Previously, we considered the case where s was constant, and we found that s made no difference in the laws of motion, since ds/dt=0; ultimately, therefore, the laws of physics were the same in bothsystems. But another case we can take is that s=ut , where u is a uniform velocity in a straight line. Then s is not constant, and ds/dt is not zero, but is u , a constant. However, the acceleration d2x/dt2is still the same as d2x′/dt2, because du/dt=0 . This proves the law that we used in Chapter 10,namely, that if we move in a straight line with uniform velocity the laws ofphysics will look the same to us as when we are standing still. That is theGalilean transformation. But we wish to discuss the interesting case where sis still more complicated, say s=at2/2 . Then ds/dt=at and d2s/dt2=a , a uniform acceleration; or in a still more complicated case, theacceleration might be a function of time. This means that although the laws ofmotion from the point of view of Joe would look like
md2x/dt2=Fx,
the laws of motion as looked upon by Moe would appear as
md2x′/dt2=Fx′=Fx−ma.
That is, since Moe’s coordinate system is accelerating with respect toJoe’s, the extra term ma comes in, and Moe will have to correct his forces by that amount inorder to get Newton’s laws to work. In other words, here is an apparent, mysteriousnew force of unknown origin which arises, of course, because Moe has the wrongcoordinate system. This is an example of a pseudo force; other examples occurin coordinate systems that are rotating.
前面，在我们考虑的情况中，s是一个常数，我们发现，对于运动规律，s不会带来什么不同，由ds/dt=0；因此，最终，物理规律，在两个坐标系中是一样的。但是，我们可以考虑的另外一种情况则是，s=ut，这里u是一条直线上的匀速。因此，s就不是常数，而ds/dt不是零，而是u，一个常数。然而，加速度d2x/dt2 与d2x′/dt2 仍一样，因为 du/dt=0 。这就证明了我们在第10章所用的规律，亦即，如果我们在一条直线上，做匀速运动，那么，物理规律对我们来说，与我们静止时，看上去一样。这就是伽利略转换。但是，我们希望讨论更有趣的情况，就是s要更复杂些，比如s=at2/2 。因此ds/dt=at 和d2s/dt2=a, 即加速度是均匀的；或者，在一种更复杂的情况中，加速度是时间的函数。 这就是说，虽然运动规律，从Joe的观点看上去，是这样：
md2x/dt2=Fx，
但从Moe看上去，则是：
md2x′/dt2=Fx′=Fx−ma。
也就是说，由于Moe的坐标系，相对于Joe的坐标系，是加速的，那么，额外的项 ma，就来了，且Moe必须通过这个量，来更改他的力，以让牛顿规律，能够工作。换句话说，这里，一个明显的、神秘的新力，产生了，其起源未知，当然，这是因为Moe有一个错误的坐标系。这就是一个伪力的例子；其他的例子，出现在正在旋转的坐标系中。

Another example of pseudo force is what isoften called “centrifugal force.” An observer in a rotating coordinate system,e.g., in a rotating box, will find mysterious forces, not accounted for by anyknown origin of force, throwing things outward toward the walls. These forcesare due merely to the fact that the observer does not have Newton’s coordinatesystem, which is the simplest coordinate system.
另外一个伪力的例子，就是我们称为“离心力”的东西。一个观察者在一个旋转中的坐标系中，例如，在一个旋转的盒子中，他将会发现神秘的力，把东西往外面的墙上抛；任何已知的力的起源，都无法解释。这些力，只能归于如下事实，即观察者并没有任何牛顿的坐标系，它是最简单的坐标系。

Pseudo force can be illustrated by aninteresting experiment in which we push a jar of water along a table, withacceleration. Gravity, of course, acts downward on the water, but because ofthe horizontal acceleration there is also a pseudo force acting horizontallyand in a direction opposite to the acceleration. The resultant of gravity andpseudo force makes an angle with the vertical, and during the acceleration thesurface of the water will be perpendicular to the resultant force, i.e.,inclined at an angle with the table, with the water standing higher in the rearwardside of the jar. When the push on the jar stops and the jar decelerates becauseof friction, the pseudo force is reversed, and the water stands higher in theforward side of the jar (Fig. 12–4).
可以通过一个有趣的实验，来图示伪力，在实验中，我们沿着桌子，推一瓶水，有加速度。重力当然也作用于水，向下，但是，因为水平加速度，就有一个伪力，作用于水平方向，且与加速度的方向相反。重力与伪力，形成合力，合力与垂直方向，有一个角度，在加速的时候，水的表面，会垂直于合力，亦即，与桌面形成一个角度，水瓶后部的水，就会高一些。当停止推瓶子时，由于摩擦力，瓶子会减速，伪力就被反向了，这样，在水瓶前部的水，就会高一些。

