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Küp Şeklindeki Bir Cismin Civarindaki Bir Kütleye Etkiyen, Kuvvetin Değişiminin Sonlu Elemanlarla Hesaplanmasi

Küp Şeklindeki Bir Cismin Civarindaki Bir Kütleye Etkiyen, Kuvvetin Değişiminin Sonlu Elemanlarla Hesaplanmasi

##### Dosyalar

##### Tarih

1996

##### Yazarlar

Arslanalp, Mustafa

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bütün ömrümüz boyunca hep ayaklarımız üzerinde durmağa çalışırız. Birşey bizi hep yerde tutmaktadır. Çağlar boyunca insanlar bu konuyu merak etmiş olmalılar ki, bu konuya pek çok açıklama getirmişlerdir. Bu açıklamaları: (1) Newton öncesi, (2) Newton sonrası, diye guruplandırabiliriz. Newton öncesine göre: Kainattaki herşeyin ait oldukları bir ilk yer, orijin, vardır. Cisimler bu ilk oluştukları yere dönmek istemektedirler. Suyun akışı, havaya atılan taşın yere düşmesi, bu maddelerin kainatın merkezine gitme eğilimlerinden dolayıdır. Kainatın merkezi ile dünyanın merkezi çakışmaktadır. Newton ile birlikte gravitasyon kavramı değişmiş, "Bütün maddeler biri birlerini, kütleleri ile doğru ve aralarındaki uzaklığın karesi ile ters orantılı olarak çekerler " şekline gelmiştir. Maddelerin aralarındaki uzaklık kavramı ile, kütle merkezleri arasındaki uzaklık alınması, yalnızca noktasal yükler ve küresel simetrik cisimler için geçerlidir. Gök cisimlerini, noktasal bir kütle gibi kabul adabiliriz. Bunun isbatı verilmiştir. Kübik bir cismi noktasal bir kütle gibi düşünemeyiz. Bu çalışma ile kübik bir cisim, alt küp elemanlara bölünmüş, civarındaki bir noktasal m kütlesine, bir doğrultu boyunca uyguladığı kuvvet bulunarak, kübün kütlesine eşdeğer küresel bir kütle için bulunan sonuca oranlanmışlar. Bilahare vektörler yardımı ile hesaplama yapılmış, istenen herhangi bir doğrultu için benzer hesaplatın yapılabileceği bulunmuştur.

