Let us consider one nice example ofcompound motion in a plane. We shall take a motion in which a ball moveshorizontally with a constant velocity u , and at the same time goes vertically downward with a constant acceleration −g; what is the motion? We can say dx/dt= vx= u . Since the velocity vx is constant,
我们考虑一个例子：平面上的复合运动。我们看一个运动，一个球，水平方向，以匀速u运动，垂直方向，以匀加速度−g运动；这是什么运动？我们可以说dx/dt= vx= u。由于速度vx是常数，
x=ut,(8.17)
and since the downward acceleration −g is constant, the distance y the object falls can be written as
且由于向下的加速度是−g是常数，对象下落的距离y可写为：
y=−1/2gt2.(8.18)
What is the curve of its path, i.e., what is the relation between yand x ? We can eliminate t from Eq. (8.18),since t=x/u . When we make this substitution we find that
它的曲线是什么，亦即，y与x之间的关系是什么？由于t=x/u，我们可以从方程（8.18）中剔除t。做此替换后，我们发现：
y=−gx2/(2u2).(8.19)
This relation between y and x may be considered as the equation of the path of the moving ball. Whenthis equation is plotted we obtain a curve that is called a parabola; any freelyfalling body that is shot out in any direction will travel in a parabola, as shownin Fig. 8–4.
这个y与x之间的关系，可以被认为是移动球的路径的方程。把这个方程画成曲线，我们就得到一个抛物线；在任何方向被射出的物体，其自由下落，都会按抛物线来走，如图8-4所示。

Fig. 8–4.The parabola described by afalling body with an initial horizontal velocity. 图8-4 由一个初始水平速度的下落物体所描绘的抛物线。

Chapter9. Newton’s Laws of Dynamics第9章牛顿的力学规律
9–1Momentum and force 9-1 动量与力
The discovery of the laws of dynamics, orthe laws of motion, was a dramatic moment in the history of science. BeforeNewton’s time, the motions of things like the planets were a mystery, but afterNewton there was complete understanding. Even the slight deviations fromKepler’s laws, due to the perturbations of the planets, were computable. Themotions of pendulums, oscillators with springs and weights in them, and so on,could all be analyzed completely after Newton’s laws were enunciated. So it iswith this chapter: before this chapter we could not calculate how a mass on aspring would move; much less could we calculate the perturbations on the planetUranus due to Jupiter and Saturn. After this chapter we will be able tocompute not only the motion of the oscillating mass, but also the perturbationson the planet Uranus produced by Jupiter and Saturn!
力学或运动规律的发现，在科学史上，是激动人心的。在牛顿之前，像行星这种事物的运动，是神秘的，但在牛顿之后，就有了完整的理解。甚至，来自开普特规律的轻微的偏差，也是可以计算的；此偏差，归于行星间的干扰。在牛顿规律被阐明之后，单摆的运动、由弹簧和重量构成的振荡器的运动等，都可以被完整地分析了。所以，这一章要讲的就是：在这一章之前，一个弹簧上的质量如何移动，我们无法计算；更不用说，计算木星和土星对天王星的扰动了。在这一章之后，我们将不仅能够计算‘正在振荡着的质量’的运动，而且，也可以计算：由木星和土星所产生的对天王星的干扰。

Of course, the next thing which is neededis a rule for finding how an object changes its speed if something isaffecting it. That is, the contribution of Newton. Newton wrote down threelaws: The First Law was a mere restatement of the Galilean principle of inertiajust described. The Second Law gave a specific way of determining how thevelocity changes under different influences called forces. The Third Lawdescribes the forces to some extent, and we shall discuss that at another time.Here we shall discuss only the Second Law, which asserts that the motion of anobject is changed by forces in this way: the time-rate-of-change of aquantity called momentum is proportional to the force. We shall state thismathematically shortly, but let us first explain the idea.
自然，下面需要的，就是一条规则，用来找出，当一个对象被某物影响时，其速度如何变化？这就是牛顿的贡献。牛顿写了三条规律，第一规律，就是刚刚讲过的伽利略惯性定律的重述。第二规律，在被称为力的影响下，速度会发生变化, 如何定义这种变化呢？第二规律给出了具体的方法。第三规律，在某种程度上，描述了力，我们将在另外的时间讨论它。这里，我们将只讨论第二规律，它断言，一个对象的运动，在受力后，是以这样的方式，发生改变的：一个量的时间变化率，它被称为动量，正比于力。我们先解释这个想法，等一会儿，再用数学的方式来表达。

