2020년 3월 21일 토요일

01.07 - 뉴튼 중력법칙(Newtonian Gravity)

01.07 - 뉴튼 중력법칙(Newtonian Gravity) [커세라 강의페이지]



Gravity is the force that keeps us standing on Earth's surface. It's the reason that a ball thrown upwards falls back down towards the ground. It was Newton who first realized that this force, gravity, doesn't just affect physical objects here on earth, but is also responsible for the motion of the stars and planets. Gravity keeps the earth moving in orbit around the sun and the sun in orbit around the supermassive black hole at the center of the Milky Way galaxy. Gravity is a central principle in black hole physics because it's gravity that gives black holes their extreme properties.

뉴튼의 중력법칙(또는 만유인력의 법칙) 모르는 사람이 없을 것이다. 우리가 지구에 서있게 하는 힘의 근원이자 지구가 태양을 돌게 하는 원리이고 은하 중심에 엄청난 질량의 블랙홀(은하의 모든 별들을 붙들고 회전하게 만들 만큼)을 설명하는 원리는 바로 만유인력이다. 블랙홀 연구의 핵심 이론이다.

It was Newton who provided the first empirical description of how gravity works. Although Newton was the first to explain gravity mathematically, almost exactly 100 years earlier in 1589, Galileo Galilei was busy investigating gravity and his observations greatly advanced our understanding of the interaction between objects and their masses.

뉴튼은 중력이 어떻게 작용하는지 실증적으로 보여줬다. 뉴튼 이전에 100년전에 갈릴레오 갈릴레이는 질량을 물체 사이에 상호작용이 있다는 사실을 증명하려 애썼다.

Galileo theorized that falling objects of different masses would fall at the same rate contrary to the Aristotelian belief that heavy objects fall faster than light objects.

갈릴레오는 무거운 물체가 빠르게 낙하한다는 아리스토텔레스 적인 믿음에 반하여 질량이 다른 물체가 동일한 비율로 낙하한다는 이론을 주장했다.

It's famously claimed that to prove this idea, Galileo climbed up the Leaning Tower of Pisa and dropped two cannon balls with different masses one heavier and lighter. He observed that if both cannonballs were dropped simultaneously they hit the ground at precisely the same time independent of their weights. Galileo made the mistake of assuming that the gravitational force was a constant between two objects with no relationship to the distance between them. Historians disagree whether this experiment really took place because it's first mentioned almost 65 years after it's supposedly took place in a biography of Galileo by Vincenzo Viviane.

갈릴레오는 피사의 기울어진 탑에 올라 이 낙하 실험을 했다고 한다. 갈릴레오의 두 물체사이의 거리와 상관 없이 중력은 동일하다는 가정은 사실 틀렸다. [뉴튼의 만유인력법칙: 중력은 거리의 제곱에 반비례한다.] 실제로 갈릴레오가 피사의 탑 실험을 했는지 역사학자들이 의문을 가지고 있긴하다. 갈릴레오의 전기에 따르면 시기가 맞지 않다고 주장한다.



One experiment done during Apollo 15's mission to the moon demonstrates the principle that Galileo addressed. At the end of the last moon walk, astronaut David Scott performed the same demonstration that Galileo did with a hammer and a feather in the vacuum of space. The result of course is visible in this famous video.

아폴로 15호의 임무중 이 낙하 실험이 포함 되었다. 망치와 독수리 깃털을 양손에 각각 들고 진공에서 떨어트렸고 동시에 달표면에 떨어지는 것을 실증해 줬었다.

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"In my left hand I have a feather, in my right hand, a hammer. I guess one of the reasons we got here today was because of a gentleman named Galileo a long time ago who made a rather significant discovery about falling objects in gravity fields. And we thought that where would a better place to confirm his findings than on the moon. So we thought we'd try it here for you and the feather happens to be appropriately a falcon feather or a falcon and I'll drop the two of them here and hopefully they'll hit the ground at the same time. How about that? So, Mr. Galileo was correct in his findings."
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Shortly after Galileo's death, mathematician and astronomer Johannes Kepler observed that planets trace ellipses through the solar system as they orbit the sun. Kepler famously described the motion of the planets mathematically, laying the groundwork for the second last piece of the gravity puzzle which was solved by Christian Huygens, who in the 1660's described the law of centrifugal force. Together with the help of Edmund Halley, Christopher Wren and Robert Hooke, Isaac Newton had all the clues he needed to piece together the mathematical description of gravity.

갈릴레오는 중력에 대한 숙제를 남기고 죽었다. 케플러는 평생 관측을 기반으로 행성의 운행법칙을 만들었다. 하지만 수학적인 증명하진 못했다. 하위헌스의 원심력(centrifugal force)을 수학적으로 기술함으로써 행성의 운행의 원리가 중력일지 모른다는 수수께끼 단서를 제공했다. 헬리, 렌, 후크와 함께 뉴튼은 그동안의 파편들을 모아 중력법칙을 수학적으로 기술했다.



