Chapter18.Rotation in Two Dimensions第18章 2维中的旋转
18–1The center of mass 18-1 质量的中心
In the previous chapters we have beenstudying the mechanics of points, or small particles whose internal structuredoes not concern us. For the next few chapters we shall study the application ofNewton’s laws to more complicated things. When the world becomes morecomplicated, it also becomes more interesting, and we shall find that thephenomena associated with the mechanics of a more complex object than just apoint are really quite striking. Of course these phenomena involve nothing butcombinations of Newton’s laws, but it is sometimes hard to believe thatonly F=ma is at work.
在前面的几章，我们研究的，都是点的力学，所谓点，也叫小的粒子，其内部结构，我们并不关心。在接下来的几章中，我们将研究，牛顿规律应用于更复杂的事物。当世界变得更复杂时，也变得更加有趣，我们将找出，现象与更复杂的对象的力学的联系，而不只是一个点的力学，这种复杂的联系，确实让人相当震惊。当然，这些现象，牵扯到的，无非只是牛顿规律的组合，有时候，确实很难相信，只有F=ma，在起作用。

The more complicated objects we deal withcan be of several kinds: water flowing, galaxies whirling, and so on. Thesimplest “complicated” object to analyze, at the start, is what we call a rigidbody, a solid object that is turning as it moves about. However, even sucha simple object may have a most complex motion, and we shall therefore first considerthe simplest aspects of such motion, in which an extended body rotates about a fixedaxis. A given point on such a body then moves in a plane perpendicular tothis axis. Such rotation of a body about a fixed axis is called planerotation or rotation in two dimensions. We shall later generalize theresults to three dimensions, but in doing so we shall find that, unlike thecase of ordinary particle mechanics, rotations are subtle and hard tounderstand unless we first get a solid grounding in two dimensions.
我们要处理的对象，更为复杂，可以分为几类：水流，银河系的旋转，等等。在开始的时候，我们要分析的最简单的“复杂的”对象，就是我们称为刚体的东西，一个固体，运动时，在旋转。然而，即便是这样一种简单的对象，也可以有最费解的运动，因此，我们将首先考虑这种运动的最简单的方面，在其中，一个扩展的物体，绕着一个固定的轴旋转。因此，这样一个物体上的一个给定的点，就在一个垂直于轴的平面上运动。这种物体绕着一个固定轴的旋转，被称为平面旋转或二维中的旋转。稍后，我们将把这个结果，推广到三维，但是，要这样做，我们将发现，与普通粒子力学的情况不同，旋转是精巧的，且难以理解，除非我们首先在二维中，得到一个坚实的基础。

The first interesting theorem concerningthe motion of complicated objects can be observed at work if we throw an objectmade of a lot of blocks and spokes, held together by strings, into the air. Ofcourse we know it goes in a parabola, because we studied that for a particle.But now our object is not a particle; it wobbles and it jiggles, and soon. It does go in a parabola though; one can see that. What goes in aparabola? Certainly not the point on the corner of the block, because that isjiggling about; neither is it the end of the wooden stick, or the middle of thewooden stick, or the middle of the block. But something goes in a parabola,there is an effective “center” which moves in a parabola. So our first theoremabout complicated objects is to demonstrate that there is a meanposition which is mathematically definable, but not necessarily a point of thematerial itself, which goes in a parabola. That is called the theorem of thecenter of the mass, and the proof of it is as follows.
复杂对象运动时，有个现象，非常有趣，这也是第一个定理，它可通过下面方式，被观察到：我们把很多木块和木棍，用绳子捆在一起，然后，把它抛向空中。当然，我们知道，它走的是抛物线，因为，对于粒子，我们研究过此事。但是现在，我们的对象，不是粒子，它会摇摇摆摆。尽管如此，它走的确实是一条抛物线；这可看到。那么究竟是什么，在走抛物线呢？当然，不是木块的角上的那个点，因为，它在摆动；也不是木棍的尾端，或者木棍的中间，或者木块的中心。但是，确有某物，在走抛物线，有一个有效的“中心”，在一条抛物线中移动。于是，关于复杂对象，我们的第一条定理，就是有一个平均位置，它可以用数学方法，来定义，但此点，并不必然就在这个材料本身上，且它走的，是抛物线。这就被称为质量中心定理，证明如下。