Fig. 12–4.Illustration of a pseudo force. 图12-4 伪力的图示。

One very important feature of pseudo forcesis that they are always proportional to the masses; the same is true ofgravity. The possibility exists, therefore, that gravity itself is a pseudoforce. Is it not possible that perhaps gravitation is due simply to thefact that we do not have the right coordinate system? After all, we can alwaysget a force proportional to the mass if we imagine that a body is accelerating.For instance, a man shut up in a box that is standing still on the earth findshimself held to the floor of the box with a certain force that is proportionalto his mass. But if there were no earth at all and the box were standing still,the man inside would float in space. On the other hand, if there were no earthat all and something were pulling the box along with anacceleration g , then the man in the box, analyzing physics, would find a pseudoforce which would pull him to the floor, just as gravity does.
伪力的一个重要特质，就是它们总是正比于质量，重力也同样。因此，存在这种可能性，即重力本身也是一种伪力。或许万有引力，可以简单地归于一个事实，即我们没有正确的坐标系，这可能吗？毕竟，如果我们想象一个物体是在加速的话，那么，我们总是可以得到一个力，正比于质量。例如，一个人，被关在一个盒子里，而盒子，是静止的地面上，他发现他自己被一定的力，吸引到这个盒子的地板上，此力正比于他的质量。但是，如果根本没有地球，且盒子是静止的，那么，里面的人将会飘在空中。另一方面，如果根本没有地球，而是某物，用加速度g，吸引着盒子，那么，盒子里的人，分析物理，将会发现一个伪力，把他吸向地板，正如重力所做那样。

Einstein put forward the famous hypothesisthat accelerations give an imitation of gravitation, that the forces ofacceleration (the pseudo forces) cannot be distinguished from those ofgravity; it is not possible to tell how much of a given force is gravity andhow much is pseudo force.
爱因斯坦提出了著名的假设，加速度给出了万有引力的一个模拟，即加速度的力（即伪力），无法与重力的力，区分开来；对于一个被给予的力，要说出：它的多少是重力，它的多少又是伪力，是不可能的。

It might seem all right to consider gravityto be a pseudo force, to say that we are all held down because we areaccelerating upward, but how about the people in Madagascar, on the other sideof the earth—are they accelerating too? 把重力，考虑为一种伪力，似乎没问题，说我们被向下吸引，是因为我们被向前加速着，但是，在马达加斯加的人呢？他们在地球另一侧，他们也在加速吗？Einstein found that gravity could be considered a pseudo force onlyat one point at a time, and was led by his considerations to suggest that the geometryof the world is more complicated than ordinary Euclidean geometry. 爱因斯坦发现，重力只在某个点、某一时间上，才可以被考虑为是一种伪力，且被他的考虑引导而建议说：世界的几何学，要比通常的欧几里得几何学，复杂的多。The present discussion is only qualitative, and does not pretend toconvey anything more than the general idea. 现在的讨论只是定性的，所传达的事情，还是在普通想法的范围内，并不打算超此范围。To give a rough idea of how gravitation could be the result ofpseudo forces, we present an illustration which is purely geometrical and doesnot represent the real situation.万有引力会是伪力的结果，关于这点，要给出一个粗略的想法，我们给出一个说明，它是纯几何的，并不代表真实情况。 Suppose that we all lived in two dimensions, and knew nothing of athird. We think we are on a plane, but suppose we are really on the surface ofa sphere. And suppose that we shoot an object along the ground, with no forceson it. Where will it go? 假设我们都生活在二维空间，且对第三维，一无所知。假设我们沿着地面，发射一个对象，它不受力。它会去向哪里呢？It will appear to go in a straight line, but it has to remain on thesurface of a sphere, where the shortest distance between two points is along agreat circle; so it goes along a great circle. 它似乎会走一条直线，但是，它必须保持在一个球形的表面中，在球上，两点之间的最短距离，就是沿着一个巨大的圆；所以，它沿着一个大的圆走。If we shoot another object similarly, but in another direction, it goesalong another great circle. 如果我们类似地发射另一个对象，在另一个方向，那么，它沿着另一个大圆走。Because we think we are on a plane, we expect that these two bodieswill continue to diverge linearly with time, but careful observation will showthat if they go far enough they move closer together again, as though they wereattracting each other.因为，我们认为我们是在一个平面上，我们期待，这两个物体，随着时间，继续线性地偏离，但是，仔细的观察将会指出，如果它们走的足够远，它们又会相互靠近，就好像它们相互吸引一样。 But they are not attracting each other—there is justsomething “weird” about this geometry. 但是，它们并不相互吸引，这个几何学中，有某种“怪异”的事情。This particular illustration does not describe correctly the way inwhich Einstein’s geometry is “weird,” but it illustrates that if we distort thegeometry sufficiently it is possible that all gravitation is related in someway to pseudo forces; that is the general idea of the Einsteinian theory ofgravitation.
这个具体的说明，并不能正确地描述：爱因斯坦几何学中的那种“怪异”方式，但是，它说明了，如果我们充分地扭曲几何学，那么，所有万有引力，都以某种方式，而与伪力有关；这就是爱因斯坦的万有引力理论的一般想法。

