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

나무에서 방출된 사과는 자연적으로 땅에 떨어지는 경향이 있습니다. 이 하향 힘은 지구에만 있는 것으로 생각되었습니다.
그러나 아이작 뉴턴 경은 이 힘으로 인해 달이 지구 주위를 도는 경로를 따라가게 된다고 추론했습니다.
지구와 달이 구형 대칭 질량 분포를 가지고 있다고 가정하면 질량은 각각의 중심에 집중되어 있는 것으로 간주될 수 있습니다.
지구의 질량은 5.97 × 1024kg이고 달의 질량은 7.35 × 1022kg입니다. 중심 사이의 평균 거리는 3.84 × 108 미터입니다.
중력 상수는 킬로그램 제곱당 6.67 × 10-11 뉴턴 미터 제곱과 같다는 것을 기억하십시오.
중력 방정식에서 알려진 값을 대체하면 지구와 달 사이의 중력은 1.98 × 1020뉴턴으로 추정됩니다.
마찬가지로, 태양을 구형 대칭체로 간주하고 질량과 지구로부터의 거리를 대체하여 태양과 지구 사이의 중력을 계산할 수 있습니다.

구형체 사이의 중력

뉴턴의 중력 법칙은 두 점 질량 사이의 중력을 설명합니다. 그러나 지구, 달, 기타 행성과 같이 확장된 구형 물체의 경우 구형 물체의 질량이 해당 중심에 집중되어 있다는 가정 하에 법칙이 유지됩니다.

이 가정은 속이 빈 질량 구(M)와 점 질량(m) 사이의 중력 위치 에너지에 대한 표현이 한 쌍의 확장된 구형 고체 물체에 대한 표현과 동일하다는 것을 보여줌으로써 쉽게 증명될 수 있습니다.

폭이 Rdϕ 인 작은 고리를 생각해 보세요. 그림 1(a)에 표시된 것처럼 점 질량으로부터 s 거리에 있는 구형 중공 구 표면의 질량 dM입니다.

<img alt="Figure1" src="/files/ftp_upload/12745/12745_Figure_1.PNG" style="text-align: center; height: 275px;" />

고리와 점질량 사이의 중력 위치에너지(m)  다음과 같이 표현됩니다:

<img alt="Equation1" src="/files/ftp_upload/12745/12745_Equation_1.png" style="text-align: center; height: 30px;" />

전체 껍질 질량에 대한 고리 질량의 비율은 껍질 면적에 대한 고리 면적의 비율과 같습니다. 따라서 단순화하면 링의 질량은 다음과 같이 표현될 수 있습니다.

<img alt="Equation2" src="/files/ftp_upload/12745/12745_Equation_2.png" style="text-align: center; height: 30px;" />

이제 거리의 제곱(s)은 그림 1(b)에서 볼 수 있듯이 삼각형의 다른 두 변의 제곱의 합으로 표현할 수 있습니다.

더 단순화하고 양쪽에 차이를 두어

<img alt="Equation3" src="/files/ftp_upload/12745/12745_Equation_3.png" style="text-align: center; height: 14px;" />

전체 셸을 고려할 때 s는 r − R 및 r + R(그림 1(c) 참조).

따라서 위치 에너지 방정식에 dM과 s를 대입하고 r − R에서 r + R로, 얻은 관계는 점 질량 m과 M 사이의 위치 에너지입니다. r 거리에서.

<img alt="Equation4" src="/files/ftp_upload/12745/12745_Equation_4.png" style="text-align: center; height: 30px;" />

따라서 가정이 입증되었습니다.

힘은 위치 에너지에서 파생되므로 지구와 달과 같은 구형의 두 물체 사이의 중력에 대한 가정이 유지됩니다. 따라서 뉴턴의 중력 법칙을 이용하면 지구와 달, 지구와 태양 사이의 중력을 결정할 수 있다.

뉴턴의 중력 법칙은 두 점 질량 사이의 중력을 설명합니다. 그러나 지구와 같은 확장된 구형 물체의 경우 ...

Education

교육

JoVE Core

Physics

물리학

Unit 1

제1단원

Gravitation

중력

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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...

/files/media/science-education/science-education-thumbs/14717.jpg

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

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...

/files/media/science-education/science-education-thumbs/14740.jpg

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

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

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.

motion-and-stability-forces-and-interactions

Motion and Stability- Forces and Interactions

PS.Phy.2.4

ohios-learning-standards-and-model-curriculum

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

south-carolina-college-and-career-ready-science-standards

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.

Kepler's First Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. He ...

Kepler's Second Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. His ...

Kepler's Third Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. In ...

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ESS.EES.1.2

Light as Energy

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First Law of Thermodynamics

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Use models to illustrate how the sun produces energy.

ESS.EES.1.3

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

Point-Mass Approximation in Spherical Gravity

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