Everything we have just said is containedin the formula W=∫F⋅ds . It is all very well to say that it is a marvelous formula, but it isanother thing to understand what it means, or what some of the consequencesare.
我们刚才所说的一切，都包含在公式W=∫F⋅ds中。说它是一个了不起的公式，很好，完全没有问题，但是，理解其意义、或一些后果，则是另一回事，

The word “work” in physics has a meaning sodifferent from that of the word as it is used in ordinary circumstances that itmust be observed carefully that there are some peculiar circumstances in whichit appears not to be the same. For example, according to the physicaldefinition of work, if one holds a hundred-pound weight off the ground for awhile, he is doing no work. Nevertheless, everyone knows that he begins to sweat,shake, and breathe harder, as if he were running up a flight of stairs. Yetrunning upstairs is considered as doing work (in running downstairs,one gets work out of the world, according to physics), but in simply holding anobject in a fixed position, no work is done. Clearly, the physical definitionof work differs from the physiological definition, for reasons we shall brieflyexplore.
“功”这个词，在物理学中的意义，与在普通情况中的意义，完全不同，所以应该仔细观察，因为在有些具体情况下，意义似乎不一样。例如，依据物理学对功的定义，如果有人把100磅重量，（从地面上抱起来），抱一会儿，他并没有做功。尽管如此，每个人都知道，他要开始出汗，发抖，和气喘吁吁了，就好像他跑了一段楼梯一样。当然，向上跑楼梯，被认为是做了功（向下跑楼梯，依据物理学，是从世界上拿走功），但是，拿着一个东西，静止不动，没有做功。很清楚，功的物理定义，与生理定义不同，其原因，我们将做简单探索。

It is a fact that when one holds a weighthe has to do “physiological” work. Why should he sweat? Why should he need to consumefood to hold the weight up? Why is the machinery inside him operating at fullthrottle, just to hold the weight up? Actually, the weight could be held up withno effort by just placing it on a table; then the table, quietly and calmly,without any supply of energy, is able to maintain the same weight at the sameheight! The physiological situation is something like the following. There aretwo kinds of muscles in the human body and in other animals: one kind, called striatedor skeletal muscle, is the type of muscle we have in our arms, forexample, which is under voluntary control; the other kind, called smoothmuscle, is like the muscle in the intestines or, in the clam, the greater adductormuscle that closes the shell. The smooth muscles work very slowly, but they canhold a “set”; that is to say, if the clam tries to close its shell in a certainposition, it will hold that position, even if there is a very great forcetrying to change it. It will hold a position under load for hours and hours withoutgetting tired because it is very much like a table holding up a weight, it “sets”into a certain position, and the molecules just lock there temporarily with nowork being done, no effort being generated by the clam. The fact that we haveto generate effort to hold up a weight is simply due to the design of striatedmuscle. What happens is that when a nerve impulse reaches a muscle fiber, thefiber gives a little twitch and then relaxes, so that when we hold something up,enormous volleys of nerve impulses are coming in to the muscle, large numbersof twitches are maintaining the weight, while the other fibers relax. We can seethis, of course: when we hold a heavy weight and get tired, we begin to shake.The reason is that the volleys are coming irregularly, and the muscle is tiredand not reacting fast enough. Why such an inefficient scheme? We do not knowexactly why, but evolution has not been able to develop fast smoothmuscle. Smooth muscle would be much more effective for holding up weights becauseyou could just stand there and it would lock in; there would be no work involvedand no energy would be required. However, it has the disadvantage that it isvery slow-operating.
当一个人提起一个重量时，他必须要做生理上的功，这是一个事实。为什么他会出汗？为什么他需要消耗食物，以提起这个重量？为什么它里面的那个机制，要开足马力，只是为了拿起这个重量？实际上，这个重量，可以被拿起来，放在桌子上，没有任何其他效果；然后，这个桌子，可以默默无闻地，把这个重量，保持在这个高度，无需任何能量支持！生理的情况，大致如下。在人类和其他动物的身体中，有两类肌肉，一类被称为条纹肌或骨骼肌，这类肌肉，可随意控制，例如，我们的膀子上的肌肉；另一类，被称为平滑肌，如肠中的肌肉，或者，蚌的肌肉，最大的闭壳肌，用来关闭壳的。平滑肌工作很慢，但是，它们可以举起一个“集”{？}，也就是说，如果蚌尝试在一个位置，关掉它的壳，它会保持那个位置，即便那些非常大的力量，想改变之{也不行}。这种肌肉，在负载下，能保持这个位置，达数小时而不会变得疲倦，因为，这很像桌子，载着重量，它会设置到一定的位置，分子只是暂时地锁在那里，没做任何功，蚌并没有做任何努力。我们必须努力，才能举起一个重量，这一事实，可简单地要归于条纹肌的设计安排。实际发生的是这样，当神经脉冲达到肌纤维的时候，纤维有一个小的抽动，然后放松，这样当我们提起某物时，巨大的神经脉冲齐射，就会来到肌肉，大量的抽动，就会维持这个重量，而其他的纤维则会放松。当然，我们可以看到这一点：当我们拿起一个非常重的东西时，就会变得疲劳，然后，就会颤抖。原因就是，齐射不规则地到来，而肌肉变得疲劳，反应也不够快。为什么是这样一种低效的方案？我们并不准确地知道为什么，但是，进化还没有发展出快的平滑肌。对于拿起重量来说，平滑肌可以更有效，因为，你只需站在那里，他就会锁住{？}，不会牵扯到功，对能量也没有要求。然而，它也有的不利的一点，即它是慢操作型的。

