal or written descriptions, energy diagrams and mathematical equations. Translations can be made between these representations. The conservation of energy principle applies to any defined system and time interval within a situation or event in which there are no nuclear changes that involve mass-energy equivalency. The system and time interval may be defined to focus on one particular aspect of the event. The defined system and time interval may then be changed to obtain information about different aspects of the same event.

Conservation of energy

Radioactivity and Nuclear Equations

Nuclear chemistry is the study of reactions that involve changes in nuclear structure. The nucleus of an atom is composed of protons and, except for ...

Types of Radioactivity

The most common types of radioactivity are &#945; decay, &#946; decay, &#947; decay, neutron emission, and electron capture.
Alpha (&#945;) decay is the ...

Nuclear Stability

Protons and neutrons, collectively called nucleons, are packed together tightly in a nucleus. With a radius of about 10&#8722;15 meters, a nucleus is ...

Nuclear Binding Energy

The difference between the calculated and experimentally measured masses is known as the mass defect of the atom. In the case of helium-4, the mass defect ...

Radioactive Decay and Radiometric Dating

Radioactivity is a spontaneous disintegration of an unstable nuclide and is a random process, as all the nuclei in the sample do not decay simultaneously. ...

Nuclear Fission

Many heavier elements with smaller binding energies per nucleon can decompose into more stable elements that have intermediate mass numbers and larger ...

Nuclear Power

Controlled nuclear fission reactions are used to generate electricity. Any nuclear reactor that produces power via the fission of uranium or plutonium by ...

Nuclear Fusion

The process of converting very light nuclei into heavier nuclei is also accompanied by the conversion of mass into large amounts of energy, a process ...

Nuclear energy

Content elaboration: Alpha, beta, gamma and positron emission each have different properties and result in different changes to the nucleus. The identity of new elements can be predicted for radioisotopes that undergo alpha or beta decay. Nuclear reactions, such as fission and fusion, are accompanied by large energy changes that are much greater than those that accompany chemical reactions. Nuclear fission reactions are used as a controlled source of energy in nuclear power plants. There are advantages and disadvantages of generating electricity from fission and fusion. During nuclear interactions, the transfer of energy out of a system is directly proportional to the change in mass of the system as expressed by $E=mc^{2}$ which is known as the equation for mass-energy equivalence. A very small loss in mass is accompanied by a release of a large amount of energy. In nuclear processes such as nuclear decay, fission and fusion, the mass of the product is less than the mass of the original nuclei. The missing mass appears as energy. This energy can be calculated for fission and fusion when given the masses of the particle(s) formed and the masses of the particle(s) that interacted to produce them.

Nuclear energy

Gravitational Binding and Potential Energy

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