Fig. 7–10.A cluster of galaxies. 一个星系的簇。
Gravity appears to exist at even biggerdimensions, as indicated by Fig. 7–10,which shows many “little” things clustered together. This is a cluster ofgalaxies, just like a star cluster. Thus galaxies attract each other atsuch distances that they too are agglomerated into clusters. Perhapsgravitation exists even over distances of tens of millions of lightyears; so far as we now know, gravity seems to go out forever inversely as thesquare of the distance.
万有引力似乎存在于更大的尺寸中，如图7-10所示，其中有很多“小的”东西，聚集在一起。这是一个星系的簇。这样，星系就在这么大的尺寸中，相互吸引，所以它们也是聚集成簇。或许万有引力，还存在于几千万光年的尺寸之中；只要我们现在知道，万有引力，反比于距离的平方，向外无限扩展。

Fig. 7–11.An interstellar dust cloud. 图7-11 一个星际灰尘云。
Not only can we understand the nebulae, butfrom the law of gravitation we can even get some ideas about the origin of thestars. If we have a big cloud of dust and gas, as indicated in Fig. 7–11, thegravitational attractions of the pieces of dust for one another might make themform little lumps. Barely visible in the figure are “little” black spots whichmay be the beginning of the accumulations of dust and gases which, due to theirgravitation, begin to form stars. Whether we have ever seen a star form or notis still debatable. Figure 7–12 showsthe one piece of evidence which suggests that we have. At the left is a pictureof a region of gas with some stars in it taken in 1947, and at the right isanother picture, taken only 7 years later, which shows two new bright spots. Has gasaccumulated, has gravity acted hard enough and collected it into a ball bigenough that the stellar nuclear reaction starts in the interior and turns itinto a star? Perhaps, and perhaps not. It is unreasonable that in only seven yearswe should be so lucky as to see a star change itself into visible form; it ismuch less probable that we should see two!
从万有引力规律出发，我们不仅能够理解星云，甚至对于恒星的起源，也能有些想法。如图7-11所示，如果我们有一大团尘埃和气体，尘埃相互之间的万有引力的吸吸引，没准儿会让它们形成小块。在图中，有些小的黑点，几乎看不到，它们或许就是尘埃和气体的最初累积，由于万有引力，尘埃和气体开始形成恒星。我们是否曾看到过一个恒星的形成，尚有争议。图7-12所示证据，认为我们看到过。在左边，是一个气体区域的照片，里面有一些恒星，摄于1947年，在右边，是另外一张照片，摄于7年前，它显示了两个新的亮点。气体是否累积了，万有引力是否扮演了一个强的角色，把气体收集进一个球中，此球足够大，以至于在其中，开始了星际的原子核反应，并把它变成了恒星？或许如此，或许并非。只用了七年，恒星就把自己变成了可见的形式，且我们如此幸运地看到了，这不合理；更别说看到两个了。