Returning now to physics, we may ask whywe want to calculate the work done. The answer is that it is interesting anduseful to do so, since the work done on a particle by the resultant of all theforces acting on it is exactly equal to the change in kinetic energy of thatparticle. That is, if an object is being pushed, it picks up speed, and
Δ(v2)=2F⋅Δs /m.
现在，返回物理学。我们可以问，为什么我们想计算所做的功。答案是，这样做，既有用又有趣，由于一个粒子，可以受很多力，这些力的合力所做的功，准确地等于其动能的变化。也就是说，如果一个对象被推了，它会得到速度，且
Δ(v2)=2F⋅Δs /m.

14–2Constrained motion 14-2 受限的运动
Another interesting feature of forces andwork is this: suppose that we have a sloping or a curved track, and a particlethat must move along the track, but without friction. Or we may have a pendulumwith a string and a weight; the string constrains the weight to move in acircle about the pivot point. The pivot point may be changed by having thestring hit a peg, so that the path of the weight is along two circles of differentradii. These are examples of what we call fixed, frictionless constraints.
力和功的另外一个有趣特征就是：假设我们有一个斜的或弯曲的轨道，一个粒子必须沿着这个轨道运动，但没有摩擦。或者，我们可以有一个单摆，由一根绳子和一个重量组成，这个绳子限制着重量，让重量绕着一个轴心点，做圆运动。通过让轴心点撞一个钉子，可以让轴心点发生改变，于是，重量的路径，就是沿着两个不同半径的圆。这些就是我们称为固定的、无摩擦限制的例子。

In motion with a fixed frictionlessconstraint, no work is done by the constraint because the forces of constraintare always at right angles to the motion. By the “forces of constraint” we meanthose forces which are applied to the object directly by the constraintitself—the contact force with the track, or the tension in the string.
在些运动，是有限制{条件}的，这种限制，是固定的和无摩擦的，限制并未做功，因为限制的力，总垂直于运动。所谓“限制的力”，我们的意思是指：那些通过限制本身，而被直接应用到对象上的力—即与轨道接触的力，或绳子的张力。

The forces involved in the motion of aparticle on a slope moving under the influence of gravity are quite complicated,since there is a constraint force, a gravitational force, and so on. However,if we base our calculation of the motion on conservation of energy and thegravitational force alone, we get the right result. This seems ratherstrange, because it is not strictly the right way to do it—we should use the resultantforce. Nevertheless, the work done by the gravitational force alone will turnout to be the change in the kinetic energy, because the work done by the constraintpart of the force is zero (Fig. 14–1).
一个粒子，在重力的影响下，在一个斜坡上运动，所牵扯到的力，相当复杂，由于有一个限制的力、一个重力等。然而，如果让我们对运动的计算，只基于能量守恒和万有引力，那么，我们还是会得到正确的结果。这似乎相当奇怪，因为，严格讲，这并不是做此事的正确方法--我们应该用合力。尽管如此，将会证明，仅万有引力做的功，将会是动能变化的原因，因为限制部分的力，所做功为零（图14-1）。