12–6Nuclear forces 12-6 原子核力
We conclude this chapter with a brief discussionof the only other known forces, which are called nuclear forces. Theseforces are within the nuclei of atoms, and although they are much discussed, noone has ever calculated the force between two nuclei, and indeed at presentthere is no known law for nuclear forces. These forces have a very tiny rangewhich is just about the same as the size of the nucleus, perhaps 10−13 centimeter. With particles so small and at such a tiny distance,only the quantum-mechanical laws are valid, not the Newtonian laws. In nuclearanalysis we no longer think in terms of forces, and in fact we can replace theforce concept with a concept of the energy of interaction of two particles, asubject that will be discussed later. Any formula that can be written for nuclearforces is a rather crude approximation which omits many complications; onemight be somewhat as follows: forces within a nucleus do not vary inversely asthe square of the distance, but die off exponentially over a certain distance r, as expressed by F=(1/r2)exp(−r/r0), where the distance r0 is of the order of 10−13 centimeter. In other words, the forces disappear as soon as theparticles are any great distance apart, although they are very strong withinthe 10−13 centimeter range. So far as they are understood today, the lawsof nuclear force are very complex; we do not understand them in any simple way,and the whole problem of analyzing the fundamental machinery behind nuclearforces is unsolved. Attempts at a solution have led to the discovery of numerousstrange particles, the π -mesons, for example, but the origin of these forces remains obscure.
作为本章的总结，我们简单地讨论另外唯一的力，原子核力。这些力，在原子核中，虽然它们被讨论过很多，但没有人计算出过两个原子核之间的力，确实，现在，原子核力的规律，尚属未知。这些力的范围，非常小，大约与原子核的尺寸一样，或许10−13 厘米。对于这么小的粒子，在这么微小的距离，有效的，只有量子力学，而非牛顿规律。在原子核的分析中，我们不再用力这种词汇来思考，事实上，我们可以用两个粒子间交互作用的能量这一概念，来替换力的概念，这个主题，稍后讨论。对于原子核力，任何可以写出的公式，都是一个相当粗糙的近似，因为它忽略了很多复杂的情况；一个公式，大概会是这样：在一个原子核中的力，其变化，并不与距离的平方成反比，而是在一定距离r内，程指数衰减，即 F=(1/r2)exp(−r/r0)，这里r0是10−13厘米的级别。换句话说，当粒子之间的距离，大到一定程度时，力就会消失，虽然在10−13厘米之内，它们很强。就我们今天的理解而言，原子核力的规律，还是很复杂；我们不能以任何简单的方式，来理解它们，分析原子核力的基础机制这一问题，尚未解决。对解决方案的探索，导致发现了一系列奇怪的例子，例如π -介子，但这些力的起源，依然模糊。

Chapter13.Work and Potential Energy(A)第13章 功与势能（A）
13–1Energy of a falling body 13-1 一个下落物体的能量
In Chapter 4 we discussed the conservationof energy. In that discussion, we did not use Newton’s laws, but it is, ofcourse, of great interest to see how it comes about that energy is in factconserved in accordance with these laws. For clarity we shall start with thesimplest possible example, and then develop harder and harder examples.
在第4章，我们讨论了能量守恒。在那个讨论中，我们没有用牛顿规律，但当然，对于‘能量事实上是守恒的’这件事，依据这些规律，来看它是如何产生的，将会很有趣。