Fig. 7–12.The formation of new stars? 图7-12 新星的形成？

7–6Cavendish’s experiment 7-6 卡文迪什的实验

Fig. 7–13.A simplified diagram of theapparatus used by Cavendish to verify the law of universal gravitation for smallobjects and to measure the gravitational constant G . 图7-13 卡文迪什使用的仪器的简化图，为小的对象，证实宇宙间万有引力的规律，并测量万有引力常数G。
Gravitation, therefore, extends overenormous distances. But if there is a force between any pair of objects,we ought to be able to measure the force between our own objects. Instead of havingto watch the stars go around each other, why can we not take a ball of lead anda marble and watch the marble go toward the ball of lead? The difficulty ofthis experiment when done in such a simple manner is the very weakness or delicacyof the force. It must be done with extreme care, which means covering theapparatus to keep the air out, making sure it is not electrically charged, andso on; then the force can be measured. It was first measured by Cavendish withan apparatus which is schematically indicated in Fig. 7–13. Thisfirst demonstrated the direct force between two large, fixed balls of lead andtwo smaller balls of lead on the ends of an arm supported by a very fine fiber,called a torsion fiber. By measuring how much the fiber gets twisted, one canmeasure the strength of the force, verify that it is inversely proportional tothe square of the distance, and determine how strong it is. Thus, one mayaccurately determine the coefficient G in the formula
F=Gmm/r2.
因此,万有引力可以扩展到非常遥远的距离。但是,如果在任何一对对象之间，都有力，那么，我们就应该可以测量我们自己的对象之间的力。除了可以观察恒星转动之外，为什么我们就不能拿一个铅球和一个石球，来观察石球是如何向铅球走的呢？以这种简单的方式，来做这个实验时所遇到的困难，就是这个力非常弱、非常微妙。做这个实验，必须非常仔细，也就是说，仪器要被盖起来，以防止空气出来，保证没有充电等等，然后，就可以测这个力了。这个实验，最初是由卡文迪什做的，仪器的原理，如图7-23所示。此第一个实验，是这样的：一个支杆，一端是两个大的固定的铅球，一端是两个较小的铅球，支杆由一个非常精细的纤维吊着，它被称为可扭转纤维，待测量的，是支杆两端的力。通过测量纤维的扭曲度，我们就可以测量出力的强度，证实它与距离的平方成反比，从而定义它有多强。这样，就可以准确地得出方程F=Gmm/（rr）中的系数G。

All the masses and distances are known. You say, “We knew it already for the earth.” Yes, but we did not know the mass of the earth. By knowing G from this experiment and by knowing how strongly the earth attracts, we can indirectly learn how great is the mass of the earth! This experiment has been called “weighing the earth” by some people, and it can be used to determine the coefficient G of the gravity law. This is the only way in which the mass of the earth can be determined. G turns out to be
6.670×10（−11次方） newton⋅m*m/kg*kg.
{公式中}所有的质量和距离，都是已知的。你说：“我们已经知道地球的半径了”。是的，但我们还不知道地球的质量。通过这个实验，可以知道G，及通过知道地球的吸引有多强，我们就可以间接地得到地球的质量！这个实验被一些人称为“称地球的重量”，它也可以被用来决定万有引力规律的系数G。这是唯一决定地球质量的方法。得到的G就是：6.670×10−11 newton⋅m2/kg2。

It is hard to exaggerate the importance of the effect on the historyof science produced by this great success of the theory of gravitation. Comparethe confusion, the lack of confidence, the incomplete knowledge that prevailedin the earlier ages, when there were endless debates and paradoxes, with the clarityand simplicity of this law—this fact that all the moons and planets and starshave such a simple rule to govern them, and further that man could understandit and deduce how the planets should move! This is the reason for the successof the sciences in following years, for it gave hope that the other phenomenaof the world might also have such beautifully simple laws.
这一成果，是万有引力理论的巨大成功，在科学史上，有着巨大的影响，其重要性，已经很难被夸大了。现在，做个比较，在以前，不完整的知识普遍流行，没完没了的辩论和各种悖论，带来的是缺乏自信、困惑，而这个规律，带来的是清晰和简单，因为这个规律告诉了我们这一事实，所有的月亮、行恒和恒星，都由这么简单的规则统治着，更进一步，人们应该理解它，并且推出行星应该如何运动！在接下来的岁月中，科学取得了巨大的成功，这就是原因，因为它带来了希望：即世界上其他的现象，或许也有这种漂亮而简单的规律。

