Chapter4. Conservationof Energy第四章 能量守恒
4–1What is energy?什么是能量
In this chapter, we begin our more detailedstudy of the different aspects of physics, having finished our description ofthings in general. To illustrate the ideas and the kind of reasoning that mightbe used in theoretical physics, we shall now examine one of the most basic lawsof physics, the conservation of energy.
前面已经把概论讲完了，在这一章，我们将更仔细地研究物理学的各个方面。理论物理学中，会用到一些想法和推理，要展示它们，我们将考察物理学中的一条最基本的定律，即能量守恒定律。

There is a fact, or if you wish, a law,governing all natural phenomena that are known to date. There is no known exceptionto this law—it is exact so far as we know. The law is called the conservationof energy. It states that there is a certain quantity, which we call energy,that does not change in the manifold changes which nature undergoes. That is amost abstract idea, because it is a mathematical principle; it says that thereis a numerical quantity which does not change when something happens. It is nota description of a mechanism, or anything concrete; it is just a strange fact thatwe can calculate some number and when we finish watching nature go through hertricks and calculate the number again, it is the same. (Something like the bishopon a red square, and after a number of moves—details unknown—it is still on somered square. It is a law of this nature.) Since it is an abstract idea, we shallillustrate the meaning of it by an analogy.
有这样一个事实，或者，你也可以称其为规律，即它统治着目前所知道的所有的自然现象。对于这条规律，目前还没有例外，就我们所知，该规律非常准确。这条规律，就被称为能量守恒规律。该规律声称，有某种确定的量，我们称之为能量，它在自然的多种变化中，保持不变。这是一个最抽象的想法，因为它是一种数学的原理；它说：有一种数字的量，当某事发生的时候，它并不变化。它并不是一种机械的描述，或者任何具体的东西；它是一种奇怪的事实：即我们可以计算出某些数字，当我们观察自然完成了其变化的把戏之后，我们重新计算这些数字，结果一样。（它就像国际象棋中一个红方块上的相一样，在一系列移动之后，尽管细节我们并不知道，但它始终是在一个红方块上。它就是这种本质的规律）。鉴于它是一种抽象的想法，所以我们将通过类比，来展示其意义。

想象有一个小孩，他可能叫“有威胁的丹尼斯”，他有一些积木，这些积木绝对不会坏，也不可能被分成小快，每个都与其他的一样。假定有28个。在某天开始的时候，他妈妈把他与这28个积木，放在一个房间。在这天结束的时候，出于好奇，她非常仔细地数了一下，发现了一个现象，无论她怎么做，积木总是28个。这个现象持续了若干天，直到有一天，她发现只有27个了，小小地调查之后，她发现有一个在垫子下面，这意味着，她应该在各处寻找，以保证的积木的数目，没有发生变化。然而，某一天，数目好像变了，只有26个了，经过仔细的研究，发现窗户是开着的，往外一看，有两个在外面。又一天，发现有30个！这让人相当诧异，转念一想，意识到布鲁斯来玩，他的积木也带来了，然后，在丹尼斯的房间留了几个。在他处置了额外的积木后，她关上了窗户，不让布鲁斯进来，然后，一切又恢复了正常，直到某一天，她又发现，只有25个积木了。然而，在房间里，有一个盒子，一个玩具盒子，妈妈想去打开这个玩具盒子，但孩子说，“不，不要打开我的玩具盒子”，然后惊叫。母亲没有被允许去打开这个盒子。由于非常好奇，且有一些天才，所以她发明了一个计划。她知道每个积木，重三盎司，所以，她就在看到有28个积木的时候，称了一下盒子的重量，16盎司。下次，当她想检查的时候，她就称一下盒子，然后减去16盎司，再除以三。她发现了下面的公式：
（看到的积木数）+（盒子的重量-16盎司）/3 盎司=常数。（4.1）

