Name: gravitational potential inside a shell
Uploaded: 2023-05-16T08:18:41.000Z
Duration: 1 min 14 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Physics

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Motion and Stability- Forces and Interactions

PS.Phy.2.4

gravitation-between-spherically-symmetric-masses

Gravitation Between Spherically Symmetric Masses

L'énergie potentielle gravitationnelle entre deux objets à symétrie sphérique est donnée par les masses des objets et la distance entre eux, en supposant que la masse est concentrée au centre.
Considérons une masse ponctuelle placée à une distance d’une coquille sphérique. La distribution de masse à symétrie sphérique comprend plusieurs couches concentriques, et toutes les particules d’un anneau spécifique sont à des distances égales de la masse ponctuelle.
Le potentiel entre l'anneau et la masse ponctuelle peut être obtenu lorsque la masse de l'anneau est connue. Le rapport entre l'aire de l'anneau et l'aire de la coquille est égal au rapport de leurs masses. 
En résolvant la relation entre les distances, en l’appliquant et en intégrant l’expression, on obtient le potentiel entre la coquille et la masse ponctuelle. Cette expression est la même que le potentiel entre deux masses ponctuelles.
Si la masse ponctuelle se trouve à l’intérieur de la coquille, alors le potentiel gravitationnel est le même partout pour tous les points à l’intérieur de la cavité.
Comme la force est une dérivée du potentiel, l’hypothèse est également valable pour la force gravitationnelle.

<ol>
	<li>Young, H.D. et Freedman, R.A. (2012). La physique universitaire avec la physique moderne. San Francisco, Californie : Pearson. p. 418 à 420.</li>
	<li>Verma, H.C. (2008). Concepts de physique Partie I, quatrième édition. p. 207 à 209.</li>
</ol>

gravitation entre des masses sphériquement symétriques

La énergie potentielle gravitationnelle entre deux corps sphériquement symétriques peut être calculée à partir des masses et de la distance entre les corps, en supposant que le centre de masse soit concentré aux centres respectifs des corps.

<img alt="Equation1" src="/files/ftp_upload/14717/14717_Figure_1.png" style="text-align: center; height: 150px;" />

Considérez que la répartition de masse sphériquement symétrique comprend plusieurs couches sphériques concentriques. Une masse ponctuelle est placée à une distance 'r' du centre de masse de la coquille sphérique. Toutes les particules dans un anneau sphérique donné à la surface de la coquille sont à des distances égales de la masse ponctuelle.

<img alt="Equation1" src="/files/ftp_upload/14717/14717_Equation_1.svg" style="text-align: center; height: 32px;" />

Le potentiel entre la masse ponctuelle et l'anneau peut être obtenu à partir de la masse de l'anneau. En intégrant l'expression de l'énergie potentielle entre une masse ponctuelle et un anneau sur les limites de distance, on obtient l'énergie potentielle gravitationnelle, qui est la même que l'énergie potentielle entre deux masses ponctuelles.

<img alt="Equation2" src="/files/ftp_upload/14717/14717_Equation_2.svg" style="text-align: center; height: 32px;" />

Si la masse ponctuelle est à l'intérieur de la coquille, alors les limites d'intégration changent. Cela montre que l'énergie potentielle ne dépend pas de la distance et est la même partout pour tous les points à l'intérieur de la coquille. Cependant, aucun travail n'est effectué sur la masse ponctuelle si elle est à l'intérieur de la coquille.

<img alt="Equation3" src="/files/ftp_upload/14717/14717_Equation_3.svg" style="text-align: center; height: 32px;" />

Gravitation entre des masses sphériquement symétriques

L’énergie potentielle gravitationnelle entre deux corps sphériquement symétriques peut être calculée à partir des masses et de la distance entre les corps, ...

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Education

Enseignement

JoVE Core

Physique

Unit 1

Unité 1

Gravitation

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 mass. The potential from that ring can be found from the ring’s mass.

When the expression for the point mass and ring is integrated over the distance limits, the result is the same gravitational potential energy as for two point masses. If the point mass is inside the shell, the limits of integration change. The potential energy then does not depend on distance and is the same everywhere inside the shell. In that case, no work is done on the point mass.

Gravitational Potential Inside a Shell

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

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...

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

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

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How Earth’s Shape Affects Apparent Weight

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potential energy due to gravitation

Gravitational Binding and Potential Energy

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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

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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

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kepler's third law of planetary motion

Kepler’s Planet Distance Law

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tidal forces

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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

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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

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key terms and concepts

Learn how gravitational potential energy is found for spherically symmetric bodies, including why it stays constant inside a shell and when no work is done.

Gravitational Potential Inside a Shell | JoVE K12

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JoVE Core Physique présente plusieurs branches de la physique fondamentale grâce à des vidéos animées concises et faciles à comprendre ; cela inclut: la cinématique, la dynamique, la mécanique des fluides, la mécanique statique, les ondes et l'acoustique. En parallèle, les vidéos de scientifiques à l'action démontrent l'application de concepts associés à ces domaines dans des expériences de recherche classiques et originales réalisées dans des laboratoires modernes du monde entier.

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Ohio’s Learning Standards and Model Curriculum

Forces, Momentum And Motion

Gravitational force and fields

Energy

Gravitational potential energy

Tennessee Academic Standards

High School

Motion and Stability: Forces and Interactions

PHYS.PS2.9

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HIgh School

Earth Science

ES.3

ES.3.b

PH.3

PH.3.b

PH.7

PH.7.a

PH.7.b

Minnesota Academic Standards in Science

2.2.1 9P.1.2.1.2

Arizona Science Standards

Earth and Space Science

HS.E2

HS.E2U1.16

HS+E.E2

HS+E.E2U1.13

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New Jersey Student Learning Standards for Science

Physical Science

HS-PS2-4

HS+Phy.P3

HS+Phy.P3U1.2

Alabama Course of Study Science

9-12.P.1.5.b

Next Generation Science Standards

Earth’s Place in the Universe

HS-ESS1-4

Florida Department of Education

SC.912.E.5

SC.912.E.5.10

SC.912.E.5.11

SC.912.P.10

SC.912.P.10.10

SC.912.P.12

SC.912.P.12.4

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.H

Chemistry and Physics

IPC.5

IPC.5.E

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South Carolina College and Career Ready Science Standards

P-PS2-4

Earth and Space Science (ESS)

E-ESS1-6.

new-york-state-science-learning-standards

New York State Science Learning Standards

HS. Forces and Interactions

HS-PS2-4.

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Use models (graphs, equations, diagrams) to infer motion in one dimension.

PS.Phy.1.1

Analyze the motion of objects using time, distance, displacement, speed, velocity, and acceleration.

Gravitational Potential Inside a Shell

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