In 1687 Newton's book Philosophiae Naturalis Principia Mathematica which translate to the Mathematical Principles of Natural Philosophy. Newton laid the mathematical foundations to explain all of gravitationally related phenomenon including apples falling from trees and planets in orbit around stars.

1687년 뉴튼은 자연철학을 수학적으로 풀어내기 위해 이책을 냈다. 뉴튼은 이책에 중력과 관련된 설명을 담았는데 나무에서 사과가 떨어지는 것부터 행성이 태양 주위를 공전하는 월리를 포함하였다.




Gravity is an attractive force between two objects that have mass. Any object that we talk about in this course with the exception of light has mass.

질량을 가진 두 물체 사이에 인력(끌어당기는 힘)이 있다는 것이 만유인력이다. '광자'를 제외하고 모든 물체는 질량이 있다.

[그런데 질량이 없는 '광자'는 왜 무거운 질량에 끌리는가? 질량에 끌리는 이상한 힘이 작용한것이 아니라 무거운 질량 때문에 공간이 휘었기 때문이라고 밝혔다. 빛은 늘 그랬듯이 가장 가까운 경로를 택했을 뿐이다.]

The earth has mass, I have mass, and you have mass. There's therefore a gravitational attraction between the earth and me, the earth and you, but also between you and I at any given time. [질량을 가진것 들 사이에 인력(끌림힘)이 작용한다. 주어진 '시간'에!]

The mathematical description of the force of gravity needs to take into account the mass of both objects, and also the distance between them. In order to get useful information out of any equation, we also need a universal gravitational constant to tell us how strong the force will be given the masses and the distances.

[두 질량 사이에 끄는힘이 존재한다. 이 힘은 두 질량체의 거리와도 상관있다. 거리 역제곱 법칙은 물리공식에 자주 찾아볼 수 있다. 두 전하사이의 전기력.]

Let's call the mass of the larger object capital M, and the mass of the smaller object little m. The distance between the two objects will be measured by a lowercase r and the universal gravitational constant will be denoted as a capital G.



The force of attraction between two objects will be directly proportional to their masses, but inversely proportional to the square of the distances separating them. Direct proportionality means that the force F will be equal to the universal gravitational constant G times capital M times little m. Finally, because of the inverse-square relationship, we divide the whole right hand side of the equation by r to the power of two. This equation is called Newton's universal law of gravitation, and calculates the force between two objects no matter what their masses.



In order to use this equation, we need to consider the units of each term. G, the universal gravitation constant has a value of 6.67 times 10 to the minus 11 in units of Newton meters squared per kilogram squared, and that's a mouthful.

[물리 공식에 늘 따라붙는 '상수'들이 있다. 아주 작은 값이라는 점 외에 단위(혹은 차원)을 유심히 살펴보라. 이 복잡한 단위들은 어떻게 붙었을까? 차원해석요약 및 연습문제]



To make these units cancel out, you can see that capital M and little m will cancel out the kilograms squared term, and that the distance squared cancels out the meters squared term leaving behind Newtons which are a measurement of force. Notice how tiny the gravitational constant is.

If we ask ourselves how much attractive forces felt between two objects each weighing one kilogram and separated by one meter, the answer of course is G times one kilogram, times one kilogram divided by one meter squared. So 6.67 times 10 to the minus 11 Newtons or 66.7 picoNewtons. For comparison 67 picoNewtons is about how hard you have to pull the two ends of a DNA molecule in order to have them unravel. But gravity acts on much larger scales and is therefore comparatively weak.

Let's compare 66 picoNewton's to the force of gravity that I feel due to the Earth. Since Earth weighs 5.97 times 10 to the 24 kilograms and I weigh about 75 kilograms, in order to calculate the force of gravitational attraction, we'll replace capital M with Earth's mass and little m with my mass. We also need to know how far apart the center of the Earth is from the center of me.



Let's take the radius of the Earth's surface to be r and replace it with a value of 6,378.1 kilometers, which we have to convert into meters. So, 6,378,100 meters, which we then square. Finally, we replaced the universal gravitation constant G with its value of 6.67 times 10 to the minus 11. And its units Newton meter squared divided by kilograms squared. Together, the units of meters cancel each other out as do the units of kilograms leaving Newtons in the result. I'll get my calculator out and plug in the math and I get the result of 735 Newtons.