Fig. 14–1.Forces acting on a sliding body(no friction). 图14-1 一个滑动物体（无摩擦力）上的力。

The important feature here is that if aforce can be analyzed as the sum of two or more “pieces” then the work done bythe resultant force in going along a certain curve is the sum of the works doneby the various “component” forces into which the force is analyzed. Thus if we analyzethe force as being the vector sum of several effects, gravitational plusconstraint forces, etc., or the x -component of all forces and the y -component of all forces, or any other way that we wish to split itup, then the work done by the net force is equal to the sum of the works doneby all the parts into which we have divided the force in making the analysis.
这里的重要特征就是，如果一个力，可以被分析为多个部分的总和，那么，合力沿着曲线所做的功，就是‘力所分解成的各分力’所做功的总和。这样，如果我们把力，分析为几个效力的矢量总和，如：重力加上限制力等等，或者是所有力在x方向的分量、在y方向的分量，或任何其他我们可以把它分开的方法，那么，净力所做的功，就等于所有分力所做的功；分力就是在分析时，我们把力所分成的部分。

14–3 Conservative forces 14-3 保守的力
In nature there are certain forces, that ofgravity, for example, which have a very remarkable property which we call“conservative” (no political ideas involved, it is again one of those “crazywords”). If we calculate how much work is done by a force in moving an objectfrom one point to another along some curved path, in general the work depends uponthe curve, but in special cases it does not. If it does not depend upon thecurve, we say that the force is a conservative force. In other words, if theintegral of the force times the distance in going from position 1 to position 2 in Fig. 14–2 iscalculated along curve A and then along B , we get the same number of joules, and if this is true for this pairof points on every curve, and if the same proposition works no matterwhich pair of points we use, then we say the force is conservative. In suchcircumstances, the work integral going from 1 to 2 can be evaluated in a simple manner, and we can give a formula for theresult. Ordinarily it is not this easy, because we also have to specify the curve,but when we have a case where the work does not depend on the curve, then, ofcourse, the work depends only upon the positions of 1 and 2 .
在自然中，有某些列，例如重力，它们又一个非常著名的性质，我们称之为“保守的”（没有牵扯到政治意义，这又是那种“疯狂词汇”。）一个对象，被一个力作用，沿着某个曲线路径，从一个点运动到另一个点，如果我们计算，此力做了多少功，那么，通常此功依赖于曲线，但在特殊案例中，则不然。如果它不依赖于曲线，那么，我们就说，此力是一个保守的力。换句话说，如图14-2，从位置1到位置2，对时间乘以距离进行积分，如果此积分，无论是沿着曲线A，还是沿着曲线B，我们都得到同样的焦耳数，如果对于曲线上的每对点来说，这个积分都为真，且如果无论我们使用的这一对点是什么，这个命题都成立，那么，我们就说，力是保守的。在这种情形下，从1到2，积分出来的功，可以用一种简单的方式来估算，且我们可以为结果，给出一个公式。通常，这并不容易，因为，我们还必须具体指定曲线，但是，当我们的案例不依赖于曲线时，那么，当然，功就只依赖于位置1和2.

Fig. 14–2.Possible paths between two pointsin a field of force. 图14-2 一个力场中的两个点之间的可能路径

We shall call this function of position −U(x,y,z), and when we wish to refer to some particular point 2 whose coordinates are (x2,y2,z2), we shall write U(2) , as an abbreviation for U(x2,y2,z2). The work done in going from point 1 to point P can be written also by going the other way along the integral,reversing all the ds ’s. That is, the work done in going from 1 to P is minus the work done in going from the point P to 1 :
Thus the work done in going from P to 1 is −U(1) , and from P to 2 the work is −U(2) . Therefore the integral from 1 to 2 is equal to −U(2) plus [−U(1) backwards], or +U(1)−U(2) :
(14.1)
The quantity U(1)−U(2) is called the change in the potential energy, and we call Uthe potential energy. We shall say that when the object is located atposition 2 , it has potential energy U(2) and at position 1 it has potential energy U(1) . If it is located at position P , it has zero potential energy. If we had used any other point,say Q , instead of P , it would turn out (and we shall leave it to you to demonstrate) thatthe potential energy is changed only by the addition of a constant.Since the conservation of energy depends only upon changes, it does notmatter if we add a constant to the potential energy. Thus the point Pis arbitrary.
我们把这个位置的函数，称为−U(x,y,z)，设某个具体的点2的坐标为(x2,y2,z2)，当我们希望引用它时，我们将写成 U(2)，作为 U(x2,y2,z2)的缩写。从点1到点P所做的功，也按积分线的反方向来写，所有的ds都取反了。也就是说，从点到P所做的功，就是负的从点P到点1所做的功：