Let us find out directly fromNewton’s Second Law how the kinetic energy should change, by taking thederivative of the kinetic energy with respect to time and then using Newton’slaws. When we differentiate mv2/2 with respect to time,we obtain
dT/dt= d(mv2/2)/dt=(1/2)(m2v)dv/dt= mvdv/dt, (13.2)
since m is assumed constant. Butfrom Newton’s Second Law, m(dv/dt) = F, so that
dT/dt = Fv. (13.3)
In general, it will come out to be F ·v, but in our one-dimensional case let us leave it as the forcetimes the velocity.
让我们直接从牛顿第二规律出发，找出动能应该如何变化，即通过动能对时间求导，然后使用牛顿规律。当我们对mv2 /2求时间的导数时，我们得到：
dT/dt= d(mv2/2)/dt=(1/2)(m2v)dv/dt= mvdv/dt, (13.2)
由于被假定为常数。m。但是，从牛顿第二规律，m(dv/dt) = F,于是：
dT/dt = Fv. (13.3)
通常，结果会是F · v，但是，在我们一维的案例中，我们就把它当作力乘以速度吧。

Now in our simple example the force isconstant, equal to −mg, a vertical force (the minus sign means that itacts downward), and the velocity, of course,is the rate of change of thevertical position, or height h, with time. Thus the rate of change ofthe kinetic energy is −mg(dh/dt), which quantity, miracle of miracles,is minus the rate of change of something else! It is minus the time rate ofchange of mgh! Therefore, as time goes on, the changes in kinetic energyand in the quantity mgh are equal and opposite, so that the sum of thetwo quantities remains constant. Q.E.D.
现在，在我们的简单例子中，力是常数，等于−mg，一个垂直的力（负号的意思是向下作用），当然，矢速就是垂直位置随着时间的变化率，或者h。这样，动能随着时间的变化率就是−mg(dh/dt)，这个量，奇迹中的奇迹，就是负的另外某物的变化率！它是负的mgh随着时间的变化率！因此，随着时间的变化，动能的变化和量mgh的变化，相等且相反，所以，这两个量的和不变。证明完毕。

We have shown, from Newton’s second law ofmotion, that energy is conserved for constant forces when we add the potentialenergy mgh to the kinetic energy 12 mv2. Now let us look intothis further and see whether it can be generalized, and thus advance ourunderstanding. Does it work only for a freely falling body, or is it moregeneral? We expect from our discussion of the conservation of energy that itwould work for an object moving from one point to another in some kind offrictionless curve, under the influence of gravity (Fig. 13-1). If the objectreaches a certain height h from the original height H, then thesame formula should again be right, even though the velocity is now in somedirection other than the vertical. We would like to understand why thelaw is still correct. Let us follow the same analysis, finding the time rate ofchange of the kinetic energy. This will again be mv(dv/dt), but m(dv/dt)is the rate of change of the magnitude of the momentum, i.e., the force inthe direction of motion—the tangential force Ft. Thus
dT/dt= mv dv/dt= Ftv.
Now the speed is the rate of change ofdistance along the curve, ds/dt, and the tangential force Ftis not −mg but is weaker by the ratio of the vertical distance dhto the distance ds along the path. In other words,
Ft = −mg sinθ =−mg dh/ds,
so that
Ftds/dt= −mg(dh/ds)(ds/dt)=−mg dh/dt,
since the ds’s cancel. Thus we get −mg(dh/dt),which is equal to the rate of change of −mgh, as before.
从牛顿第二规律出发，我们已经指出，对于恒力，当我们把势能mgh，与动能mv2/2相加时，能量是守恒的。现在，让我们进一步看看这条，能否普遍化，这样，就可以扩展我们的理解。它只对自由落体成立，还是更具普遍性？从我们关于能量守恒的讨论出发，我们期待，一个对象，从一个点，沿着一条无摩擦的曲线，在重力的影响下，运动到另一个点时，该规律也能成立（图13-1）。如果对象，从最初的高度H，到达高度h，那么，同样的公式应该还正确，尽管矢速，现在并不垂直，而是在另一方向。我们希望能够理解，为什么此规律依然正确。我们现在做同样的分析，找出动能随时间的变化率。这将又是mv(dv/dt)，但m(dv/dt) 是动量大小的变化率，亦即，是运动方向上的力—切线方向力Ft。这样：
dT/dt= mv dv/dt= Ftv.
现在，速度是距离沿着曲线的变化率ds/dt, 切力Ft不是−mg ，而是要弱一些，是垂直距离dh，与沿着路径的距离ds 的比率。换句话说:
Ft = −mg sinθ =−mg dh/ds,
于是：
Ftds/dt= −mg(dh/ds)(ds/dt)=−mg dh/dt,
由于ds消掉了。这样，我们得到−mg(dh/dt)，它等于−mgh的变化率，与以前一样。

