物理：牛顿三定律

Newton’s first law of motion

In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line).

In simpler terms, we can say that when no force acts on an objects, the acceleration of the object is zero.

牛一定理实际上已经把力和加速度联系起来了。只是这里用的是否定的表达式，无外力则无加速度。

Inertial Frames

An inertial frame of reference is one that is not acceleration. Because Newton’s first law deals only with objects that are not accelerating, it holds only in inertial frames. Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame. The Galilean transformations relate positions and velocities between two inertial frames.

Mass

The tendency of an object to resist any attempt to change its velocity is called inertia of the object. Inertia is a measure of how an object responds to an external force. Mass is that property of an object that specifies how much inertia the object has. The greater the mass of an object, the less that object accelerates under the action of an applied force.

The SI unit of mass is the kilogram.

Mass is an inherent property of an object and is independent of the object’s surroundings and of the method used to measure it.

Newton’s second law

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
$$\sum F=ma$$
The SI unit of force is the newton, which is defined as the force that, when acting on a 1-kg mass, produces an acceleration of $1m/s^2$ .

The force of gravity and weight
The attractive force exerted by the Earth on an object is called the force of gravity F_{g} . This force is directed toward the center of the Earth, and its magnitude is called the weight of the object.

$$F_g=mg$$

Because weight is depends on g , it varies with geographic location. Hence, weight, unlike mass, is not an inherent property of an object.

Newton’s third law

If two objects interact, the force $F_{12}$ exerted by object 1 on object 2 is equal in magnitude to and opposite in direction to the force $F_{21}$ exerted by object 2 on object 1:
$$F_{12}=-F_{21}$$
This is equivalent to stating that a single isolated force cannot exist. 上述两种力，也可以称为作用力和反作用力。In all cases, the action and reaction forces act on different objects.

The force exerted by the hammer on the nail (the action force $F_{hn}$) is equal in magnitude and opposite in the direction to the force exerted by the nail on the hammer (the reaction force $F_{nh}$). It is the latter force that causes the hammer to stop its rapid forward motion when it strikes the nail.

一个放置在桌上的物体，受到重力和桌子对它的支撑力而静止不动，但是这两个力并非作用力和反作用力。因为这两个力都作用在同一个物体上。

在应用牛顿三定律时，一个重要的方法是：

When we apply Newton’s laws to an object, we are interested only in external forces that act on the object.

另外，滑轮是个神奇的存在。当你通过一个滑轮向下发力时，绳索通过滑轮将另一边的物体往上拉。你确确实实向下拉绳了，物体也确确实实受到你的作用力了，然而同时它也确确实实向上加速了。这意味着，假设物体向上的加速度为正，那么你向下的作用力也必须是正，因为： $F=ma$ 。滑轮即改变了实际的方向，也改变了坐标系的方向。

Forces of friction

$$\begin{gathered} f_s\le {\mu }_{s}n \\ {f}_k={\mu }_{k}n \end{gathered}$$
注意：The coefficients of friction are nearly independent of the area of contact between the surfaces. 但是接触面积和normal force还是有关系的。

（注：这里的normal意思是垂直的。）

对ABS的介绍（Automobile Antilock Braking Systems）

简单的说，当汽车轮胎不打滑时，轮胎和地面的摩擦力是 $f_s$，而当汽车轮胎打滑时，摩擦力为 $f_k$。Thus, to maximize the frictional force and minimize stopping distance, the wheels must maintain pure rolling motion and not skid. An additional benefit of maintaining wheel rotation is that directional control is not lost as it is in skidding. Un fortunately, in emergency situations drivers typically press down as hard as they can on the brake pedal, “locking the brakes.” This stops the wheels from rotating, ensuring a skid and reducing the frictional force from the static to the kinetic case. To address this problem, automotive engineers have developed antilock braking systems (ABS) that very briefly release the brakes the brakes when a wheel is just about to stop turning…through the use of computer control, the “brake-off” time is kept to a minimum. As a result, the stopping distance is much less than what it would be if the wheels were to skid.

说明：这里实际上是个结论，具体数据是有实验数据的。即使静止摩擦力没有达到最大，汽车运行过程中的静止摩擦力依然大于运动摩擦力。

关于ABS的讨论还引发了另一个有趣的问题，就是汽车也好，人走路也好，前进的力是什么？然而细想之下，发现这个问题居然不是现在的我可以说的清楚的！这个。。。人前进的方式仔细考虑也是很神奇的。之前我以为是通过脚向后蹬地，地面提供向前的摩擦力，使得人往前走。然而写下不久就觉得这个想法很蠢，就好像你向前推一个箱子，箱子受到地面向后摩擦力的作用反而向后走一样荒谬。那么在摩擦力和蹬地的力在水平方向相当的情况下，究竟是什么导致人前进的呢？好像也不能很通透的说清楚。这个。。。

Newton’s second law applied to uniform circular motion

公式很简单：
$$\begin{gathered} a=v^2/r \\ {F}_r=ma=m{v}^2/r \end{gathered}$$
但是这个被称之为向心力的力，并非一种新类型的力。

We are familiar with a variety of forces in nature–frictions, gravity, normal forces, tension, and so forth. But centripetal force should not be added to this list… A common mistake in force diagrams is to draw all the usual forces and then to add another vector for the centripetal force. But it is not a separate force–it is simply one of our familiar forces acting in the role of a force that causes a circular motion.