这样，从P到1所做的功，就是−U(1)，从P到2所做的功，就是 −U(2)。因此，从1到2的积分，就等于−U(2) 加上[−U(1) 反向], or +U(1)−U(2):

(14.1)
量U(1)−U(2)，被称为势能的变化，我们称U为势能。当对象处于位置2时，我们说，其势能为U(2)，处于位置1时，我们说，其势能为U(1)。如果在位置P，其势能为零，如果我们用另一个点，比如说Q，来代替P，那么,结果就是，势能的变化，只是加了一个常数。（我们将把这个留给读者去演证）。由于势能的守恒，只依赖于变化，与加不加一个常数无关。这样，点P就是任意的。

Now, we have the following twopropositions: (1) that the work done by a force is equal to the change inkinetic energy of the particle, but (2) mathematically, for a conservativeforce, the work done is minus the change in a function U which we call the potential energy. As a consequence of these two, wearrive at the proposition that if only conservative forces act, the kineticenergy T plus the potential energy U remains constant:
T+U=constant. (14.2)
现在，我们就有下面两个命题：（1）一个力所做的功，等于这个粒子的动能的变化，但是，（2）从数学上看，对于一个保守的力，它所做的功，就是负的函数U的变化，U我们称为势能。作为这两个命题的结果，我们得到一个命题：如果只有保守力在做功，动能T加上势能U，恒为常数：
T+U=constant. (14.2)

Let us now discuss the formulas for the potential energy for a numberof cases. If we have a gravitational field that is uniform, if we are not goingto heights comparable with the radius of the earth, then the force is a constantvertical force and the work done is simply the force times the verticaldistance. Thus
U(z)=mgz, (14.3)
and the point P which corresponds to zero potential energy happens to be any point inthe plane z=0 . We could also have said that the potential energy is mg(z−6)if we had wanted to—all the results would, of course, be the same inour analysis except that the value of the potential energy at z=0 would be −6mg . It makes no difference, because only differences in potentialenergy count.
现在，我们来看几个案例，讨论它们势能的公式。如果我们有一个均匀的万有引力场，如果我们所达到的高度，不足以与这个地球的半径相比，那么，力就恒为垂直的，而功就只是力乘以垂直距离。这样：
U(z)=mgz, (14.3)
且相应于零势能的点P，刚好就在平面z=0处。如果我们想的话，我们也可以说，势能是mg(z−6)，--当然，在我们的分析中，所有的结果，将是同样的，除了在z=0处的势能的值，将是−6mg。但这并没有任何区别，因为，只有势能的差，才被用到。

The energy needed to compress a linearspring a distance x from an equilibrium point is
U(x)= kx2/2 (14.4)
and the zero of potential energy is at the point x=0 , the equilibrium position of the spring. Again we could add anyconstant we wish.
把一个线性的弹簧，从平衡点压缩距离x，所需的能量就是：
U(x)= kx2/2 (14.4)
且势能为零的位置，就是点x=0，即弹簧的平衡位置。我们还是加上任何我们希望的常数。

The potential energy of gravitation forpoint masses M and m , a distance r apart, is
U(r)=−GMm/r. (14.5)
The constant has been chosen here so that the potential is zero atinfinity. Of course the same formula applies to electrical charges, because itis the same law:
U(r)=q1q2/4πϵ0r. (14.6)
在万有引力场中，有质量点M和 m，其距离为r，那么，势能就是：
U(r)=−GMm/r。 (14.5)
这里，常数已经被选择了，这样在无穷远处，势能为零。当然，同样的公式，可以用于电荷，因为规律一样：
U(r)=q1q2/4πϵ0r. (14.6)