Imagine a child, perhaps “Dennis the Menace,”who has blocks which are absolutely indestructible, and cannot be divided intopieces. Each is the same as the other. Let us suppose that he has 28 blocks. His mother puts him with his 28 blocks into a room at the beginning of the day. At the end of theday, being curious, she counts the blocks very carefully, and discovers a phenomenallaw—no matter what he does with the blocks, there are always 28 remaining! This continues for a number of days, until one day there areonly 27 blocks, but a little investigating shows that there is one underthe rug—she must look everywhere to be sure that the number of blocks has notchanged. One day, however, the number appears to change—there are only 26 blocks. Careful investigation indicates that the window was open,and upon looking outside, the other two blocks are found. Another day, careful countindicates that there are 30 blocks! This causes considerable consternation, until it is realizedthat Bruce came to visit, bringing his blocks with him, and he left a few at Dennis’house. After she has disposed of the extra blocks, she closes the window, doesnot let Bruce in, and then everything is going along all right, until one timeshe counts and finds only 25 blocks. However, there is a box in the room, a toy box, and the mothergoes to open the toy box, but the boy says “No, do not open my toy box,” andscreams. Mother is not allowed to open the toy box. Being extremely curious, andsomewhat ingenious, she invents a scheme! She knows that a block weighs three ounces,so she weighs the box at a time when she sees 28 blocks, and it weighs 16 ounces. The next time she wishes to check, she weighs the boxagain, subtracts sixteen ounces and divides by three. She discovers the following:
(number ofblocks seen)+(weight of box−16 ounces )/3ounces=constant.(4.1)

There then appear to be some new deviations, but careful study indicatesthat the dirty water in the bathtub is changing its level. The child isthrowing blocks into the water, and she cannot see them because it is so dirty,but she can find out how many blocks are in the water by adding another term toher formula. Since the original height of the water was 6 inches and each block raises the water a quarter of an inch, thisnew formula would be:
(number ofblocks seen)+(weight of box−16 ounces )/3ounces +(height of water−6 inches)/(1/4 inch)=constant.(4.2)
然后，又出现了偏差，但仔细的研究指出，浴盆里脏水的水平面改变了。孩子把一些积木，扔到了水里，虽然因为水很脏，她看不到这些积木，但是，通过在她的公式里增加一项，她就可以找出，水里有多少个积木。因为，原先水的高度是六英寸，而每一个积木，可以把水的高度，抬升1/4英寸，所以新的公式就是：
(看到的积木个数)+（盒子的重量-16盎司）/3 盎司+(水的高度−6 英寸)/(1/4 英寸)=常数。(4.2)

In the gradual increase in the complexity of her world, she finds awhole series of terms representing ways of calculating how many blocks are inplaces where she is not allowed to look. As a result, she finds a complexformula, a quantity which has to be computed, which always stays thesame in her situation.
她的世界，逐步地变的复杂，在此期间，有些地方，她没法看到，但她还是找到了一系列的办法，用来计算这些地方的积木数。最终，她发现了一个公式，和一个量，这个量，总要去算，且在她的情况下，该量总保持一样。

What is the analogy of this to theconservation of energy? The most remarkable aspect that must be abstracted fromthis picture is that there are no blocks. Take away the first terms in (4.1)and (4.2)and we find ourselves calculating more or less abstract things. The analogy hasthe following points. First, when we are calculating the energy, sometimes someof it leaves the system and goes away, or sometimes some comes in. In order toverify the conservation of energy, we must be careful that we have not put anyin or taken any out. Second, the energy has a large number of differentforms, and there is a formula for each one. These are: gravitationalenergy, kinetic energy, heat energy, elastic energy, electrical energy,chemical energy, radiant energy, nuclear energy, mass energy. If we total upthe formulas for each of these contributions, it will not change except forenergy going in and out.
这与能量守恒，有何可类比之处？从这幅图画中，应该抽出来的、最值得说明的一点，就是没有积木。把公式（4.1）和（4.2）中的第一项拿掉，我们发现，我们或多或少是在计算抽象的事物。这个类比有以下几点。第一，当我们计算能量时，有时候，有些能量会离开这个系统，消失掉，而有时，则会有一些能量进来。为了证实能量守恒，我们就应该注意，既不要把任何能量放进来，也不要把任何能量拿出去。第二，能量有多种不同的形式，对于每种形式，都有一个公式。这些能量就是重力能量、动能、热能、弹性能量、电子能量、化学能量、辐射能、原子能、质能。如果我们把这些能量公式都统计起来，就会发现的，除了能量的此出彼进外，没有何变化。