例如打球绕太阳旋转，向心力实际上是重力。将一个小球系在绳上，让小球做圆周运动，向心力是绳子的拉力。For an amusement-park patron pressed against the inner wall of a rapidly rotating circular room, the centripetal force is the normal force exerted by the wall.

nonuniform circular motion

无事可记。

motion in accelerated frames

当你坐在一辆正在左转弯的车内，你会感觉到身体向右侧倾斜，一般用离心力来解释这种现象。然而离心力是一种fictitious force. 正确的理解应该是：当车辆拐弯时，人也会随着车子拐弯。这个时候，人也开始做圆周运动。而圆周运动是需要一个向心力的，否则，人会依照惯性继续做直线运动。那么这个向心力哪里来呢？首先是座位和人体间的摩擦力。因为人基于惯性想做直线运动，而车辆处在圆周运动过程中，因此二者开始出现偏差，这个时候必然会有摩擦力。当摩擦力足够大时，就形成了向心力。否则，就需要通过抓住把手等提供额外的力。

那么，所谓的离心力是怎么产生的呢？

These forces “invented” by the observer in the accelerating frame appear to be real. However, we emphasize that these fictitious forces do not exist when the motion is observed in an inertial frame.

原来，是观察的frame不同，导致了对同一个现象的不同解释。

Fictitious forces in linear motions

在火车车厢内，一个小球被绳子悬挂在顶部。当火车停止时，绳子对小球的拉力等于小球的重力，外力为0，小球也静止不动。当火车加速时，小球也不得不随之加速。但加速需要外力，这个时候小球首先基于惯性，倾向于保持静止。车子向前，小球想要保持静止，从车内的角度（the accelerating frame）观察，小球会向后方摆动。从车厢外的人观察（the inertial frame），车子向前，小球保持不动。而当小球向后摆动时，绳子会产生比原来额外的拉力，分解成垂直方向的力就是 $Tsin\theta =mg$ ，而水平方向的力，就是让小球产生加速度的力。但是从车内人的角度看，小球相对于车厢是静止状态的。因此，小球本身并没有加速度。但是，小球因为后摆，绳子和垂直方向形成一个角度。绳子拉力在水平方向有一个分力，因为在车厢内视角，小球是静止的，因此必须要有一个力来抵消这个分力，这个时候就会和前面提到的离心力一样，产生一个虚构力，即 $Tsin\theta -F_{fictitious}=0$ 。实际上，这个虚构力，就等于 $ma$ 。

Fictitious force in a rotating system

在餐桌的一个转盘上（假设没有摩擦力），一个物体被绳子固定在转盘上，当物体随转盘旋转时，实际上是因为绳子拉力对小球形成向心力。但是如果你同时坐在转盘上，那么对你而言，小球并没有转动（地球在自转，但是从人的视角看，周围的物体都是不动的，这是同一个道理）。这个时候，因为绳子有额外的拉力，因此必须有一个虚构力，来抵消掉绳子的拉力，从而让小球维持静止状态。这个力就是所谓的离心力。

Motion in the presence of resistive forces
If we assume that the resistive force acting on an object moving through a liquid or gas is proportional to the object’s speed, then the magnitude of the resistive force can be expressed as

$$R=bv$$
where v is the speed of the object and b is a constant whose value depends on the properties of the medium and on the shape and dimensions of the object. If the object is a sphere of radius r , then b is proportional to r .

Consider a small sphere of mass m released from rest in a liquid. Assuming that the only forces acting on the sphere are the resistive force bv and the force of gravity $F_g$ let us describe its motion.
$$\begin{gathered} mg-bv=ma=mdv/dt \\ dv/dt=g-bv/m \end{gathered}$$
这是一个正儿八经的微分方程，需要解的。然而我不想。

然而分析两个特殊的情况。

情况一，当小球开始放入液体中，初始速度为0。这个时候， $dv/dt=g=a$ 。加速度等于 g ，就是自由落体，也就是说，这个时候，没有向上的阻力。

情况二，当速度因为重力加速度逐渐增大时， R 也随之增大，直到 $R=F_g$。这个时候，因为没有额外向下的力让小球产生加速度，小球最终达到最大速度，这个速度，称为terminal speed。

Air drag at high speeds

For objects moving at high speeds through air, such as airplanes, sky divers, cars, and baseballs, the resistive force is approximately proportional to the square of the speed.
$$R=D\rho A{v}^2/2$$
where $\rho$ is the density of air, A is the cross-sectional area of the falling object measured in a plane perpendicular to its motion, and D is a dimensionless empirical quantity called the drag coefficient.

当物体自由落体时， ${\sum }_{}^{}F=mg-D\rho A{v}^2/2$。

可得加速度： $$a=g-D\rho A{v}^2/2m$$

可以计算出terminal speed，即 a=0 时，可得： $$g=D\rho A{v}^2/2m$$$$v_t=\sqrt{2mg/D\rho A}$$

Using this expression, we can determine how the terminal speed depends on the dimensions of the object. Suppose the object is a sphere of radius r . In this case, $A\sim r^2$($fromA=\pi r^2$ ) and $m\sim R^3$(because the mass is proportional to the volume of the sphere). Therefore, $v_t\sim \sqrt{r}$.