We may consider any object as being made oflots of little particles, the atoms, with various forces among them. Let irepresent an index which defines one of the particles. (There aremillions of them, so i goes to 1023 , or something.) Then the force on the i th particle is, of course, the mass times the acceleration ofthat particle:
Fi=mi (d2ri /dt2). (18.1)
任何对象，都可被考虑为，是由很多小的粒子、原子构成，它们之间，有力的作用。让i代表着一个索引，它定义了一个粒子（粒子有数百万，所以，可以达到1023之类）。因此，作用于第i个粒子上的力，当然，就是质量，乘以那个粒子的加速度：
Fi=mi (d2ri /dt2). (18.1)

In the next few chapters our moving objects will be ones in which allthe parts are moving at speeds very much slower than the speed of light, and weshall use the nonrelativistic approximation for all quantities. In these circumstancesthe mass is constant, so that
Fi=d2 (miri) / dt2. (18.2)
If we now add the force on all the particles, that is, if we take thesum of all the Fi ’s for all the different indexes, we get the total force, F. On the other side of the equation, we get the same thing as thoughwe added before the differentiation:
(18.3)
Therefore the total force is the second derivative of the masses timestheir positions, added together.
接下来的几章中，对于我们的移动中的对象，其各部分的速度，都是远小于光速，所以，对于所有的量，我们都将用非相对论的近似。在这些情形中，质量就是一个常数，于是：
Fi=d2 (miri) / dt2 (18.2)
如果我们现在，把作用于所有粒子上的力，全加起来，也就是说，如果对于Fi，我们对其所有不同的索引，求和，那么，我们就会得到，总的力F。在方程的另一面，我们会得到同样的事情，就好像，我们是在微分前，做的加法：
(18.3)
因此，总的力，就是质量乘以它们的位置，求二次微分，再全加起来。

Now the total force on all the particles isthe same as the external force. Why? Although there are all kinds offorces on the particles because of the strings, the wigglings, the pullings andpushings, and the atomic forces, and who knows what, and we have to add allthese together, we are rescued by Newton’s Third Law. Between any two particlesthe action and reaction are equal, so that when we add all the equationstogether, if any two particles have forces between them it cancels out in thesum; therefore the net result is only those forces which arise from otherparticles which are not included in whatever object we decide to sum over. Soif Eq. (18.3)is the sum over a certain number of the particles, which together are called“the object,” then the external force on the total object is equal tothe sum of all the forces on all its constituent particles.
现在，作用于所有粒子上的总的力，与外部的力，是一样的。为什么？虽然，由于绳子、摇摆、推和拉、原子的力、及鬼知道什么原因等的作用，使得粒子上，有各种各样的力，且我们还必须把这些力加起来，{感谢上帝}，我们被牛顿第三规律拯救了。在任何两个粒子之间，作用力与反作用力，是相等的，于是，当我们把所有的方程，都加起来时，那么，任何两个粒子，如果它们之间有力，则在总和中，此力就会被抵消掉；因此，净结果就是，从其他的粒子中所产生的力，这里的其他粒子，是指在我们求总和的对象中，所不包含的粒子。于是，如果方程是8.3，覆盖了一定数量的粒子，这些粒子，被称为“此对象”，那么，作用于整个对象上的外部的力，就等于，作用于所有这些粒子上的力的总和。

Now it would be nice if we could writeEq. (18.3)as the total mass times some acceleration. We can. Let us say M is the sum of all the masses, i.e., the total mass. Then if we definea certain vector R to be
(18.4)
Eq. (18.3)will be simply
F=d2(MR)/dt2=M(d2R/dt2), (18.5)
since M is a constant. Thus we find that the external force is the total masstimes the acceleration of an imaginary point whose location is R. This point is called the center of mass of the body. It is apoint somewhere in the “middle” of the object, a kind of average rin which the different ri ’s have weights or importances proportional to the masses.
现在，如果我们可以把方程（18.3），写作：总的质量，乘以某个加速度，那会很好。我们可以这样做。让我们说M是所有质量的总和，亦即，总的质量。因此，如果我们定义一个特定的矢量R：为
(18.4)
由于M是一个常数，方程 (18.3)就会简化为：
F=d2(MR)/dt2=M(d2R/dt2), (18.5)
这样，我们就发现，外部的力，就是总的质量，乘以一个想象出来的点的加速度，该点的位置，就是R。这个点，被称为物体的质量中心。它是一个点，在对象的“中部”某处，是某种r的平均值，在此值中，不同的ri，有正比于质量的重量或重要性{？}。