It is important to realize that in physicstoday, we have no knowledge of what energy is. We do not have a picturethat energy comes in little blobs of a definite amount. It is not that way.However, there are formulas for calculating some numerical quantity, and whenwe add it all together it gives “28 ”—always the same number. It is an abstract thing in that it does not tellus the mechanism or the reasons for the various formulas.
有一点很重要，那就是在今天的物理学中，我们要意识到，对于能量究竟是什么，我们并没有任何知识。我们并没有一幅图像嗯，在其中，能量是以一定量的块，一块一块地到来。事情并不是那样。然而，是有一些公式，可用来计算一些数字的量，当我们把这些量加起来时，会得到“28”--总是这个数。计算能量，是一个抽象的事情，它不能告诉我们各种不同公式的机制或原因。

4–2Gravitational potential energy 4-2重力势能
Conservation of energy can be understoodonly if we have the formula for all of its forms. I wish to discuss the formulafor gravitational energy near the surface of the Earth, and I wish to derivethis formula in a way which has nothing to do with history but is simply a lineof reasoning invented for this particular lecture to give you an illustrationof the remarkable fact that a great deal about nature can be extracted from afew facts and close reasoning. It is an illustration of the kind of worktheoretical physicists become involved in. It is patterned after a mostexcellent argument by Mr. Carnot on the efficiency of steam engines.1
能量有多种形式，每种形式都有其公式，只有当我们拥有了所有能量的公式只时，能量守恒才可被理解。我希望讨论接近地球表面的重力能量的公式，我希望这样来派生它：即与历史性，没有任何关系，而只是一种简单的推理过程，此过程是专为本讲座发明的，目的是给你们展示一个引人注目的事实，即关于自然的大量东西，可以从少数几个事实和严密的推理得出。它所展示的，就是理论物理学家正在开始从事的工作。卡诺先生，对于蒸汽机的效率，做过一个最杰出的论证，在此之后，上述工作，就变成了一种模式。

Consider weight-lifting machines—machineswhich have the property that they lift one weight by lowering another. Let usalso make a hypothesis: that there is no such thing as perpetual motionwith these weight-lifting machines. (In fact, that there is no perpetual motionat all is a general statement of the law of conservation of energy.) We must becareful to define perpetual motion. First, let us do it for weight-liftingmachines. If, when we have lifted and lowered a lot of weights and restored themachine to the original condition, we find that the net result is to have lifteda weight, then we have a perpetual motion machine because we can use thatlifted weight to run something else. That is, provided the machine whichlifted the weight is brought back to its exact original condition, andfurthermore that it is completely self-contained—that it has notreceived the energy to lift that weight from some external source—like Bruce’sblocks.
我们考虑起重机，它有这样一个特性，即可以通过降低一个重量，来举起另一个重量。我们先做一个假设：对于这些起重机来说，并没有永动机这种东西。（事实上，根本没有任何永动，是能量守恒规律的一个一般声明。）要定义永动，我们必须很仔细。首先，让我们为起重机做此事。如果，在我们举起并降下很多重量之后，把机器恢复到它的起始状态，我们发现，最终结果，就是举起了一些重量，然后，我们就有一种永动机，因为我们可以用此被举起的重量，来运行其他的事情。也就是说，假设举起重量的这个机器，被带回了起始的状态，另外，它也是完全自我包容的，--它并没有从任何外部的源泉，接收能量，以举起重量，就像布鲁斯的积木。