7–7What is gravity? 7-7 什么是万有引力？
But is this such a simple law? What aboutthe machinery of it? All we have done is to describe how the earth movesaround the sun, but we have not said what makes it go. Newton made nohypotheses about this; he was satisfied to find what it did withoutgetting into the machinery of it. No one has since given any machinery.It is characteristic of the physical laws that they have this abstractcharacter. The law of conservation of energy is a theorem concerning quantitiesthat have to be calculated and added together, with no mention of themachinery, and likewise the great laws of mechanics are quantitativemathematical laws for which no machinery is available. Why can we usemathematics to describe nature without a mechanism behind it? No one knows. Wehave to keep going because we find out more that way.
但是，这个规律，就是这样一种简单的规律吗？它的机制是什么呢？我们现在所完成的，就是描述了地球是如何绕着太阳转的，但是，我们并没有说，是什么使得它如此。牛顿对此，没有做任何假设；只发现了地球在做什么，但并没有涉及到后面的机制，对此他很满意。从那以后，没有人给出过任何机制。物理学规律的特点就是，它们拥有这种抽象的特性。能量守恒规律就是这样一种理论，它所关心的，是那些应该被计算和加起来的量，而不用提到机制，同样地，伟大的力学规律，就是定量的数学的规律，对于它们，没有任何机制可以得到。为什么我们可以用数学来描述自然，而却不知道自然后面的机制是什么？这没人知道。我们必须往前走，因为以这种方式，我们找到了更多的东西。

Next we shall discuss the possible relationof gravitation to other forces. There is no explanation of gravitation in termsof other forces at the present time. It is not an aspect of electricity or anythinglike that, so we have no explanation. However, gravitation and other forces arevery similar, and it is interesting to note analogies. For example, the forceof electricity between two charged objects looks just like the law of gravitation:the force of electricity is a constant, with a minus sign, times the product ofthe charges, and varies inversely as the square of the distance. It is in theopposite direction—likes repel. But is it still not very remarkable that thetwo laws involve the same function of distance? Perhaps gravitation andelectricity are much more closely related than we think. Many attempts havebeen made to unify them; the so-called unified field theory is only a veryelegant attempt to combine electricity and gravitation; but, in comparing gravitationand electricity, the most interesting thing is the relative strengths ofthe forces. Any theory that contains them both must also deduce how strong thegravity is.
万有引力，与其他的力之间，可能会有关系，下面我们将讨论之。当前，依据其他力，而对万有引力的解释，还没有。它并不是电力的某一方面，或任何类似的事情，所以，我们没有这种解释。然而，万有引力与其它的力，非常类似，所以，做些类比，也很有趣。例如，两个充电对象之间的电力，看上去，很像万有引力的规律：电力是一个常数，有一个负号，乘以电荷数，变化与距离的平方成反比。它在相反的方向，好像是排斥。这两个规律，有一个共同点，即都包含了距离的功能，但这一点，还不是非常值得注意。或许，万有引力与电力的关系，要比我们想的，更为紧密。人们做了很多努力，尝试把它两，联合起来；所谓的统一场理论，就是想把万有引力与电力，结合起来，只是一个非常优美的尝试；但是，在万有引力与电力的比较中，最有趣的事情，是诸力的相对强度。任何包含它们两个的理论，都应该能推出：万有引力有多强。