We use the term mass as aquantitative measure of inertia, and we may measure mass, for example, by swingingan object in a circle at a certain speed and measuring how much force we needto keep it in the circle. In this way we find a certain quantity of mass forevery object. Now the momentum of an object is a product of two parts:its mass and its velocity. Thus Newton’s Second Law may be writtenmathematically this way:
F=d(mv)/dt. (9.1)
我们用词汇质量，作为惯性的一个定量测量，例如，我们可以这样测量质量：用绳子绑住一个对象，让它以一定的速度，绕一个点旋转，然后测量，要让它保持这种圆运动，需要多少力。以这种方式，我们就为每种对象，都找到了质量。现在，一个对象的动量，就是两部分的乘积：其质量与其矢量速度。这样，牛顿第二规律，用数学写就是：
F=d(mv)/dt. (9.1)

Now there are several points to be considered. In writing down any lawsuch as this, we use many intuitive ideas, implications, and assumptions whichare at first combined approximately into our “law.” Later we may have to comeback and study in greater detail exactly what each term means, but if we try todo this too soon we shall get confused. Thus at the beginning we take several thingsfor granted. First, that the mass of an object is constant; it isn’t really,but we shall start out with the Newtonian approximation that mass is constant,the same all the time, and that, further, when we put two objects together,their masses add. These ideas were of course implied by Newton when hewrote his equation, for otherwise it is meaningless. For example, suppose themass varied inversely as the velocity; then the momentum would never changein any circumstance, so the law means nothing unless you know how the masschanges with velocity. At first we say, it does not change.
现在，有几点需要考虑。在写下任何这种规律时，我们都会有用到很多直观的想法、暗示和假设，它们最初，都是以近似的值，混在我们的“规律”中。稍后。我们会回来，更仔细地研究每个词语的意思，但是，如果这些事情做的太早，我们就会迷惑。这样，在开始的时候，我们把有些事情，认为是当然的。首先，一个对象的质量是常数，它并不真是如此，但是，我们将从牛顿的近似出发，认为质量是常数，在所有时间都一样，另外，当我们把两个对象放在一起的时候，它们的质量是相加的。当牛顿写下这个方程时，他也就暗示了这些想法，否则的话，方程就没有意义。例如，假设质量与速度成反比，那么，动量在任何情况下都不会改变，于是这个规律，就毫无意义，除非你知道质量是如何随速度而变化的。所以首先我们说：它并不变化。

Then there are some implications concerningforce. As a rough approximation we think of force as a kind of push or pullthat we make with our muscles, but we can define it more accurately now that wehave this law of motion. The most important thing to realize is that thisrelationship involves not only changes in the magnitude of the momentumor of the velocity but also in their direction. If the mass is constant,then Eq. (9.1)can also be written as
F=mdv/dt=ma. (9.2)
因此，有些暗示，牵扯到力。作为一种粗略的近似，我们把力，思考为某种我们肌肉所做的推或拉，现在，我们有了这个运动的规律，我们就可以更准确地定义它。应该意识到的。可意识到的最重要的事情，就是这个关系，不仅包含着‘动量或矢量速度’的大小的变化，而且也包含着它们方向的变化。如果质量是常数，则公式（9.1）也可写为：
F=mdv/dt=ma. (9.2)