Fig. 4–1.Simple weight-lifting machine. 简单的起重机
A very simple weight-lifting machine is shownin Fig. 4–1.This machine lifts weights three units “strong.” We place three units on onebalance pan, and one unit on the other. However, in order to get it actually towork, we must lift a little weight off the left pan. On the other hand, wecould lift a one-unit weight by lowering the three-unit weight, if we cheat alittle by lifting a little weight off the other pan. Of course, we realize thatwith any actual lifting machine, we must add a little extra to get it torun. This we disregard, temporarily. Ideal machines, although they donot exist, do not require anything extra. A machine that we actually use canbe, in a sense, almost reversible: that is, if it will lift the weightof three by lowering a weight of one, then it will also lift nearly the weightof one the same amount by lowering the weight of three.
图4-1所示，是一种简单的起重机。这台机器可以举起三个单位“强”的重量。然而，为了让它真正地能够工作，我们必须从左边的盘子中，抬起一点重量。另一方面，我们可以通过降低三个单位的重量，来抬起一个单位的重量，如果我们小小地欺骗一下，即把另一端的盘子，往上抬一点。当然，我们也意识到，对于任何实际的起重机来说，我们应该增加一点额外的{重量}，以让它能够运行。这点我们暂时先不管。理想的起重机，虽然不存在，但它们并不要求任何额外的{重量}。我们实际上使用的起重机，在某种意义上，几乎是可以反转的，也就是说，如果它可以通过降低一个单位的重量，来抬起三个单位的重量，那么同样，它也可以通过降低三个单位的重量，来抬起一个单位的重量。

We imagine that there are two classes ofmachines, those that are not reversible, which includes all realmachines, and those that are reversible, which of course are actuallynot attainable no matter how careful we may be in our design of bearings,levers, etc. We suppose, however, that there is such a thing—a reversible machine—whichlowers one unit of weight (a pound or any other unit) by one unit of distance,and at the same time lifts a three-unit weight. Call this reversible machine,Machine A . Suppose this particular reversible machine lifts the three-unit weighta distance X . Then suppose we have another machine, Machine B , which is not necessarily reversible, which also lowers a unit weighta unit distance, but which lifts three units a distance Y . We can now prove that Y is not higher than X ; that is, it is impossible to build a machine that will lift a weightany higher than it will be lifted by a reversible machine. Let us seewhy. Let us suppose that Y were higher than X . We take a one-unit weight and lower it one unit height withMachine B , and that lifts the three-unit weight up a distance Y . Then we could lower the weight from Y to X , obtaining free power, and use the reversible Machine A, running backwards, to lower the three-unit weight a distance Xand lift the one-unit weight by one unit height. This will put theone-unit weight back where it was before, and leave both machines ready to beused again! We would therefore have perpetual motion if Y were higher than X , which we assumed was impossible. With those assumptions, we thus deducethat Y is not higher than X , so that of all machines that can be designed, the reversible machineis the best.
我们可以想象，我们有两类机器，一类是不可反转的，包括所有真正的机器，另一类是可反转的，但实际上，不论当我们用轴承、杠杆来设计时，是如何地仔细，这种机器，都是得不到的。然而，我们可以假设，存在这样的东西，即一个可反转的机器，它通过降低一个单位的重量（一磅或任何其它单位）一个单位的距离，来抬起三个单位的重量。这个可反转的机器，被称为机器A。假设这个可反转的机器，可以把三个单位的重量，提高一个X距离。然后，假设我们还有另外一台机器B，它并不必然是可反转的，同样，它也可以把一个单位的重量，降低一个单位的距离，但是，却可以把三个单位的重量，提高一个Y的距离。我们现在可以证明，Y并不比X高；也就是说，把一定重量举高，新建的机器所能做到的，与可反转机器所能做到的相比，并不更高。现在看为什么。先假设Y比X高。我们用机器B，把一个单位的重量，降低一个单位的高度，这样，就可以把三个单位的重量，提高Y的高度。然后，我们把重量从Y降到X，就能获得力量，然后，再使用可反转的机器A，反向运转，把三个单位的重量，降低X距离，并且，把一个单位的重量，提高一个单位的距离。这样，就把一个单位的重量，放回了其原处，从而让两台机器，都可被重新使用！如果Y比X高的话，我们就可以有一台永动机了，而这我们假设是不可能的。根据这些假设，我们就可以推出，Y不可能比X高，所以，在我们所能设计的所有机器中，可反转的机器，是最好的。

