Name: point-mass approximation in spherical gravity
Uploaded: 2025-05-22T09:05:41.000Z
Duration: 1 min 27 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

Earth and Environmental Science

Earth's Place in the Universe

ESS.EES.1.1

Gravity between Spherical Bodies

Ein Apfel, der von einem Baum gelöst wird, neigt natürlich dazu, auf den Boden zu fallen. Man nahm an, dass diese nach unten gerichtete Kraft nur auf der Erde zu finden ist.
Sir Isaac Newton argumentierte jedoch, dass diese Kraft auch dazu führt, dass der Mond seiner Bahn um die Erde folgt.
Unter der Annahme, dass die Erde und der Mond kugelsymmetrische Massenverteilungen haben, kann davon ausgegangen werden, dass ihre Massen in ihren jeweiligen Zentren konzentriert sind.
Die Masse der Erde beträgt 5,97 × 1024 Kilogramm und die Masse des Mondes 7,35 × 1022 Kilogramm. Der durchschnittliche Abstand zwischen ihren Mittelpunkten beträgt 3,84 ×10,8 Meter.
Erinnern Sie sich, dass die Gravitationskonstante 6,67 × 10-11 Newtonmeter zum Quadrat pro Kilogramm zum Quadrat beträgt.
Setzt man die bekannten Werte in der Gravitationskraftgleichung ein, so wird die Gravitationskraft zwischen Erde und Mond auf 1,98 × 1020 Newton geschätzt.
Betrachtet man die Sonne als einen kugelsymmetrischen Körper, der ihre Masse und ihre Entfernung von der Erde ersetzt, so kann die Gravitationskraft zwischen Sonne und Erde berechnet werden.

gravitation zwischen kugelförmigen körpern

Newton's Gesetz der Gravitation beschreibt die Gravitationskraft zwischen zwei Punktmassen. Jedoch gilt dieses Gesetz auch für ausgedehnte kugelförmige Objekte wie die Erde, den Mond und andere Planeten, unter der Annahme, dass die Massen der kugelförmigen Objekte in ihren jeweiligen Zentren konzentriert sind.

Diese Annahme kann leicht bewiesen werden, indem gezeigt wird, dass der Ausdruck für die gravitationelle potentielle Energie zwischen einer hohlen Kugel mit der Masse (M) und einer Punktmasse (m) die gleiche ist wie für ein Paar ausgedehnter kugelförmiger fester Objekte.

Betrachten wir einen winzigen Ring mit einer Breite von Rdθ und einer Masse dM auf der Oberfläche einer hohlen Kugel im Abstand s von der Punktmasse, wie in Abbildung 1(a) gezeigt.

<img alt="Figure1" src="/files/ftp_upload/12745/12745_Figure_1.PNG" style="text-align: center; height: 275px;" />

Die gravitationelle potentielle Energie zwischen dem Ring und der Punktmasse (m) wird wie folgt ausgedrückt:

<img alt="Equation1" src="/files/ftp_upload/12745/12745_Equation_1.png" style="text-align: center; height: 30px;" />

Das Verhältnis der Masse des Rings zur gesamten Schalenmasse entspricht dem Verhältnis der Fläche des Rings zur Fläche der Schale. Daher kann die Masse des Rings nach Vereinfachung wie folgt ausgedrückt werden:

<img alt="Equation2" src="/files/ftp_upload/12745/12745_Equation_2.png" style="text-align: center; height: 30px;" />

Jetzt kann das Quadrat des Abstands (s) als die Summe der Quadrate der anderen beiden Seiten des Dreiecks ausgedrückt werden, wie in Abbildung 1(b) zu sehen ist.

Durch weitere Vereinfachung und Differentiation auf beiden Seiten ergibt sich:

<img alt="Equation3" src="/files/ftp_upload/12745/12745_Equation_3.png" style="text-align: center; height: 14px;" />

Wenn man die gesamte Schale betrachtet, kann s zwischen r - R und r + R variieren, wie in Abbildung 1(c) zu sehen ist.

Daher ergibt sich durch Substitution von dM und s in die Potentiellenergiegleichung und Integration innerhalb der Grenzen von r - R bis r + R die folgende Beziehung für die potentielle Energie zwischen den Punktmassen m und M im Abstand r:

<img alt="Equation4" src="/files/ftp_upload/12745/12745_Equation_4.png" style="text-align: center; height: 30px;" />

Daher ist die Annahme bewiesen.

Da Kraft eine Ableitung der potentiellen Energie ist, gilt die Annahme auch für die gravitationelle Kraft zwischen zwei kugelförmigen festen Objekten wie der Erde und dem Mond. Daher kann das Gravitationsgesetz von Newton verwendet werden, um die gravitationelle Kraft zwischen der Erde und dem Mond sowie zwischen der Erde und der Sonne zu bestimmen.

Gravitation zwischen Kugelförmigen Körpern

Das Newtonsche Gravitationsgesetz beschreibt die Gravitationskraft zwischen zwei beliebigen Punktmassen. Für ausgedehnte kugelförmige Objekte wie die Erde ...

Education

Lehre

JoVE Core

Physics

Physik

Unit 1

Einheit 1

Gravitation

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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 is considered. The ring has width R dc6 and mass dM.

The ring’s mass is found from its share of the shell’s surface area. The ratio of the ring’s mass to the whole shell’s mass equals the ratio of their areas. The distance s from the ring to the point mass is then written using the sides of the triangle in the diagram. After simplifying, differentials are taken, and the shell is considered over the full range from r − R to r + R.

When dM and s are substituted into the potential energy expression and the result is integrated across those limits, the final relation matches the potential energy for point masses m and M separated by distance r. Since force is the derivative of potential energy, the same assumption works for gravitational force between two solid spherical bodies. That is why Newton’s law can be used for systems such as the Earth and the Moon, and the Earth and the Sun.

Point-Mass Approximation in Spherical Gravity

point-mass-approximation-in-spherical-gravity

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

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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

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

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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 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...

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

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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

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detection of black holes

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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

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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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key terms and concepts

See how spherical bodies can be treated as point masses in gravity problems. The proof uses potential energy to justify Earth-Moon and Earth-Sun calculations.

Point-Mass Approximation in Spherical Gravity | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

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Motion and Stability- Forces and Interactions

PS.Phy.2.4

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Ohio’s Learning Standards and Model Curriculum

Forces, Momentum And Motion

Gravitational force and fields

Energy

Gravitational potential energy

Physical Geology

Plate Tectonics

Structure of Earth

Tennessee Academic Standards

High School

Motion and Stability: Forces and Interactions

PHYS.PS2.9

Virginia  Standards of Learning

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+E.E2

HS+E.E2U1.13

Physical Science

HS+Phy.P3

HS+Phy.P3U1.2

Alabama Course of Study Science

9-12.P.1.5.b

Next Generation Science Standards

HS-PS2-4

HS-PS2-5

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

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South Carolina College and Career Ready Science Standards

P-PS2-4

Earth and Space Science (ESS)

E-ESS1-6.

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New York State Science Learning Standards

HS. Forces and Interactions

HS-PS2-4.

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Kepler's Third Law of Planetary Motion

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Use models to illustrate how the sun produces energy.

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Explain how Earth's position relative to the sun influences conditions on Earth.

Point-Mass Approximation in Spherical Gravity

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