지구와 지표면에 서있는 나 사이에 인력은 얼마나 될지 계산을 해보자. 만유인력을 계산할때 거리는 질량체의 무게중심 사이의 거리다. 질량이 넓은 부피에 분포 된 경우 힘은 무게중심에 집중된것으로 간주한다. 지구의 반지름은 6,378.1 킬로미터다. 지표면에 서있는 나와 지구라는 질량체의 무게 중심 거리는 지구의 반지름이다. 나와 지구 사이의 인력은 735N(뉴튼) 이다.



So, I'm being pulled towards the center of the Earth with a force of 735 Newtons. The unit of force Newtons is sometimes difficult to put into context. It's related classically with the acceleration of a mass by Newton's second law F equals ma, which relates the force on a mass to how quickly the mass accelerates. Since I feel the force of gravity as 735 Newtons, I can calculate my acceleration due to gravity by dividing my mass 75 kilograms which results in acceleration of 9.798 meters per second squared. You might recognize the coincidence.

뉴튼 N은 힘의 단위다. 힘(F)은 질량(m) 곱하기 가속도(a) 다. [F=ma] 나의 몸무게 75킬로그램인 질량을 가속 시키면 735뉴튼이 될까? 735뉴튼을 75킬로그램으로 나누자. 9.798 미터/초 제곱이다. 이를 지구 질량이 끄는 인력(중력) 때문에 생긴 힘에서 얻은 가속도이므로 '중력 가속도'라고 부르자.

The acceleration I feel is very close to the value of Earth's acceleration due to gravity, which is often denoted as a little g, and has an average value of 9.807 meters per second squared. The reason that these two numbers are different is because the strength of Earth's gravity varies over a surface. For example, you weigh about half a percent heavier, when you're at the Earth's poles than you do when you're along its equator. In fact, Earth's gravity varies a lot over its surface because of the different densities of rocks and the different geography of regions.

지표면에서 평균 중력 가속도는 9.807 미터/초의 제곱이다. 위에서 계산한 값과 다른 이유는 지표면의 위치에따라 지형의 고도가 다르기 때문이다.

Earth's gravity diminishes by about one fifth of one percent from earth's surface to an altitude of five kilometers. So, your height above or below sea level is also a factor. But geology can account for another one 100th of a percent difference in gravity. This map of the globe represents the difference in Earth's gravity from the average value. Red indicates stronger gravity and blue indicates weaker gravity.

지형적 특성에 따라 지표면 고도가 다르기 때문에 지구 중심에서 반경이 다르므로 중력 가속도의 차이를 보인다.

The data was collected by a pair of satellites called GRACE, the Gravity Recovery and Climate Experiment. GRACE uses changes in Earth's gravity to measure changes to huge masses of ice in our polar regions. If gravity there decreases, scientists can determine how much of the glacial ice is melting in those regions and this data can even tell where vast underground reservoirs of water are filling up.

두개의 GRACE 위성을 띄워 중력 가속도 분포 지도를 작성하였다. 지형의 고도뿐만 아니라 극지의 무거운 빙산에 의해서도 중력 가속도 값이 달라지기도 한다. 과학자들은 중력 가속도의 변화를 측정하여 극지의 빙산이 줄어드는지 측정한다[빙산이 녹는 이유는 기후변화 때문이다. GRACE 위성은 중력 가속도를 측정하여 지구의 기후변화를 관측한다].



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If Newton had accomplished nothing but the mathematical formulation for the law of gravity, he would still go down as one of history's greatest physicists. But he contributed much more to our understanding of the universe. He revolutionized our understanding of motion, forces, and mechanics with his three laws of motion.

만일 뉴튼이 그저 중력법칙을 수식으로 남겨두기만 했다면 그는 그저 역사속의 위대한 물리학자로 남았을 것이다. 하지만 뉴튼의 운동법칙은 우주를 이해하는데 혁명적으로 기여하였다.

Newton's three laws can be stated in the following way:

뉴튼의 세가지 법칙,
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 Newton's first law states, an object at rest will stay at rest unless a force acts upon it. An object in motion especially uniform motion, will stay in motion unless a force acts upon it as well. It's interesting that we distinguish between an object at rest and an object moving with a uniform velocity.



As we get deeper into this course, you'll understand that these two examples, an object at rest and an object in uniform motion are themselves within what we call an inertial frame of reference.

뉴튼 제1법칙은 정속운동에 관한 법칙. 정지한 물체도 속도가 0인 정속운동과 같다. 정지 또는 정속운동하는 물체는 관성좌표계(inertial frame of reference)에서 기술된다.

We could also think about a rocket moving in outer space at a constant speed. Unless the rocket were to fire its thrusters to exert a force in the opposite direction, the rocket will continue moving at a constant speed forever.

Newton's first law is also called the law of inertia. Inertia is the resistance an object has to changing its state of motion.