We can also see that all reversiblemachines must lift to exactly the same height. Suppose that B were really reversible also. The argument that Y is not higher than X is, of course, just as good as it was before, but we can also make ourargument the other way around, using the machines in the opposite order, andprove that X is not higher than Y . This, then, is a very remarkable observation because it permits us toanalyze the height to which different machines are going to lift something withoutlooking at the interior mechanism. We know at once that if somebody makesan enormously elaborate series of levers that lift three units a certaindistance by lowering one unit by one unit distance, and we compare it with asimple lever which does the same thing and is fundamentally reversible, hismachine will lift it no higher, but perhaps less high. If his machine isreversible, we also know exactly how high it will lift. To summarize:every reversible machine, no matter how it operates, which drops one pound onefoot and lifts a three-pound weight always lifts it the same distance, X. This is clearly a universal law of great utility. The next questionis, of course, what is X ?
我们还可以看到，所有可反转的机器，都应该举到同样的高度。假设B也确实是可反转的。关于Y并不比X高的论证就是，以相反的顺序使用起重机，从而证明X并不比Y高；当然，这种论证，正如以前一样好，但是，我们也可以用另外的方式，给出我们的论证。因此，这一观察，引人注目，因为，不同的起重机，都会把某物提升到一个高度，而此观察，允许我们去分析这些高度，且不用去查看内部的机制。所以，如果某人制造了巨大的复杂的杠杆系列，可以通过把一个单位的重量降低一个单位的距离，来把三个单位的重量提高一定的距离，假设还有另外一种简单的杠杆，也可以做同样的事情，且是可反转的，那么，我们比较这两种机械，马上就会知道，前一机器，并不会举的更高，或许还更低。如果他的机器是可反转的，那么我们也会准确地知道，它能举多高。总结：每种可反转的机器，如果它是通过把一磅的重量降低一英尺，来抬高三磅的重量，那么，不论它是怎么运行的，总是抬高同样的距离：X。很清楚，这是普遍的伟大实用的规律，下一个问题当然就是：X是什么？