The acceleration a is the rate of change of the velocity, and Newton’s Second Law says morethan that the effect of a given force varies inversely as the mass; it says alsothat the direction of the change in the velocity and the directionof the force are the same. Thus we must understand that a change in a velocity,or an acceleration, has a wider meaning than in common language: The velocityof a moving object can change by its speeding up, slowing down (when it slowsdown, we say it accelerates with a negative acceleration), or changing itsdirection of motion. An acceleration at right angles to the velocity was discussedin Chapter 7. There we saw that an object moving in a circle ofradius R with a certain speed v along the circle falls away from a straightline path by a distanceequal to 1/2(v2/R)t2 if t is very small. Thus the formula for acceleration at right angles to themotion is
a=v2/R, (9.3)
and a force at right angles to the velocity will cause an object tomove in a curved path whose radius of curvature can be found by dividing the forceby the mass to get the acceleration, and then using (9.3).
加速度a，是矢量速度变化的比率，对于一个被给予的力，其所造成的影响，与质量成反比，牛顿第二规律所说的，还要更多；它还说了，矢量速度的方向的改变，与力的方向的改变，是一样的。这样，我们就应该理解：在矢量速度或加速度中，改变这个词的意义，要比普通语言中的，更广泛些。一个移动着的对象的矢量速度，可以通过其加速或减速，（当它减速时，我们说它用一个负的加速度，来加速），或通过改变其移动方向，来改变。在第7章，讨论过一个矢量速度的直角方向的加速度。在那里，我们看的一个对象，绕一个半径为 R的圆，以一定的速度v运动时，会从一条直线路径下落，如果t非常小的话，下落的距离就是 1/2(v2/R)t2。这样，在直角方向，对圆的加速度就是：
a=v2/R, (9.3)
对矢量速度的在直角方向的一个力，将会导致对象，沿着一个弯曲的路径移动，它的曲率半径，可以这样得到：力除以质量，得到加速度，然后使用（9.3）。

9–2Speed and velocity 9-2 速度与矢量速度
In order to make our language more precise,we shall make one further definition in our use of the words speed and velocity.Ordinarily we think of speed and velocity as being the same, and in ordinarylanguage they are the same. But in physics we have taken advantage of the factthat there are two words and have chosen to use them to distinguish twoideas. We carefully distinguish velocity, which has both magnitude and direction,from speed, which we choose to mean the magnitude of the velocity, but whichdoes not include the direction. We can formulate this more precisely by describinghow the x -, y -, and z -coordinates of an object change with time. Suppose, for example, thatat a certain instant an object is moving as shown in Fig. 9–1. In a givensmall interval of time Δt it will move a certain distance Δx in the x -direction, Δy in the y -direction, and Δz in the z -direction. The total effect of these three coordinate changes is adisplacement Δs along the diagonal of a parallelepiped whose sides are Δx , Δy , and Δz . In terms of the velocity, the displacement Δx is the x -component of the velocity times Δt , and similarly for Δy and Δz :
Δx=vxΔt, Δy=vyΔt, Δz=vzΔt. (9.4)
为了让我们的语言更精确，我们要对速度和矢量速度，做进一步的定义，以便使用。平常，我们认为速度和矢量速度，是同样的，且在平常的语言中，它们就是一样的。但在物理学中，我们要利用这一事实：即这两个词，可以用来区别两个观念。我们把速度矢量速度，仔细地区别开来，矢量速度，既有大小又有方向，而速度，则只意味着矢量速度的大小，不包含方向。我们可以把这一点，用公式更精确地表达，即通过描述，一个对象的x、y、z坐标，是如何变化随着时间而改变的。例如，假设在某一瞬间，一个对象的移动，如图9-1所示。在一个被给予的、小的时间间隔Δt内，它在x方向，移动一定的距离Δx，在y方向，移动Δy，z方向，Δz。这三个坐标变化的总的效果，就是位移Δs，它沿着平行六面体的对角线，此六面体的边长，就是Δx , Δy和 Δz。用矢量速度的话说，位移Δx，就是矢量速度的x分量，乘以Δt；Δy和 Δz类似：
Δx=vxΔt, Δy=vyΔt, Δz=vzΔt. (9.4)