Until Newton's findings, around 1665 it was not realised that both the movement of celestial bodies and the free fall of objects on Earth are determined by the same force. The classical Greek philosophers for example, did not consider the celestial bodies to be affected by gravity because the bodies were observed to follow perpetually repeating, non- descending trajectories in the sky. Thus, Aristotle considered that each heavenly body followed a particular "natural" motion, unaffected by external causes or agents. Aristotle also believed that massive earthly objects possess a natural tendency to move toward the Earth's centre. These ideas have prevailed for centuries that a body moving at constant speed requires a continuous force acting on it and that force must be applied by contact rather than interaction at a distance. These views impeded the understanding of the principles of motion and precluded the development of ideas about universal gravitation. The 17th-century German astronomer Johannes Kepler accepted the Copernican perspective in which the planets' orbit the Sun rather than the Danish astronomer Tycho. Kepler described the planetary orbits with simple geometric and arithmetic relations. Kepler's three quantitative laws of well known planetary motion are: (1) the planets describe elliptic orbits with the Sun. at one focus (one of two points inside an ellipse: rays coming from one of the points bounce off any side): (2) the line joining a planet to the Sun sweeps equal areas in equal times; and (3) the square of the period of revolution of a planet is proportional to the cube of its average distance from the Sun. During these same period the Italian astronomer and physicist Galileo made progress in understanding "natural" motion for earthly objects. He realised that bodies that are uninfluenced by forces continue indefinitely to move and that force is necessary to change motion. In studying how objects M toward the Earth, Galileo discovered that the motion is one of constant acceleration. He was able to show that the distance a falling body travels from rest in this way varies as the square of the time. Newton discovered the relationship between the motion of the Moon and the motion of any felling body on Earth. His gravitational theory explained Kepler's laws and established the modern quantitative science of gravitation. Newton assumed the presence of an attractive force between all massive bodies, one which does not require bodily vm contact and acts at a distance. By invoking his law of Newton observed that a force exerted by the Earth on the Moon is needed to keep it in a circular motion rather than moving in a straight line. He realised that this force could be at long range, the same as the force with which the Earth pulls objects on its surface downward. When Newton discovered that the acceleration of the Moon is 1/3.600 smaller than the acceleration at the surface of the Earth he related the number 3.600 to the radius of the Earth squared. He calculated that the circular orbital motion of radius R and period T requires a constant inward acceleration A equal to the product of 4. n2 and the ratio of the radius to the square of the time: *.*£? m Newton applied his result to the Moon's orbit, which has a period of 27.3 days and a radius of about 384.000 kilometres (approximately 60 Earth radii). He found the Moon's inward acceleration in its orbit to be 0.0027 metre per square second, the same as (1/60)2 of the acceleration of a falling object at the surface of the Earth. Because Newton could relate the two accelerations to a common interaction. He deduced that the gravitational force between bodies diminishes as the inverse square of the distance between the bodies. A further result, that the mass of the Earth acts gravitationally on the outside world as if the mass were concentrated at the planet's centre. Newton proved it to be true for all spherically symmetric bodies. Newton saw that the gravitational force between bodies must depend on the masses of the bodies. Since a body of mass M experiencing a force F accelerates at a rate F/M, a force of gravity proportional to M would be consistent with Galileo's observation that all bodies accelerate under gravity toward the Earth at the same rate. In Newton's equation. Fn is the magnitude of the gravitational force acting between masses Mi and M2 separated by distance ri2. _GMX.M2 A2 ~ i (2) The constant G is a quantity with the physical dimensions (length)/(mass)(time)2 its numerical value depends on the physical units of length mass and time used. The attractive force of a number of bodies of masses Mi on a body of mass M is F = ^Ş- LI (3) Where Si means that the vector forces due to all the attracting bodies must be added together. This is Newton's gravitational law essentially in its original form. IX Equations (1) and (2) above con be used to derive Kepler's third law for the case of circular planetary orbits. By using the expression for the acceleration A in equation (1) for he force of gravity for the planet G.Mp.M, / R2 divided by the planet's mass M the following equation, in which Ms is the mass of the Sun. is obtained: GMS _ 4u2R R2 ~ T2 or R3 = GMS V4k2 ) (4) Newton was able to show that all three of Kepler's observationally derived laws follow mathematically from the assumption of his own laws of motion and gravity. In all observations of the motion of a celestial body, only the product of G and the mass can be found. Newton first estimated the magnitude of G by assuming the Earth's average mass density to be about 5.5 times that of water (somewhat greater than the Earth's surface rock density) and by calculating the Earth's mass from this. Then, taking Me and rE as the Earth's mass and radius, respectively, the value of G was that numerically comes close to the accepted value of 6.6726x10- ms'2/kg ME Comparing g = - ^- above for the Earth's surface acceleration g with the R3/T2 r* ratio for the planets, a formula for the ratio of the Sun's mass. Ms to Earth's mass. Me, was obtained in terms of known quantities. Re being the radius of the Earth's orbit: - *- = ^ = 325.000 (6) MB gTB\ E It was already known in Newton's day that the Moon does not move in a simple Keplerian orbit. Later, more accurate observations of the planets also showed discrepancies from Kepler's laws. The motion of the Moon is particularly complex: however, apart from a long-term acceleration due to tides on the Earth, it can be accounted for by the gravitational attraction of the Sun, the Earth, and the other planets. In addition, the gravitational attraction of the planets for each other explains almost all the features of their motions. The exceptions are nonetheless important. Uranus the seventh planet from the Sun, was observed to undergo variations in its motion that could not be explained by perturbations from Saturn, Jupiter, and the other planets. Measurements of the motion of the innermost planet, Mercury,, over an extended period led astronomers to conclude that the major axis of this planet's elliptical orbit processes in space at a rate 43 are seconds per century faster than could be accounted for from perturbations of the other planets. In this case, however, no other bodies could be found that could produce this discrepancy, and very slight modification of Newton's law of gravitation seemed to be needed. Einstein's theory of relativity precisely predicts this observed behaviour of Mercury's orbit. In the last three hundred years long, the development of gravitation understanding became a section of the classical mechanics. When somebody would like to find out how a body produces gravitational force onto another one three things must be in consideration: First, the shape of the body. Second, the element shape that it is easy to locate the centre of gravity of the element And third, the coordinate system that would be helpful demonstrating the calculations. As long as I see when I was studying on the subject I found out that the traditional mass element is very small (differential) elements. The selection of the elements will determine whether the resulting integral be a single, a double, or a triple. If we would like to have an exact solution, for any body that doesn't have symmetrical planes then we need to know the orientation of the gravitational force on an element In my study I tried to find out a cube's gravitational attraction on a very small body that can be accepted as a point mass, by dividing the cube into different number of finite cube elements. The least of the element number may be one. By trying to calculate it we will find out it is like a point mass. After this attempt we can conclude that the least number is eight and if we continue using integers, next possible element number is twenty- seven and it goes on the cube of four, five etc. I went on trying the higher number of cube elements as much as one million. I found out that it has no difference by using two- thousand or one million. Another view is to make the results compare with a point mass that as if it was located on the cube's centre of gravity. I divided the resultant force by the point mass' gravitational force that had already been mentioned above. At the beginning of my study I accepted the cube dimension 1, but I saw that the higher the number of elements the smaller the cube elements dimensions. All the time I had to use ratios. I decided to use the cube elements dimensions one each time that I took more elements. The dimensions of the cube become larger and larger. The location of the mass m was bigger each time. I divided the distances by the radius of the sphere that it's mass is equal to the mass of the cube. So my results weren't affected by the dimensions of the cubes. It is easy to use the cube element dimensions as 1 but I fount out it would be easy if I accept the dimensions 2 because of the location of the element's centre of gravity would be integer. Xi I also decided to calculate the desired results for any directions instead of just for x directions. So I applied vectors on finding the element's centre of gravity. In summary we can accept a cube as a point mass if some approximations can be convenient. As an example, if one percent approximation is acceptable, than we can accept the cube as a point mass after two and half times of equal sphere radius.