Fig. 4–2.A reversible machine. 一个可逆的机器
Suppose we have a reversible machine[A1] which is going to lift this distance X , three for one. We set up three balls in a rack which does not move,as shown in Fig. 4–2. Oneball is held on a stage at a distance one foot above the ground. The machinecan lift three balls, lowering one by a distance 1 . Now, we have arranged that the platform which holds three balls hasa floor and two shelves, exactly spaced at distance X , and further, that the rack which holds the balls is spaced atdistance X , (a). First we roll the balls horizontally from the rack to theshelves, (b), and we suppose that this takes no energy because we do notchange the height. The reversible machine then operates: it lowers the singleball to the floor, and it lifts the rack a distance X , (c). Now we have ingeniously arranged the rack so that theseballs are again even with the platforms. Thus we unload the balls onto therack, (d); having unloaded the balls, we can restore the machine to itsoriginal condition. Now we have three balls on the upper three shelves and oneat the bottom. But the strange thing is that, in a certain way of speaking, wehave not lifted two of them at all because, after all, there were ballson shelves 2 and 3 before. The resulting effect has been to lift one ball adistance 3X . Now, if 3X exceeds one foot, then we can lower the ball to return themachine to the initial condition, (f), and we can run the apparatus again.Therefore 3X cannot exceed one foot, for if 3X exceeds one foot we can make perpetual motion. Likewise, we can provethat one foot cannot exceed 3X , by making the whole machine run the opposite way, since it is areversible machine. Therefore 3X is neither greater nor less than a foot, and we discover then,by argument alone, the law that X=13 foot. The generalization is clear: one pound falls a certaindistance in operating a reversible machine; then the machine can lift p pounds this distance divided by p . Another way of putting the result is that three pounds times theheight lifted, which in our problem was X , is equal to one pound times the distance lowered, which is one footin this case. If we take all the weights and multiply them by the heights atwhich they are now, above the floor, let the machine operate, and then multiplyall the weights by all the heights again, there will be no change. (Wehave to generalize the example where we moved only one weight to the case wherewhen we lower one we lift several different ones—but that is easy.)
假设我们有一个可反转的机器，它可以通过降低三个X的距离，来抬高一个X的距离。我们在一个固定的支架上，放三个球，如图4-2。另一个球，在一定距离之外被托着，比地面高一英尺。此机器，可以通过把一个球降低1英尺，来抬高三个球。现在，有个装三个球的平台，它有三层，每层高都是X，我们把它放在距支架X处，（a）。首先，我们把球，滚到平台的各层上，（b），我们假定这不消耗能量，因为我们并未改变高度。然后，可反转机器开始运行：它把那个单个的球，降到地板，从而把平台抬高一个X的距离，（c）。现在，我们巧妙地安排平台，已让这些球，与架子相平。这样，我们就把这些球，卸载到了架子上，（d）;卸载了这些球之后，我们就可以把机器，恢复到起原始状态。现在三个球在架子的上三层，一个在底层。但是，以某种方式来说，奇怪的事情就是，我们根本没有抬高其中的两个球，因为，毕竟这两个球，以前就是在第二层和第三层。实际结果就是，把一个球，抬高了3X的距离。现在，如果3X超过了1英尺，那么，我们就可以把球降低，然后让机器回到它的初始状态，（f），这样，我们就可以重新运行这个装置了。因此，3X不能超过1英尺，因为如果那样，等于我们造出了一个永久运动。同样，因为这个机器，是可反转的，所以，通过让这个机器反向运行，我们也可以证明，1英尺不能超过3X。所以，3X既不比1英尺大，也不比它小。所以，通过这个论证，我们就发现了规律，即X=1/3英尺。可以清晰地概括如下：运行一台可反转的机器，让一磅重量降下一定距离，那么，这个机器，就可以把p磅的重物，抬高此一定距离的1/p。表示这一结果的另一方法就是，三磅乘以抬高的高度，在我们这个问题中就是X，等于一磅乘以降下的距离，这里就是1英尺。如果我们让所有的重量，乘以它们现在的在地板之上的高度，然后，让机器运行，再然后，让所有的重量，再乘以所有的高度，将没有任何变化。（在我们的例子中，我们是把一个重量降低，抬高了几个重量，对此例子，我们应该归纳一下，即我们实际只移动了一个重量，然而，这种归纳，并不容易。）
[A1]

We call the sum of the weights times the heights gravitational potential energy—the energy which an object has because of its relationship in space, relative to the earth. The formula for gravitational energy, then, so long as we are not too far from the earth (the force weakens as we go higher) is
gravitationalpotential energy for one object=(weight)×(height).(4.3)
我们把重量乘以高度的总和，称为重力势能，一个对象，拥有这种能量，乃是因为，它在空间中相对于地球的关系。因此，只要我们离地球不是很远（高度越高，力量越弱），那么，重力势能的公式就是：
一个对象的重力势能=重量*高度。 （4.3）