If we take, in some natural units, therepulsion of two electrons (nature’s universal charge) due to electricity, andthe attraction of two electrons due to their masses, we can measure the ratio ofelectrical repulsion to the gravitational attraction. The ratio is independentof the distance and is a fundamental constant of nature. The ratio is shown inFig. 7–14.The gravitational attraction relative to the electrical repulsion between twoelectrons is 1 divided by 4.17×1042 ! The question is, where does such a large number come from? It is notaccidental, like the ratio of the volume of the earth to the volume of a flea.We have considered two natural aspects of the same thing, an electron. Thisfantastic number is a natural constant, so it involves something deep innature. Where could such a tremendous number come from? Some say that we shallone day find the “universal equation,” and in it, one of the roots will be thisnumber. It is very difficult to find an equation for which such a fantasticnumber is a natural root. Other possibilities have been thought of; one is torelate it to the age of the universe. Clearly, we have to find anotherlarge number somewhere. But do we mean the age of the universe in years?No, because years are not “natural”; they were devised by men. As an example ofsomething natural, let us consider the time it takes light to go across a proton,10−24 second. If we compare this time with the age of the universe,2×1010 years, the answer is 10−42 . It has about the same number of zeros going off it, so it has beenproposed that the gravitational constant is related to the age of the universe.If that were the case, the gravitational constant would change with time,because as the universe got older the ratio of the age of the universe to thetime which it takes for light to go across a proton would be graduallyincreasing. Is it possible that the gravitational constant is changingwith time? Of course the changes would be so small that it is quite difficultto be sure.
在某些自然的单元中，如果我们把两个电子（自然中的普遍电荷）之间的排斥，归于电，把它们之间的吸引，归于质量，那么，我们就可以测量电子排斥与重力吸引的比率。这个比率，独立于距离，是一个基础性的自然常数。图7-4所示，就是这个比率。万有引力的吸引，相对于两个电子间的电排斥，是1除以4.17×1042！问题是，这么大的一个数字，从何而来？它不是偶然的，比如，地球的体积，与一个虱子的体积的比率。我们已经考虑了同一个事物（即电子）的两个不同的自然方面。这个奇异的数字，是一个自然的常数，所以，它就包含着自然中某种深的东西。这样一个巨大的数字，从何而来？有人说，迟早某天，我们就会发现那个“普遍方程”，其中的一个根，就会是这个数。要找到一个其根为此奇异数字的方程，非常困难。其他可能性，也被思考过，一种就是，让它与宇宙的年龄相关。很清楚，我们必须在其他的地方，找到另外一个大的数字。但是，我们是用年，来谈论宇宙的年龄吗？不，因为年不是“自然的”，它是人发明出来的。举一个自然的例子，让我们考虑，光通过质子的时间，10−24秒。如果我们把这个时间，与宇宙的年龄2×1010相比，那么，答案就是10−42。在它后面，有几乎同样数目的零，所以，重力常数就被认为与宇宙的年龄有关。果真如此，重力常数就会随着时间而变化，因为随着宇宙变老，那么，‘宇宙的年龄’与‘光通过质子的时间’的比率，也会逐渐增加。重力常数随着时间而改变，这可能吗？当然，这个变化将会是如此之小，以至于，很难确定。

One test which we can think of is todetermine what would have been the effect of the change during the past 109 years, which is approximately the age from the earliest life onthe earth to now, and one-tenth of the age of the universe. In this time, thegravity constant would have increased by about 10 percent. It turns out that if we consider the structure of thesun—the balance between the weight of its material and the rate at whichradiant energy is generated inside it—we can deduce that if the gravity were 10 percent stronger, the sun would be much more than 10 percent brighter—by the sixth power of the gravityconstant! If we calculate what happens to the orbit of the earth when thegravity is changing, we find that the earth was then closer in.Altogether, the earth would be about 100 degrees centigrade hotter, and all of the water would not have beenin the sea, but vapor in the air, so life would not have started in the sea. Sowe do not now believe that the gravity constant is changing with the ageof the universe. But such arguments as the one we have just given are not veryconvincing, and the subject is not completely closed.
有个测试，我们能够想到，就是去决定，在过去的109年中，变化的影响是什么，此109年，大约是从地球上最早的生命开始，到现在的地球的年龄，且是宇宙年龄的1/10。在此期间，重力常数应该增长了大约10%。结果就是，如果我们考虑太阳的结构—一种平衡，即构成太阳的材料的重量，与其内部辐射能产生的速率，之间的平衡--，那么，我们就可以推出，如果万有引力增强了10%，则太阳的亮度，将会远大于10%--通过万有引力常数的六次方{计算得出}！如果我们计算，当万有引力改变时，地球的轨道发生了什么，那么就会发现，地球轨道应该更小了{？}。总而言之，地球应该更热，大约高100摄氏度，所有的水，还没有在海里，而只是空气中的蒸汽，所以，生命还没有在海中开始。所以，我们现在并不相信，万有引力常数随着宇宙的年龄而变化。但是，我们现在所提供的这种论据，说服力并不强，所以，这个话题，并未完全关闭。