Until Newton's findings, around 1665 it was not realised that both the movement of celestial bodies and the free fall of objects on Earth are determined by the same force. The classical Greek philosophers for example, did not consider the celestial bodies to be affected by gravity because the bodies were observed to follow perpetually repeating, non- descending trajectories in the sky. Thus, Aristotle considered that each heavenly body followed a particular "natural" motion, unaffected by external causes or agents. Aristotle also believed that massive earthly objects possess a natural tendency to move toward the Earth's centre. These ideas have prevailed for centuries that a body moving at constant speed requires a continuous force acting on it and that force must be applied by contact rather than interaction at a distance. These views impeded the understanding of the principles of motion and precluded the development of ideas about universal gravitation. The 17th-century German astronomer Johannes Kepler accepted the Copernican perspective in which the planets' orbit the Sun rather than the Danish astronomer Tycho. Kepler described the planetary orbits with simple geometric and arithmetic relations. Kepler's three quantitative laws of well known planetary motion are: (1) the planets describe elliptic orbits with the Sun. at one focus (one of two points inside an ellipse: rays coming from one of the points bounce off any side): (2) the line joining a planet to the Sun sweeps equal areas in equal times; and (3) the square of the period of revolution of a planet is proportional to the cube of its average distance from the Sun. During these same period the Italian astronomer and physicist Galileo made progress in understanding "natural" motion for earthly objects. He realised that bodies that are uninfluenced by forces continue indefinitely to move and that force is necessary to change motion. In studying how objects M toward the Earth, Galileo discovered that the motion is one of constant acceleration. He was able to show that the distance a falling body travels from rest in this way varies as the square of the time. Newton discovered the relationship between the motion of the Moon and the motion of any felling body on Earth. His gravitational theory explained Kepler's laws and established the modern quantitative science of gravitation. Newton assumed the presence of an attractive force between all massive bodies, one which does not require bodily vm contact and acts at a distance. By invoking his law of Newton observed that a force exerted by the Earth on the Moon is needed to keep it in a circular motion rather than moving in a straight line. He realised that this force could be at long range, the same as the force with which the Earth pulls objects on its surface downward. When Newton discovered that the acceleration of the Moon is 1/3.600 smaller than the acceleration at the surface of the Earth he related the number 3.600 to the radius of the Earth squared. He calculated that the circular orbital motion of radius R and period T requires a constant inward acceleration A equal to the product of 4. n2 and the ratio of the radius to the square of the time: *.*£? m Newton applied his result to the Moon's orbit, which has a period of 27.3 days and a radius of about 384.000 kilometres (approximately 60 Earth radii). He found the Moon's inward acceleration in its orbit to be 0.0027 metre per square second, the same as (1/60)2 of the acceleration of a falling object at the surface of the Earth. Because Newton could relate the two accelerations to a common interaction. He deduced that the gravitational force between bodies diminishes as the inverse square of the distance between the bodies. A further result, that the mass of the Earth acts gravitationally on the outside world as if the mass were concentrated at the planet's centre. Newton proved it to be true for all spherically symmetric bodies. Newton saw that the gravitational force between bodies must depend on the masses of the bodies. Since a body of mass M experiencing a force F accelerates at a rate F/M, a force of gravity proportional to M would be consistent with Galileo's observation that all bodies accelerate under gravity toward the Earth at the same rate. In Newton's equation. Fn is the magnitude of the gravitational force acting between masses Mi and M2 separated by distance ri2. _GMX.M2 A2 ~ i (2) The constant G is a quantity with the physical dimensions (length)/(mass)(time)2 its numerical value depends on the physical units of length mass and time used. The attractive force of a number of bodies of masses Mi on a body of mass M is F = ^Ş- LI (3) Where Si means that the vector forces due to all the attracting bodies must be added together. This is Newton's gravitational law essentially in its original form. IX Equations (1) and (2) above con be used to derive Kepler's third law for the case of circular planetary orbits. By using the expression for the acceleration A in equation (1) for he force of gravity for the planet G.Mp.M, / R2 divided by the planet's mass M the following equation, in which Ms is the mass of the Sun. is obtained: GMS _ 4u2R R2 ~ T2 or R3 = GMS V4k2 ) (4) Newton was able to show that all three of Kepler's observationally derived laws follow