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

K12 Standards

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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

Bir ağaçtan salınan bir elma doğal olarak yere düşme eğilimindedir. Bu aşağı doğru kuvvetin Dünya'ya özgü olduğu düşünülüyordu.
Ancak Sir Isaac Newton, bu kuvvetin aynı zamanda Ay'ın Dünya etrafındaki yolunu izlemesine de neden olduğunu düşündü.
Dünya ve Ay'ın küresel simetrik kütle dağılımlarına sahip olduğunu varsayarsak, kütlelerinin kendi merkezlerinde yoğunlaştığı düşünülebilir.
Dünya'nın kütlesi 5.97 × 1024 kilogram ve Ay'ın kütlesi 7.35 × 1022 kilogramdır. Merkezleri arasındaki ortalama mesafe 3.84 × 108 metredir.
Yerçekimi sabitinin kilogram kare başına 6.67 × 10−11 Newton metre kareye eşit olduğunu hatırlayın.
Yerçekimi kuvveti denklemindeki bilinen değerleri yerine koyarak, Dünya ile Ay arasındaki yerçekimi kuvvetinin 1.98 × 1020 Newton olduğu tahmin edilmektedir.
Benzer şekilde, Güneş'i küresel simetrik bir cisim olarak kabul ederek, kütlesini ve Dünya'ya olan uzaklığını ikame ederek, Güneş ile Dünya arasındaki yerçekimi kuvveti hesaplanabilir.

küresel cisimler arasındaki yerçekimi

Newton'un yerçekimi yasası, herhangi iki nokta kütlesi arasındaki yerçekimi kuvvetini açıklar. Ancak, Dünya, Ay ve diğer gezegenler gibi geniş küresel nesneler için, yasa, küresel nesnelerin kütlelerinin merkezlerinde yoğunlaştığı varsayımıyla geçerlidir.

Bu varsayım, kütleleri (M) olan bir boş kürenin ve bir nokta kütlesinin (m) arasındaki yerçekimi potansiyel enerjisi ifadesinin bir çift genişletilmiş küresel katı cisim için olduğu gibi olduğunu göstererek kolayca kanıtlanabilir.

Bir üzerinde genişlik Rdϕ  ve kütlesi dM olan küresel bir boş kürenin yüzeyinde nokta kütle ile mesafesi s olan bir yüzük düşünün, şekil 1(a)'de gösterildiği gibi.

<img alt="Figure1" src="/files/ftp_upload/12745/12745_Figure_1.PNG" style="text-align: center; height: 275px;" />

Yüzük ile nokta kütle (m)  arasındaki yerçekimi potansiyel enerjisi aşağıdaki gibi ifade edilir:

<img alt="Equation1" src="/files/ftp_upload/12745/12745_Equation_1.png" style="text-align: center; height: 30px;" />

Bu nedenle, yüzüğün kütlesi, yüzüğün alanının kabuğun alanına oranına eşittir. Bu nedenle, basitleştirildiğinde, yüzüğün kütlesi aşağıdaki gibi ifade edilebilir:

<img alt="Equation2" src="/files/ftp_upload/12745/12745_Equation_2.png" style="text-align: center; height: 30px;" />

şimdi, mesafenin (s) karesi, şekil 1(b)'de görüldüğü gibi üçgenin diğer iki kenarının karelerinin toplamı olarak ifade edilebilir.

Daha da basitleştirerek ve her iki tarafı farklılaştırarak:

<img alt="Equation3" src="/files/ftp_upload/12745/12745_Equation_3.png" style="text-align: center; height: 14px;" />

Tüm kabul edersek, s, r − R ve r + R arasında değişebilir, şekil 1(c) 'de görüldüğü gibi.

Bu nedenle, dM ve s'nin potansiyel enerji denklemine yerine konması ve r − R ile r + R sınırlarında entegre edilmesiyle, elde edilen ilişki nokta kütleler m ve M arasındaki potansiyel enerjidir.

<img alt="Equation4" src="/files/ftp_upload/12745/12745_Equation_4.png" style="text-align: center; height: 30px;" />

Bu nedenle, varsayım kanıtlanmıştır.

Kuvvet potansiyel enerjinin bir türevidir, bu varsayım Dünya ve Ay gibi iki küresel katı nesne arasındaki yerçekimi kuvvetleri için geçerlidir. Bu nedenle, Newton'un yerçekimi yasası, Dünya ile Ay arasındaki yerçekimi kuvvetini ve Dünya ile Güneş arasındaki yerçekimi kuvvetini belirlemek için kullanılabilir.

Küresel Cisimler Arasındaki Yerçekimi

Newton'un yerçekimi yasası, herhangi iki nokta kütlesi arasındaki yerçekimi kuvvetini tanımlar. Bununla birlikte, Dünya gibi genişletilmiş küresel nesneler için, ...

Education

JoVE Core

Physics

Fizik

Unit 1

Ünite 1

Gravitation

Kütleçekimi

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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

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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

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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

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How Earth’s Shape Affects Apparent Weight

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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

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Satellite Orbits and Energy in Space

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kepler's second law of planetary motion

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kepler's third law of planetary motion

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tidal forces

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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 Temel Fizik, kinematik, dinamik, akışkanlar mekaniği, statik, dalgalar ve akustik dahil olmak üzere temel fizik konularının teorisini ve uygulamasını kısa ve anlaşılması kolay animasyonlu videolarla tanıtıyor. Ek olarak, sektör içindeki bilim insanı videoları, dünya çapındaki günümüz laboratuvarlarında gerçekleştirilen klasik ve orijinal araştırma deneylerinde ilgili kavramların uygulanmasını göstermektedir.

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.

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ESS.EES.1.3

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Point-Mass Approximation in Spherical Gravity

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