Force and Translational Dynamics
AP Physics C: Mechanics Topic 2 9:06 English narration · English + 中文 subtitles burned in
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A cart runs around a vertical loop.
一辆小车沿着竖直圆环运动。
At the very top it is completely upside down, with nothing holding it up.
在最高点，它完全倒过来，下面什么也没有托着它。
Why does it not fall off?
它为什么不掉下来？
The answer is not some mysterious outward force. It is one equation, applied carefully.
答案并不是什么神秘的向外的力，而是一条方程，只要用得仔细。
By the end of this lesson you will be able to write it down in a single line.
到这节课结束时，你能用一行就把它写出来。
This is Unit Two: forces, and the motion they cause.
这是第二单元：力，以及力所引起的运动。
Newton's three laws, the free-body diagram that makes them usable, and the special forces you meet again and again — gravity, friction, springs and drag.
牛顿三大定律、让它们真正好用的受力图， 以及你会反复遇到的几种力——重力、摩擦力、弹簧力和阻力。
Let's begin.
让我们开始吧。
First, decide what your system is: the object, or the group of objects, you are going to analyse.
第一步，确定你的系统：你要分析的那个物体，或那一组物体。
Every system behaves as if all of its mass sat at one point — the centre of mass.
每个系统的行为都好像它的全部质量都集中在一个点上——质心。
For a set of point masses it is the average position, weighted by mass.
对一组质点，质心就是按质量加权的平均位置。
For a solid body you replace the sum with an integral over the mass elements.
对一个实心物体，把求和换成对质量元的积分。
And here is the payoff: only external forces move the centre of mass.
好处在这里：只有外力才能改变质心的运动。
Internal forces always come in pairs, so they cancel inside the system.
内力总是成对出现，所以在系统内部相互抵消。
That integral deserves a worked example, because Physics C asks for it and Physics One never does.
这个积分值得做一个例题，因为 Physics C 会考它，而 Physics 1 从来不考。
The trick is always the same: cut it into slices, and write the mass of one slice in terms of the coordinate.
方法总是一样的： 把它切成小片，并用坐标把一片的质量写出来。
For a rod that is lambda d x, where lambda is the mass per unit length.
对一根细杆就是 lambda d x，其中 lambda 是 单位长度的质量。
For a sheet it is sigma d A, and for a solid, rho d V.
对一张薄片是 sigma d A，对一个立体是 rho d V。
Take a rod of length L that gets heavier along its length, so lambda equals c times x.
取一根长为 L 的杆， 它沿着长度越来越重，所以 lambda 等于 c 乘以 x。
First the total mass: integrate lambda d x from zero to L, which gives one half c L squared.
先求总质量：把 lambda d x 从零积到 L， 得到二分之一 c L 平方。
Then weight each slice by its position: integrate x times lambda d x, which gives one third c L cubed.
然后按位置给每一片加权：积分 x 乘以 lambda d x，得到三分之一 c L 立方。
Divide, and the c and two powers of L cancel, leaving the centre of mass two thirds of the way along, over toward the heavy end, exactly as it should be.
相除，c 和两次 L 都约掉，剩下质心在全长的三分之二处，偏向重的那一端，正如它 应该在的位置。
And when the body is uniform, do not integrate at all: symmetry answers it.
而当物体是均匀的时候，根本不要积分：对称性就给出答案。
In a uniform gravitational field that same point is also the centre of gravity.
在均匀的重力场中， 同一个点也就是重心。
Now the single most useful habit in mechanics.
现在讲力学中最有用的一个习惯。
Draw the object on its own, as a dot or a box, and put an arrow on it for every force acting on it — and nothing else.
把物体单独画出来，画成一个点或一个方块， 再为作用在它上面的每一个力画一支箭头——而且只画这些。
Weight always.
重力永远有。
Then a normal force if a surface touches it, tension if a string pulls it, friction if it slides or tries to, and drag if it moves through a fluid.