Fig. 9–1.A small displacement of an object.图9-1 一个对象的小的位移。

9–3Components of velocity, acceleration,and force 9-3 矢量速度的分量，加速度，和力
In Eq. (9.4)we have resolved the velocity into components by telling how fast theobject is moving in the x -direction, the y -direction, and the z -direction. The velocity is completely specified, both as to magnitudeand direction, if we give the numerical values of its three rectangular components:
vx=dx/dt, vy=dy/dt, vz=dz/dt. (9.5)
On the other hand, the speed of the object is
ds/dt=|v|=v2x+v2y+v2z−−−−−−−−−−√.(9.6)
在方程（9.4）中，通过告知对象在x、y、z方向移动的有多快，我们把矢量速度分解成分量。矢量速度被完整地具体化化了，包括大小和方向，如果我们给出这三个成直角的分量的数值：
vx=dx/dt, vy=dy/dt, vz=dz/dt. (9.5)
另一方面，对象的速度就是：
(9.6)

Fig. 9–2.A change in velocity in which both the magnitudeand direction change. 图9-2 一个矢量速度的改变，大小与方向都变了。

Next, suppose that, because of the actionof a force, the velocity changes to some other direction and a differentmagnitude, as shown in Fig. 9–2. We cananalyze this apparently complex situation rather simply if we evaluate thechanges in the x -, y -, and z -components of velocity. The change in the component of the velocity inthe x -direction in a time Δt is Δvx=axΔt , where ax is what we call the x -component of the acceleration. Similarly, we see that Δvy=ayΔtand Δvz=azΔt . In these terms, we see that Newton’s Second Law, in saying that theforce is in the same direction as the acceleration, is really three laws, inthe sense that the component of the force in the x -, y -, or z -direction is equal to the mass times the rate of change of thecorresponding component of velocity:
FxFyFz=m(dvx=m(dvy=m(dvz/dt)=m(d2x/dt)=m(d2y/dt)=m(d2z/dt2)=max/dt2)=may/dt2)=maz,,.(9.7)
下面，假设因为一个力的作用，矢量速度的方向和大小，都改变了，如图9-2所示。这个情况，明显很复杂，但是，如果我们能够估算矢量速度的x、y、z分量的改变，那么，分析它，就会相当简单。在时间Δt内，在x方向的矢量速度的改变，就是Δvx=axΔt，这里ax就是加速度的x分量。类似地，可以看到Δvy=ayΔt，Δvz=azΔt。牛顿的第二规律说，力与加速度的方向一样；在这些项中，我们可以看到牛顿的第二规律，确实就是三条规律，意义就是，力在x、y、z方向的分量，就等于质量，乘以，相应的矢量速度的分量的变化率：

(9.7)

Just as the velocity and acceleration have been resolved intocomponents by projecting a line segment representing the quantity, and itsdirection onto three coordinate axes, so, in the same way, a force in a givendirection is represented by certain components in the x -, y -, and z -directions:
FxFyFz=Fcos(x=Fcos(y=Fcos(z,F),,F),,F),(9.8)
where F is the magnitude of the force and (x,F) represents the angle between the x -axis and the direction of F , etc.
通过把矢量速度和加速度，投影到三个坐标轴上，可以把其分解，投影的线段，代表着其量和方向，于是，以同样的方式，一个被给予方向的力，也可以用x -, y -, 和z -方向的分量，来代表：
(9.8)
这里F就是力的大小，(x,F)代表着x轴和F的方向之间的角度，如此等等。

Newton’s Second Law is given in complete formin Eq. (9.7).If we know the forces on an object and resolve them into x -, y -, and z -components, then we can find the motion of the object from theseequations. Let us consider a simple example. Suppose there are no forces in they - and z -directions, the only force being in the x -direction, say vertically. Equation (9.7)tells us that there would be changes in the velocity in the vertical direction,but no changes in the horizontal direction. This was demonstrated with aspecial apparatus in Chapter 7(see Fig. 7–3). A falling body moves horizontally without anychange in horizontal motion, while it moves vertically the same way as it wouldmove if the horizontal motion were zero. In other words, motions in the x-, y -, and z -directions are independent if the forces are not connected.
在方程（9.7）中，牛顿第二定律，以完整的形式被给予。如果我们知道作用于一个对象上的力，并把它们分解成x -, y -, 和 z -分量，那么，根据这些方程，我们就可以得出对象的运动。让我们考虑一个简单的例子。假设在x -和y -方向，没有力，唯一的力，是z -方向的，也就是说，是垂直的。方程（9.7）告诉我们，垂直方向的矢量速度，将有变化，但水平方向没有。这一点，在第七章中（图7-3），用一个特殊仪器演示了。一个下落物体，在水平运动没有任何变化时，做水平移动，当它垂直移动的时候，其方式，就好像水平运动为零一样。换句话说，如果力是没有联系的话，那么，在x -, y -, 和z -方向上的运动，相互独立。
{？两个地方}