It is a fact that the force of gravitationis proportional to the mass, the quantity which is fundamentally ameasure of inertia—of how hard it is to hold something which is goingaround in a circle. Therefore two objects, one heavy and one light, going arounda larger object in the same circle at the same speed because of gravity, willstay together because to go in a circle requires a force which isstronger for a bigger mass. That is, the gravity is stronger for a given massin just the right proportion so that the two objects will go aroundtogether. If one object were inside the other it would stay inside; itis a perfect balance. Therefore, Gagarin or Titov would find things“weightless” inside a space ship; if they happened to let go of a piece of chalk,for example, it would go around the earth in exactly the same way as the wholespace ship, and so it would appear to remain suspended before them in space. Itis very interesting that this force is exactly proportional to the masswith great precision, because if it were not exactly proportional there wouldbe some effect by which inertia and weight would differ. The absence of such aneffect has been checked with great accuracy by an experiment done first byEötvös in 1909 and more recently by Dicke. For all substances tried, the massesand weights are exactly proportional within 1 part in 1,000,000,000 , or less. This is a remarkable experiment.
万有引力的力，正比于质量，这是一个事实，这个量{力}，本质上是对惯性的测量，即有某物，要飞出一个圆，抓住它有多难。因此，两个对象，一个重一个轻，由于万有引力的作用，它们都绕着一个更大的对象、在同一个圆中、以同样的速度运动，它们将会在一起，因为在一个圆中运动，对较大的质量，要求更大的力。也就是说，对于被给予的质量，万有引力，要强一些，且与质量成正比，这样，两个对象都将会绕着转。如果一个对象在另一个的里面，那么，它就会一直在里面，这是一个完美的平衡。因此，加加林或托夫，在宇宙飞船中，会发现物体“失重；例如，如果他们碰巧让一段粉笔飞起来，那么，它就会像整个飞船一样，绕着地球飞，于是，它就像是悬挂在他们前面一样。这个力与质量精确地成正比，这非常有意思，因为，如果它不是成正比的，那么将会有些影响，借此影响，惯性和重量，就会不同。这样一种影响的缺席，通过实验，可非常精确地检查出来，此实验，厄阜（Eotvos）1909年先做，最近做的，是迪克（Dicke）。对于所有实质性的东西的测试，都指出，质量与重量，精确地成正比，差别不到1,000,000,000分之一，甚至更小。这个实验，引人注目。

7–8Gravity and relativity 7-8 万有引力与相对论
Another topic deserving discussion isEinstein’s modification of Newton’s law of gravitation. In spite of all theexcitement it created, Newton’s law of gravitation is not correct! It wasmodified by Einstein to take into account the theory of relativity. Accordingto Newton, the gravitational effect is instantaneous, that is, if we were to movea mass, we would at once feel a new force because of the new position of thatmass; by such means we could send signals at infinite speed. Einstein advancedarguments which suggest that we cannot send signals faster than the speed oflight, so the law of gravitation must be wrong. By correcting it to takethe delays into account, we have a new law, called Einstein’s law ofgravitation. One feature of this new law which is quite easy to understand isthis: In the Einstein relativity theory, anything which has energy hasmass—mass in the sense that it is attracted gravitationally. Even light, whichhas an energy, has a “mass.” When a light beam, which has energy in it, comespast the sun there is an attraction on it by the sun. Thus the light does notgo straight, but is deflected. During the eclipse of the sun, for example, thestars which are around the sun should appear displaced from where they would beif the sun were not there, and this has been observed.
另外一个值得讨论的话题，就是爱因斯坦对牛顿万有引力规律的修改。牛顿的万有引力规律，确实有让人兴奋的一面，另外一面则是，它并不正确。它被爱因斯坦修改了，引入了对相对论的考虑。依据牛顿，万有引力的影响是瞬时的，也就是说，如果我们移动一个质量，我们就会立即感觉到：质量的新的位置所带来的力量；通过这种方式，我们可以用无限的速度，发出信号。爱因斯坦发展了论据，该论据认为，我们不能发出比光速更快的信号，于是，万有引力的归规律，就应该是错的。通过让它考虑延迟，来修改它，我们就有了新的规律，这被称为爱因斯坦的万有引力规律。这个新规律有一个特性，非常容易理解，那就是：在爱因斯坦的相对论中，任何有能量的东西，都有质量，--质量的意义，就是该东西可被万有引力吸引。甚至光都有质量，因为它有能量。光束是有能量的，当一束光通过太阳的时候，太阳就会对它有吸引。这样，光走的就不是直线，它被拽偏了。例如，在太阳的椭圆中，对于绕着太阳的恒星，太阳在一个位置，它们在一个位置，如果太阳不在该位置时，它们就应该有所偏离；这已经被观察到了。