如果有表面接触它，就有法向力；如果有绳拉它，就有张力；如果它滑动或有滑动趋势，就有摩擦力； 如果它在流体中运动，就有阻力。
Then choose your axes, and for a slope or an incline, tilt them along it.
然后选定坐标轴；在斜面上，就让坐标轴沿着斜面。
Almost every mistake in this unit is a missing arrow or an extra one.
本单元几乎所有的错误，都是少画了一支箭头，或多画了一支。
The first law: with zero net force, the velocity stays constant.
第一定律：合力为零时，速度保持不变。 不一定是静止——是不变。
Not necessarily at rest — constant.
这种状态叫做平动平衡。
That is translational equilibrium, and the property behind it is inertia: mass is a measure of how strongly a body resists a change in its motion. The second law is the engine of the whole unit: the net force equals mass times acceleration, applied one axis at a time.
第二定律是整个单元的发动机：合力等于质量乘以加速度， 而且要一个轴一个轴地用。
Physics C states it more generally: the net force is the rate of change of momentum.
Physics C 用更一般的形式表述它：合力等于动量的变化率。
The third law: if A pushes B, then B pushes back, equally and oppositely.
对质量不变的物体，两者是一回事。 第三定律：如果 A 推 B，B 就以大小相等、方向相反的力推回去。
Those two forces act on different objects, so they never cancel inside one diagram.
这两个力作用在不同的物体上，所以在同一张受力图里绝不会互相抵消。
A two kilogram block slides down a frictionless ramp tilted at thirty degrees.
一个两千克的木块沿倾角三十度的光滑斜面下滑。
Tilt the axes along the ramp.
把坐标轴沿斜面倾斜过来。
Perpendicular to it, the normal force balances the perpendicular part of the weight.
垂直于斜面的方向上，法向力与重力的垂直分量平衡。
Along the ramp, the only force left is the weight component down the slope.
沿斜面方向， 唯一剩下的力就是重力沿斜面向下的分量。
So the acceleration is the gravitational field strength times the sine of thirty, which is four point nine metres per second squared.
所以加速度等于重力场强度乘以三十度的正弦， 等于四点九米每二次方秒。
Notice the mass cancelled. A heavier block would slide down exactly as fast.
注意质量约掉了：更重的木块下滑得一样快。
Now two bodies joined by a string.
现在看两个用绳连接的物体。
A six kilogram cart on a smooth table is pulled over a light pulley by a two kilogram hanging mass.
光滑桌面上一辆六千克的小车，通过跨过轻滑轮的绳与一个两千克的悬挂物相连。
First, treat the two as one system.
第一步，把两者当作一个系统。
The tension is internal now, so it disappears.
此时张力是内力，直接消失。
The only external force driving the system is the weight of the hanging mass, and the mass being accelerated is all eight kilograms.
驱动系统的唯一外力是悬挂物的重力，而被加速的质量是总共八千克， 于是加速度是二点四五米每二次方秒。
That gives two point four five metres per second squared. Then isolate the cart alone: the only horizontal force on it is the tension, so the tension is about fifteen newtons.
第二步，把小车单独隔离出来： 它在水平方向上唯一受的力就是张力，所以张力等于六乘二点四五，约十五牛顿。
System first, then one body.
先系统、后单体——这个两步法几乎能解决所有连接体问题。
Close to the ground, weight is simply mass times the field strength.
在地面附近，重力就等于质量乘以重力场强度。
But that is only a local shortcut.
但这只是一个局部的简便算法。
In general, Newton's law of gravitation says any two masses attract each other with a force proportional to both masses, and inversely proportional to the square of the distance between their centres.
一般来说，任意两个质量之间都有引力，大小与两个质量成正比， 与它们中心之间距离的平方成反比。
Double the separation and the force drops to a quarter.
距离加倍，引力就变成四分之一。
Divide that force by the mass being pulled, and you get the gravitational field: the big mass, times the constant, over the distance squared.
把这个力除以被吸引的那个质量，就得到引力场强度：大质量乘以引力常量，再除以距离的平方。
Friction opposes sliding along a surface.