mathematically from the assumption of his own laws of motion and gravity. In all observations of the motion of a celestial body, only the product of G and the mass can be found. Newton first estimated the magnitude of G by assuming the Earth's average mass density to be about 5.5 times that of water (somewhat greater than the Earth's surface rock density) and by calculating the Earth's mass from this. Then, taking Me and rE as the Earth's mass and radius, respectively, the value of G was that numerically comes close to the accepted value of 6.6726x10- ms'2/kg ME Comparing g = - ^- above for the Earth's surface acceleration g with the R3/T2 r* ratio for the planets, a formula for the ratio of the Sun's mass. Ms to Earth's mass. Me, was obtained in terms of known quantities. Re being the radius of the Earth's orbit: - *- = ^ = 325.000 (6) MB gTB\ E It was already known in Newton's day that the Moon does not move in a simple Keplerian orbit. Later, more accurate observations of the planets also showed discrepancies from Kepler's laws. The motion of the Moon is particularly complex: however, apart from a long-term acceleration due to tides on the Earth, it can be accounted for by the gravitational attraction of the Sun, the Earth, and the other planets. In addition, the gravitational attraction of the planets for each other explains almost all the features of their motions. The exceptions are nonetheless important. Uranus the seventh planet from the Sun, was observed to undergo variations in its motion that could not be explained by perturbations from Saturn, Jupiter, and the other planets. Measurements of the motion of the innermost planet, Mercury,, over an extended period led astronomers to conclude that the major axis of this planet's elliptical orbit processes in space at a rate 43 are seconds per century faster than could be accounted for from perturbations of the other planets. In this case, however, no other bodies could be found that could produce this discrepancy, and very slight modification of Newton's law of gravitation seemed to be needed. Einstein's theory of relativity precisely predicts this observed behaviour of Mercury's orbit. In the last three hundred years long, the development of gravitation understanding became a section of the classical mechanics. When somebody would like to find out how a body produces gravitational force onto another one three things must be in consideration: First, the shape of the body. Second, the element shape that it is easy to locate the centre of gravity of the element And third, the coordinate system that would be helpful demonstrating the calculations. As long as I see when I was studying on the subject I found out that the traditional mass element is very small (differential) elements. The selection of the elements will determine whether the resulting integral be a single, a double, or a triple. If we would like to have an exact solution, for any body that doesn't have symmetrical planes then we need to know the orientation of the gravitational force on an element In my study I tried to find out a cube's gravitational attraction on a very small body that can be accepted as a point mass, by dividing the cube into different number of finite cube elements. The least of the element number may be one. By trying to calculate it we will find out it is like a point mass. After this attempt we can conclude that the least number is eight and if we continue using integers, next possible element number is twenty- seven and it goes on the cube of four, five etc. I went on trying the higher number of cube elements as much as one million. I found out that it has no difference by using two- thousand or one million. Another view is to make the results compare with a point mass that as if it was located on the cube's centre of gravity. I divided the resultant force by the point mass' gravitational force that had already been mentioned above. At the beginning of my study I accepted the cube dimension 1, but I saw that the higher the number of elements the smaller the cube elements dimensions. All the time I had to use ratios. I decided to use the cube elements dimensions one each time that I took more elements. The dimensions of the cube become larger and larger. The location of the mass m was bigger each time. I divided the distances by the radius of the sphere that it's mass is equal to the mass of the cube. So my results weren't affected by the dimensions of the cubes. It is easy to use the cube element dimensions as 1 but I fount out it would be easy if I accept the dimensions 2 because of the location of the element's centre of gravity would be integer. Xi I also decided to calculate the desired results for any directions instead of just for x directions. So I applied vectors on finding the element's centre of gravity. In summary we can accept a cube as a point mass if some approximations can be convenient. As an example, if one percent approximation is acceptable, than we can accept the cube as a point mass after two and half times of equal sphere radius.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1996

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1996

##### Anahtar kelimeler

Sonlu elemanlar yöntemi,
Yer çekimi kuvveti,
Finite element method,
Gravitational force