Finally, let us compare gravitation withother theories. In recent years we have discovered that all mass is made oftiny particles and that there are several kinds of interactions, such asnuclear forces, etc. None of these nuclear or electrical forces has yet beenfound to explain gravitation. The quantum-mechanical aspects of nature have notyet been carried over to gravitation. When the scale is so small that we needthe quantum effects, the gravitational effects are so weak that the need for aquantum theory of gravitation has not yet developed. On the other hand, for consistencyin our physical theories it would be important to see whether Newton’s lawmodified to Einstein’s law can be further modified to be consistent with theuncertainty principle. This last modification has not yet been completed.
最后，我们把万有引力，与其他理论比较。近些年，我们已经发现，所有的质量，都由微粒子构成，并且还有一些交互作用，例如原子核力等。在这些原子核力或电力中，并未找到任何一个，可以用来解释万有引力。自然的量子力学方面，尚未被用于万有引力。当尺度很小，以至于我们需要量子的效果，但是，万有引力的效果是如此之弱，以至于，对一个万有引力的量子理论的需要，都还未被发展出来。另一方面，在我们的物理理论中，为了保持一致，有一点很重要，即爱因斯坦的规律，是从牛顿的规律修改而来，现在要看看，是否还能继续修改，以与测不准原理一致。这个最后的修改，尚未完成。

脚注：
1. A radius vector is a line drawn from the sun to any point in aplanet’s orbit.
1、半径向量，就是从从太阳，向行星轨道上的任一点所划的直线。
2. That is, how far the circle of the moon’s orbit falls below the straightline tangent to it at the point where the moon was one second before. 2、也就是说，一秒之前，月亮处于一个位置，一秒之后月亮轨道的圆，与月亮在前一秒那个位置的切线相比，下落了多少呢？
3. The proof is not given in this course. 3、本课程未提供证明。

Chapter8.Motion第8章 运动

8–1Description of motion 8-1 运动的描述
In order to find the laws governing thevarious changes that take place in bodies as time goes on, we must be able to describethe changes and have some way to record them. The simplest change to observe ina body is the apparent change in its position with time, which we call motion.Let us consider some solid object with a permanent mark, which we shall call apoint, that we can observe. We shall discuss the motion of the little marker,which might be the radiator cap of an automobile or the center of a falling ball,and shall try to describe the fact that it moves and how it moves.
随着时间的推移，物体会发生各种变化，为了找出统治变化的规律，我们应该描述这些变化，并有记录它们的方法。物体最简单的变化，就是位置随着时间而变，我们称之为运动。让我们考虑一些固体，它们拥有固定标记，我们称之为点，我们就观察这些点。我们将讨论小标记的运动，这些小标记，可以是汽车的散热盖，也可以是一个正在下落的球的中心，我们将尝试描述以下事实：它在运动，及它是如何运动的。