Is rocket propulsion therefore absolutelyimpossible because one cannot move the center of mass? No; but of course wefind that to propel an interesting part of the rocket, an uninteresting partmust be thrown away. In other words, if we start with a rocket at zero velocityand we spit some gas out the back end, then this little blob of gas goes oneway as the rocket ship goes the other, but the center of mass is still exactlywhere it was before. So we simply move the part that we are interested in againstthe part we are not interested in.
因此，因为人不能移动质量中心，所以，火箭推进就绝对不可能吗？不；但当然，我们发现，要把火箭的一个感兴趣的部分推进，那么，其一个不感兴趣的部分，就必须被抛弃。换句话说，如果一个火箭，初速为零，我们发动它，我们把一些气，从其后部喷出，那么，这一小团气往后走，火箭往前走，但是，质量中心，仍在原处。所以，我们只是，把我们感兴趣的部分，与我们不敢兴趣的部分，分开了。

The second point concerning the center ofmass, which is the reason we introduced it into our discussion at this time, isthat it may be treated separately from the “internal” motions of an object, andmay therefore be ignored in our discussion of rotation.
牵扯到质量中心的，还有第二点，我们在此时，把它引入我们的讨论，原因也是它，它就是：从一个对象的“内部的”运动出发，质量中心，可以分开来对待；因此，在我们关于旋转的讨论中，它也可以被忽略。{？}

18–2Rotation of a rigid body 18-2 刚体的旋转
Now let us discuss rotations. Of course anordinary object does not simply rotate, it wobbles, shakes, and bends, so tosimplify matters we shall discuss the motion of a nonexistent ideal objectwhich we call a rigid body. This means an object in which the forcesbetween the atoms are so strong, and of such character, that the little forcesthat are needed to move it do not bend it. Its shape stays essentially the sameas it moves about. If we wish to study the motion of such a body, and agree toignore the motion of its center of mass, there is only one thing left for it todo, and that is to turn. We have to describe that. How? Suppose there issome line in the body which stays put (perhaps it includes the center of massand perhaps not), and the body is rotating about this particular line as anaxis. How do we define the rotation? That is easy enough, for if we mark a pointsomewhere on the object, anywhere except on the axis, we can always tellexactly where the object is, if we only know where this point has gone to. Theonly thing needed to describe the position of that point is an angle. Sorotation consists of a study of the variations of the angle with time.
现在，我们讨论旋转。当然，普通对象，不会只旋转，它还会摇摆、震颤、和弯曲，所以，为简单起见，我们将讨论一个不存在的理想对象的运动，我们称其为刚体。这就意味着，在此对象中，原子间的力，非常强，且有如此特性，即让它运动，只需很小的力，且不会使其弯曲。当它运动时，其形状基本保持不变。对于这样一个物体，如果我们希望研究其运动，且同意忽略其质量中心的运动，那么，它能做的，只有一件事，就是旋转。我们必须描述它。但怎么描述呢？假设物体中，有某条线，原地不动，(或许其中，包括了质量中心，或许没有)，而此物体，以这条具体的线为轴，旋转。我们如何定义这个旋转呢？这很容易，因为，如果我们在对象上的某处，标记一个点，该点除了不能在这个轴上外，可在任何地方，那么，如果我们知道，这个点走到了那里，那么，我们就总是可以准确地告知，这个对象，在何处。描述这个点的位置，唯一需要的，就是角度。所以，旋转，就是对角度随时间变化的研究。

In order to study rotation, we observe theangle through which a body has turned. Of course, we are not referring to anyparticular angle inside the object itself; it is not that we draw some angleon the object. We are talking about the angular change of theposition of the whole thing, from one time to another.
为了研究旋转，我们观察物体所转过的角度。当然，我们并不是指，在对象里面的任何具体的角度；我们并不是在对象上，画某个角度。我们谈论的，是从一个时间到另一个时间，整个事物的位置的角度变化。

First, let us study the kinematics ofrotations. The angle will change with time, and just as we talked about positionand velocity in one dimension, we may talk about angular position and angularvelocity in plane rotation. In fact, there is a very interesting relationshipbetween rotation in two dimensions and one-dimensional displacement, in whichalmost every quantity has its analog. First, we have the angle θ which defines how far the body has gone around; this replaces thedistance s , which defines how far it has gone along. In the same manner,we have a velocity of turning, ω=dθ/dt , which tells us how much the angle changes in a second, just as v=ds/dtdescribes how fast a thing moves, or how far it moves in a second. Ifthe angle is measured in radians, then the angular velocity ω will be so and so many radians per second. The greater the angularvelocity, the faster the object is turning, the faster the angle changes. Wecan go on: we can differentiate the angular velocity with respect to time, andwe can call α= dω/dt= d2θ/dt2 the angular acceleration. That would be the analog of the ordinaryacceleration.
首先,让我们研究旋转的运动学。角度会随着时间变化，正如我们在一维中，所讨论过的位置和矢速那样，在平面旋转中，我们可以谈论角度的位置和角度的矢速。事实上，在二维的旋转，与一维的位移之间，有个关系，非常有趣：几乎每个量，都可类比。首先，我们有角度θ，它定义物体转了多少，代替了位移，位移定义物体走了多远。以同样的方式，我们有旋转的矢速ω=dθ/dt，它告诉我们，一秒内，角度变化了多少，正如v=ds/dt描述了一个事物运动得有多快，或者，它在一秒内运动了多远。如果角度，用弧度来表示，那么，角速度ω，就是每秒多少多少弧度。角速度越大，物体转的就越快，角度变化的就越快。我们可以继续，我们可以让角速度，对时间求微分，我们可称α= dω/dt= d2θ/dt2，为角加速度。它可类比于普通的加速度。