9–4What is the force? 9-4 什么是力？
In order to use Newton’s laws, we have tohave some formula for the force; these laws say pay attention to the forces.If an object is accelerating, some agency is at work; find it. Our program forthe future of dynamics must be to find the laws for the force. Newtonhimself went on to give some examples. In the case of gravity he gave a specificformula for the force. In the case of other forces he gave some part of theinformation in his Third Law, which we will study in the next chapter, havingto do with the equality of action and reaction.
为了使用牛顿规律，我们必须为力，找到一些公式；这些规律说，要注意这些力。如果一个对象正在被加速，那么，就是有些代理，正在工作；找到它们。我们的计划，关乎力学的未来，应该为这些力，找到规律。牛顿本人给出了一些例子。在万有引力这种情况，他给出了具体的公式。对于其他的力，他在他的第三规律中，给出了部分信息；第三规律，处理的是作用力与反作用力，我们将在下一章研习。

Fig. 9–3.A mass on a spring. 图9-3 弹簧上的一个质量
As another example, let us suppose that wehave been able to build a gadget (Fig. 9–3) whichapplies a force proportional to the distance and directed oppositely—a spring.If we forget about gravity, which is of course balanced out by the initialstretch of the spring, and talk only about excess forces, we see that ifwe pull the mass down, the spring pulls up, while if we push it up the springpulls down. This machine has been designed carefully so that the force isgreater, the more we pull it up, in exact proportion to the displacement fromthe balanced condition, and the force upward is similarly proportional to howfar we pull down. If we watch the dynamics of this machine, we see a ratherbeautiful motion—up, down, up, down, … The question is, will Newton’sequations correctly describe this motion? Let us see whether we can exactlycalculate how it moves with this periodic oscillation, by applying Newton’slaw (9.7).In the present instance, the equation is
−kx=m(dvx/dt). (9.11)
另外一个例子，我们假设，我们可以建一个小装置（图9-3），它可以应用一个力，正比于距离，指向相反的方向，就是一个弹簧。重力已经被弹簧的初始拉伸，给平衡了；如果我们忘记重力，然后，只说额外的力，我们看到，如果我们把此质量往下拉，弹簧的就往上拉，如果我们把它往上推，弹簧就往下压。这个机器，经过仔细地设计，于是，我们往上推的越多，向下的力就越大，与距平衡条件的位移，精确地成正比，并且，向上的力，类似地正比于我们拉下的距离有多远。如果我们观察这个机器的力学，我们就可以看到一个相当漂亮的运动--上，下，上，下，… … 。现在的问题是，牛顿方程能正确地描述这个运动吗？让我们看看，通过应用牛顿规律（9.7），我们能否准确地计算出：它是如何做这种周期性振动的。在当前的实例中，方程是：
−kx=m(dvx/dt). (9.11)

Here we have a situation where the velocity in the x -direction changes at a rate proportional to x . Nothing will be gained by retaining numerous constants, so we shallimagine either that the scale of time has changed or that there is an accidentin the units, so that we happen to have k/m=1 . Thus we shall try to solve the equation
dvx/dt=−x. (9.12)
To proceed, we must know what vx is, but of course we know that the velocity is the rate of change ofthe position.
我们这里的情况就是，x方向的矢量速度的变化率，正比于x。通过保持无数的常数，我们什么也不会得到，所以，我们就要想象，要么，时间的尺度变化了，要么，单位有意外，于是，我们碰巧就有k/m=1。这样，我们就可尝试解方程：
dvx/dt=−x. (9.12)
要继续前进，我们就应知道，vx是什么？但是，我们当然知道，矢量速度，就是位置的变化率。