Fig. 13-1. An object moving on africtionless curve under the influence of gravity. 图13-1 在重力的影响下，一个对象，沿着一条无摩擦曲线运动。

In order to understand exactly how theconservation of energy works in general in mechanics, we shall now discuss anumber of concepts which will help us to analyze it.
在力学中，能量守恒一般是如何工作的，为了准确地理解这点，我们现在将讨论一系列概念，它们将会帮助我们，分析之。

First, we discuss the rate of change ofkinetic energy in general in three dimensions. The kinetic energy in threedimensions is
T = m/2(v2x+v2y + v2z ).
When we differentiate this with respect totime, we get three terrifying terms:
dT/dt= m(vxdvx/dt+vydvy/dt+ vzdvz/dt). (13.4)
But m(dvx/dt) is the force Fxacting on the object in the x-direction. Thus the right side of Eq.(13.4) is Fxvx + Fyvy +Fzvz. We recall our vector analysis and recognizethis as F · v; therefore
dT/dt = F · v. (13.5)
This result can be derived more quickly asfollows: if a and b are two vectors,both of whichmay depend upon the time, the derivative of a · b is,in general,
d(a · b)/dt= a · (db/dt) + (da/dt) · b. (13.6)
We then use this in the form a =b = v:
d(mv2/2)dt= d(mv· v/2)dt= mdv/dt· v = F· v = F ·ds/dt. (13.7)
首先，我们讨论三维中的一般动能额的变化率。三维中的动能是：
T = m/2(v2x+v2y + v2z ).
当我们让它对时间求导，我们得到吓人的项：
dT/dt= m(vxdvx/dt+ vydvy/dt + vzdvz/dt). (13.4)
但是m(dvx/dt) 是力Fx 在x方向作用于对象的力。这样，方程（13.4）的右边就是Fxvx + Fyvy+ Fzvz。回忆我们的矢量分析，就可认出这就是F · v ；因此
dT/dt = F · v. (13.5)
这个结果，按照下面的方法，可以更快地导出：如果a 和 b 是两个矢量，两者都可能依赖于时间，那么，a · b的导数一般就是：
d(a · b)/dt= a · (db/dt) + (da/dt) · b. (13.6)
因此，我们按这个形式a = b = v，来使用它:
d(mv2/2)dt= d(mv· v/2)dt= mdv/dt· v = F· v = F ·ds/dt. (13.7)

Because the concepts of kinetic energy, andenergy in general, are so important,various names have been given to theimportant terms in equations such as these.mv2/2 is, as weknow, called kinetic energy. F · v is calledpower: the force acting on an object times the velocity of the object(vector “dot” product) is the power being delivered to the object by thatforce. We thus have a marvelous theorem:the rate of change of kinetic energyof an object is equal to the power expended
by the forces acting on it.
因为动能的概念，及一般能量的概念，都是如此重要，所以方程中不同的项，被赋予了不同的名字。正如我们所知，mv2/2被称为动能。F · v被称为能量：作用于一个对象的力，乘以对象的矢速（矢速“点”积），就是这个力正在传送给对象的力量。这样，我们就有了一个了不起的定理：一个对象的动能的变化率，就等于，作用于其上的力所耗费的能量。

A word about units. Since forces are measuredin newtons, and we multiply by a distance in order to obtain work, work ismeasured in newton · meters (N· m),but people do not like to saynewton-meters, they prefer to say joules (J). A newton-meter is called ajoule; work is measured in joules. Power, then, is joules per second, and thatis also called a watt (W). If we multiply watts by time, the result isthe work done. The work done by the electrical company in our houses,technically,is equal to the watts times the time. That is where we get things like kilowatthours, 1000 watts times 3600 seconds, or 3.6 × 106 joules.
关于单位，说两句。由于力的测量，用的是牛顿乘以距离，以得到功，功的测量，用的是牛顿·米 (N· m)，但是，人们不喜欢说牛顿·米，而更喜欢说焦耳(J)。1牛顿·米，被称为一个焦耳；功的测量，用焦耳。因此，功率就是焦耳每秒，这也被称为瓦特(W)。如果我们让瓦特乘以时间，结果就是所做的功。电力公司在我们房子中所做的功，确切意义上说，就等于瓦特乘以时间。我们就是这样得到千瓦时的，1000 瓦特乘以3600 秒，或者 3.6 × 106 焦耳。