Fig. 18–1.Kinematics of two-dimensionalrotation. 图18-1 2维旋转的运动学
Now of course we shall have to relate thedynamics of rotation to the laws of dynamics of the particles of which theobject is made, so we must find out how a particular particle moves when theangular velocity is such and such. To do this, let us take a certain particlewhich is located at a distance r from the axis and say it is in a certain location P(x,y)at a given instant, in the usual manner (Fig. 18–1). Ifat a moment Δt later the angle of the whole object has turned through Δθ , then this particle is carried with it. It is at the same radius awayfrom O as it was before, but is carried to Q . The first thing we would like to know is how much the distance xchanges and how much the distance y changes. If OP is called r , then the length PQ is rΔθ , because of the way angles are defined. The change in x , then, is simply the projection of rΔθ in the x -direction:
Δx=−PQsinθ=−rΔθ⋅(y/r)=−yΔθ. (18.6)
Similarly,
Δy=+xΔθ. (18.7)
If the object is turning with a given angular velocity ω , we find, by dividing both sides of (18.6)and (18.7)by Δt , that the velocity of the particle is
vx=−ωy and vy=+ωx. (18.8)
Of course if we want to find the magnitude of the velocity, we just write
(18.9)
It should not be mysterious that the value of the magnitude of thisvelocity is ωr ; in fact, it should be self-evident, because the distance that itmoves is rΔθ and the distance it moves per second is rΔθ/Δt, or rω .
现在，当然，我们将让旋转的动力学，与粒子动力学的规律相关，这里，对象是由粒子构成的，于是，我们应该找出，当角速度如此如此时，一个具体的粒子，是如何运动的。要做此事，让我们先按通常方式，取某个具体的粒子，它与轴的距离为r，比如说，在给定时刻，其位置为P(x,y)，如图18-1。如果，在时间Δt之后，整个对象的角度，转过了Δθ，因此，这个粒子，也被带着走。它被带到Q，但是，与O的距离，与前相同，都是一样的半径。我们想知道的第一件事，就是距离x，变化了多少，距离y，又变化了多少？如果OP被称为r，那么，长度PQ，就是rΔθ，因为角度就是这样被定义的。因此，x的变化，就是rΔθ在x方向的投影：
Δx=−PQsinθ=−rΔθ⋅(y/r)=−yΔθ. (18.6)
类似地：
Δy=+xΔθ. (18.7)
如果对象是以给定的角速度ω在旋转，那么，我们发现，通过让(18.6)和 (18.7)的两边都除以 Δt ，就可得到粒子的矢速：
vx=−ωy 和 vy=+ωx. (18.8)
当然，如果我们想得到矢速的大小，只需如此写：
(18.9)
这个矢速的大小，值为 ωr，这并不神秘；事实上，它应该是自证的，因为移动的距离是rΔθ，每秒移动的距离就是 rΔθ/Δt , 或 rω。

