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

K12 Standards

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Ohio’s Learning Standards and Model Curriculum

9-12

Physics

Energy

Gravitational potential energy

Potential Energy due to Gravitation

Ein Objekt mit der Masse m, das sich zunächst in einem Abstand von r1 vom Erdmittelpunkt befindet, wird nach r2 verschoben. Da sich das Objekt im Gravitationsfeld der Erde bewegt, ändert sich seine potentielle Energie von U1 zu U2. Die Änderung der potentiellen Energie ist gleich der Arbeit, die die Gravitationskraft leistet, um das Objekt zu verschieben.
Hier entspricht die geleistete Arbeit dem negativen Integral der Größen der Gravitationskraft und der Verschiebung. Wenn man die Gravitationskraft substituiert und innerhalb der Grenzen von r1 und r2 integriert, ist U2 minus U1 gleich der Gravitationskonstante multipliziert mit dem Produkt der beiden Massen multipliziert mit eins über r1 minus eins über r2.
Betrachtet man die Entfernung r2 als unendlich, wobei das Gravitationsfeld der Erde vernachlässigbar ist, so ist die potentielle Energie U2 gleich Null.
Daher wird die potentielle Gravitationsenergie für zwei beliebige Massen, die durch einen Abstand r getrennt sind, ausgedrückt als minus G mal das Produkt zweier Massen, dividiert durch r.
Typischerweise nimmt die potentielle Energie zu, wenn sich die Massen weiter voneinander entfernen. Sein Maximalwert ist Null für Massen in unendlichem Abstand zueinander.

potentielle energie aufgrund der gravitation

Da die Gravitationskraft eine konservative Kraft ist, hängt die aufgewendete Arbeit, um ein Objekt zwischen zwei Punkten im Gravitationsfeld zu bewegen, unabhängig von dem gewählten Pfad ab. Daher kann ähnlich zum Gravitationsfeld eine Funktion für die potentielle Gravitationsenergie definiert werden, die nur von den räumlichen Koordinaten abhängt.

Betrachten wir eine Masse, die gravitativ an ein anderes Objekt gebunden ist. Zum Beispiel ist die Erde gravitativ an das Gravitationsfeld der Sonne gebunden. Die potentielle Energie der Erde im Gravitationsfeld der Sonne wird so definiert, dass ihr Wert nahe der Sonne negativ ist und mit zunehmendem Abstand von der Sonne auf null ansteigt.

Da die Erde und die Sonne keine Sonderfälle sind, kann das Ergebnis für jedes beliebige Paar von Objekten verallgemeinert werden. Somit fallen unter dem Einfluss der Schwerkraft alle Massen von einem höheren zu einem niedrigeren potentiellen Energieniveau, während ihre kinetische Energien zunehmen. Daher ist die Definition im Einklang mit dem Energieerhaltungsprinzip. Ist die Gesamtenergie eines Systems positiv, so ist es nicht gravitativ gebunden.

Die Größe der potentiellen Energie nimmt mit dem Abstand zwischen den beiden Objekten ab. Sie ist umgekehrt proportional zum Abstand aufgrund der umgekehrt quadratischen Abhängigkeit der Gravitationskraft vom Abstand.

Dieser Text ist adaptiert von <a href="https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy">Openstax, University Physics Volume 1, Section 13.3: Gravitational Potential Energy and Total Energy.</a>

Potentielle Energie aufgrund der Gravitation

Da es sich bei der Gravitationskraft um eine konservative Kraft handelt, ist die Menge an Arbeit, die geleistet wird, um ein Objekt zwischen zwei Punkten im Gravitationsfeld zu bewegen, in dem es sich befindet ...

Education

Lehre

JoVE Core

Physik

Unit 1

Einheit 1

Gravitation

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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

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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

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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

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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

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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

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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

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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

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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

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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

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Gravitation Between Spherically Symmetric Masses

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Gravity between Spherical Bodies

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The Principle of Superposition and the Gravitational Field

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Gravitational potential energy

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Types of Potential Energy

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Energy in springs

Work

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Positive, Negative, and Zero Work

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Power

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Power Expended by a Constant Force

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Work and power

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Work and power

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Conservation of Energy

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Conservation of Energy: Application

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Conservation of energy

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Nuclear Binding Energy

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Radioactive Decay and Radiometric Dating

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Nuclear Fission

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Nuclear Fusion

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Nuclear energy

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Nuclear energy

Gravitational Binding and Potential Energy

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