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Newton's second law states, an object acted upon by a force will experience an acceleration in proportion to its mass. This is the famous formulation which is described by the equation F equals ma that we used earlier.



Any force acting on an object will produce an acceleration in proportion to the mass of the object. So, for any given force, a small mass will accelerate quickly but a large mass would accelerate slowly.

뉴튼 제2법칙은 힘과 가속도의 법칙이다. 힘이 가해진 물체는 가속된다. 가속의 정도는 물체의 질량에 비례한다. 갘은 힘이 주어졌을 때 가벼운 물체는 빠르게 가속하고 무거운 물체는 느리게 가속된다.

Think about it using a small motor on both a small boat and a huge ship. The motor delivers the same amount of force but the ships accelerate at much different rates.

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Newton's third law states, for every action there's an equal and opposite reaction. Newton's third Law is a little hard to wrap your head around, but it basically means this, any force which is imparted on an object must also be imparted equally upon another.



In other words, for all the force of Earth's gravity pulling upon me, I'm also exerting a force pushing down upon the Earth. This point confused me for some time as a student.  Why is it that we say earth gravity has a value of 9.81 meters per second squared? That's a measure of acceleration. Well, when I'm standing still on Earth, Earth surface is not actually accelerating anywhere. My acceleration is zero. The truth of the matter is that the strength you exert to stand is the force pushing back on the Earth. The net force between you and the Earth ends up zero.



뉴튼 제3법칙은 작용과 반작용의 법칙이다. 우리가 지구 표면에서 버티는 원리는 지구의 중력가속도로 우리몸을 끌어당기는 힘과 지표면이 이와 반대되는 힘으로 밀어내기 때문이다. 끌어들이는 힘과 버티는 힘이 상쇄되기 때문에 가속도가 0이다.

Well, what about if you aren't standing on earth's surface but you've gone sky diving and you're falling freely through the air? In this case, you are accelerating at 9.81 meters per second squared but you should also consider that Earth is accelerating towards you. The forces are the same for you and for the Earth, but the acceleration of the two is different because you and the Earth have vastly different masses. In this case, Earth would accelerate towards you at a tiny rate of about 1.23 times 10 to the minus 22 meters per second squared.



당신이 지표면보다 높은 곳에 있다고 하자. 말하자면 스카이다이빙의 경우 처럼 공기중에서 지구를 향해 떨어진다. (공기저항이 없다면 자유낙하라고 한다.) 무거운 지구가 당신을 향해 9.8 미터/초 제곱의 가속도로 당신에게 달려든다. 반대로 가벼운 당신은 지구를 향해 1.23의 10의 마이너스 22승 미터/초 제곱으로 지구로 달려든다. 즉 무거운 지구와 가벼운 당신의 가속도가 다르다.

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Newtons second law of motion means that when we apply a force to an object, the object will accelerate. Therefore, when you apply a gravitational force to an object, it will accelerate.



If we take Newton's law of universal gravitation F is equal to GMm over R squared and set the force equal to F in Newton's second Law F equals ma, then the little m masses on both sides of the equations cancel each other out resulting in the equation a is equal to GM divided by r squared.

This equation provides a simple way of calculating the acceleration due to gravity. When I stand on the surface of a planet that has a radius r and a mass capital M, then the acceleration due to gravity at the surface is simply given by G times big M divided by r squared.



If the planet is earth, we use the symbol G to represent the acceleration due to gravity. We say that a body has a gravitational field when it has the potential to accelerate nearby objects towards it. Newton's equations are robust enough to send rockets to other planetary bodies. In order to do so, we need to further tie the concept of gravitational potential energy, the energy required to climb through a gravitational field in order to calculate escape velocity.

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[케플러 행성운동 정리]
행성운행에 관한 케플러 법칙(사실은 타운원운동과 각운동량 보존법칙, 구심력(중심력)만유인력법칙등 여러 물리법칙을 모아 행성의 운동에 적용한 것이므로 정리라고 해야 맞다.) 케플러 법칙을 유도하는 방법은 다양하나 벡터 해석(이과생을 위한 벡터 미적분)을 배웠으니 이를 응용해보자.

[참고]
[1] 케플러법칙유도 (개인블로그)
[2] 케플러의 법칙 유도 (개인블로그)
[3] 케플러 법칙(나무위키)
[4] 중심력(나무위키)
[5] 케플러 행성운동법칙(위키백과)
[6] 각운동량(위키백과)
[*] 구구단만 알아도 미적분... (3. 고전역학)[구구단만 알아도 미적분]
[*] 1.2 뉴튼역학:자유낙하[마구잡이 수학]
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[스카이 다이버의 미분방정식]
[*] W1.9/8강 선형 1차 미방 응용:종말속도(Application:Terminal Velocity)




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