Let us now move on to consider the dynamicsof rotation. Here a new concept, force, must be introduced. Let usinquire whether we can invent something which we shall call the torque(L. torquere, to twist) which bears the same relationship to rotation asforce does to linear movement. A force is the thing that is needed to make linearmotion, and the thing that makes something rotate is a “rotary force” or a“twisting force,” i.e., a torque. Qualitatively, a torque is a “twist”; what isa torque quantitatively? We shall get to the theory of torques quantitativelyby studying the work done in turning an object, for one very nice way ofdefining a force is to say how much work it does when it acts through a givendisplacement. We are going to try to maintain the analogy between linear andangular quantities by equating the work that we do when we turn something alittle bit when there are forces acting on it, to the torque times the angleit turns through. In other words, the definition of the torque is going to beso arranged that the theorem of work has an absolute analog: force times distanceis work, and torque times angle is going to be work. That tells us what torqueis. Consider, for instance, a rigid body of some kind with various forcesacting on it, and an axis about which the body rotates. Let us at firstconcentrate on one force and suppose that this force is applied at a certainpoint (x,y) . How much work would be done if we were to turn the object through avery small angle? That is easy. The work done is
ΔW=FxΔx+FyΔy. (18.10)
We need only to substitute Eqs. (18.6)and (18.7)for Δx and Δy to obtain
ΔW=(xFy−yFx)Δθ. (18.11)
That is, the amount of work that we have done is, in fact, equal tothe angle through which we have turned the object, multiplied by astrange-looking combination of the force and the distance. This “strange combination”is what we call the torque. So, defining the change in work as the torque timesthe angle, we now have the formula for torque in terms of the forces.(Obviously, torque is not a completely new idea independent of Newtonianmechanics—torque must have a definite definition in terms of the force.)
现在，让我们继续考虑，旋转的动力学。这里要解释，一个新的力的概念。这里，让我们考察一下，我们是否可以发明某个东西，它被称为力矩（拉丁语torquere, 扭动），它与旋转的关系，与力与线性运动的关系，一样。要形成线性运动，需要力，而要使得某物旋转，就需要一个“旋转的力”，或一个“扭动的力”，亦即，一个力矩。定性地说，一个力矩，就是一个“扭动”；那么，定量地说，它又是什么呢？旋转一个对象，需要做功，通过研究这个，我们将得到定量的力矩的理论，因为，定义力的一个非常好的方式，就是说：当它在给定距离起作用时，做了多少功。我们将尝试，在线性的量与角度的量之间，保持类比，即对于一个物体，有力作用于其上，让它旋转一点，就会做一些功，我们让这个功，等于力距乘以它所转的角度。换句话说，力矩的定义，将如此安排，即功的定理，有一个绝对的类比：力乘以距离，是功，而力矩乘以角度，也是功。这就告诉了我们，力矩是什么。例如，考虑一个刚体，有各种力，作用于其上，且它绕一个轴旋转。我们首先集中在一个力上，并假设，这个力，被应用于某点point (x,y)。那么，如果我们让此对象，旋转一个很小的角度，会做多少功呢？这很容易。所做功就是：
ΔW=FxΔx+FyΔy. (18.10)
我们只需用方程（18.6）和（18.7）替换掉Δx 和 Δy，就会得到：
ΔW=(xFy−yFx)Δθ. (18.11)
也就是说，我们所做的功，事实上，就等于：对象所转的角度，乘以一个力与距离的组合，此组合，长的很怪。这个“奇怪的组合”，我们称之为力矩。由于把功的改变，定义为：力矩乘以角度，所以，对于力矩，我们现在就有了用力来描述的公式。（很明显，力矩这个想法，并不是一个全新的、独立于牛顿力学的想法—力矩应有一个确定的定义，是用力来描述的。）

When there are several forces acting, thework that is done is, of course, the sum of the works done by all the forces,so that ΔW will be a whole lot of terms, all added together, for all the forces, eachof which is proportional, however, to Δθ . We can take the Δθ outside and therefore can say that the change in the work is equal tothe sum of all the torques due to all the different forces that are acting,times Δθ . This sum we might call the total torque, τ . Thus torques add by the ordinary laws of algebra, but we shall latersee that this is only because we are working in a plane. It is likeone-dimensional kinematics, where the forces simply add algebraically, but onlybecause they are all in the same direction. It is more complicated in three dimensions.Thus, for two-dimensional rotation,
τi=xiFyi−yiFxi (18.12)
and
τ=∑τi . (18.13)
It must be emphasized that the torque is about a given axis. If adifferent axis is chosen, so that all the xi and yi are changed, the value of the torque is (usually) changed too.
当有几个力在起作用，所做的功，当然就是，所有力所做的功之和，因此，ΔW 将是所有力的项，加起来，然而，每一项都正比于Δθ。我们可以把Δθ提出来，从而，就可以说，功的变化，就等于所有力矩的和，乘以Δθ；这里力矩，是由各种不同的力所产生的。这个和，我们可以称之为总的力矩τ。这样，力矩就是通过普通的代数规律，被增加的，但是，稍后我们将看到，这是因为，我们的工作，是在一个平面上。{？added?}它就像是一维运动学，在那里，力被简单地以算术方式相加，那只是因为，它们都在同一个方向。在三维中，要更复杂。这样，对于二维的旋转：
τi=xiFyi−yiFxi (18.12)
和
τ=∑τi . (18.13)
必须强调一点，力矩是相对于一个给定的轴说的。如果轴变了，那么，所有的xi 和yi，就都变了，力矩的值（通常），也就变了。