摩擦力阻碍物体沿表面滑动。
While a surface is sliding, the kinetic friction is the coefficient of kinetic friction times the normal force.
当表面正在滑动时，动摩擦力等于动摩擦系数乘以法向力。
Before it slides, static friction is different — and this catches people out.
在滑动之前，静摩擦力不一样——这一点常常被搞错。
Static friction is only as big as it needs to be, up to a ceiling, and that ceiling is the static coefficient times the normal force.
静摩擦力只有它需要多大就多大， 但有一个上限，这个上限是静摩擦系数乘以法向力。
So it is an inequality, not an equation.
所以它的公式是一个不等式，不是等式。
Put friction on our ramp, with a coefficient of nought point two.
给我们的斜面加上摩擦，系数取零点二。
The acceleration becomes three point two metres per second squared.
加速度变成重力场强度乘以三十度的正弦， 减去零点二乘三十度的余弦，等于三点二米每二次方秒。
The mass cancels once again.
质量又一次约掉了。
Hooke's law: an ideal spring pulls back with a force proportional to how far you stretch it, and always toward the natural length.
理想弹簧的回复力与你拉伸的距离成正比，方向总是指向原长位置。
That minus sign is what makes it a restoring force.
那个负号正是它成为回复力的原因。
The energy stored is one half times the spring constant times the extension squared.
储存的能量等于二分之一乘以劲度系数，再乘以伸长量的平方。
Now combine springs into one equivalent spring.
现在把弹簧组合起来。
In parallel — side by side — they share the load, and their constants simply add, so the pair is stiffer than either one.
并排的弹簧共同分担负载，它们的劲度系数直接相加—— 这一对比其中任何一根都更硬。
In series — end to end — each one stretches, so the reciprocals add and the pair is softer than the softer spring.
首尾相接的弹簧各自都会伸长，所以是倒数相加—— 这一对比更软的那根还软。
Either way, use the equivalent constant in the oscillation formulas.
无论哪一种，在振动公式里都要用合成后的劲度系数。
A resistive force — drag — opposes motion through a fluid, and it grows with speed.
阻力反抗物体在流体中的运动，而且随速度增大。
Put that into the second law for a falling object and you no longer get a number.
常用的模型是阻力与速率成正比。
You get a differential equation: mass times the rate of change of velocity equals weight minus drag.
把它代入落体的第二定律，你得到的不再是一个数，而是一个微分方程： 质量乘以速度的变化率，等于重力减去阻力。
Set the acceleration to zero and you get the terminal velocity: weight divided by the drag constant.
令加速度为零，就得到终极速度： 重力除以阻力系数。
For a mass of nought point one kilograms and a constant of nought point five, that is two metres per second.
质量为零点一千克、系数为零点五时，终极速度是两米每秒。
To get the whole story, separate the variables and integrate from rest.
要得到完整过程，就分离变量并从静止开始积分。
The speed climbs exponentially toward the terminal value.
速度按指数规律趋近终极速度，但永远达不到它。
Now circular motion. Move in a circle at constant speed and the velocity is still changing, because its direction keeps turning.
以恒定速率做圆周运动时，速度仍在变化，因为它的方向不断改变。
So there is an acceleration, and it points straight at the centre.
所以存在加速度，而且它直指圆心。
Its size is the speed squared divided by the radius.
它的大小等于速率的平方除以半径。
But be careful with the words. Centripetal force is not a new force you add to the diagram.
这个加速度需要一个指向圆心的合力——质量乘以速率的平方，再除以半径。
It is the net inward force that real forces — tension, gravity, friction, the normal force — are already providing.
但用词要小心：向心力并不是你额外加到受力图上的新力， 而是张力、重力、摩擦力、法向力这些真实的力所承担的任务。
Now we can answer the opening question.
现在可以回答开头的问题了。
At the very top of the loop, only two forces act on the cart: its weight, and the track pushing down on it.
在圆环的最高点，小车只受两个力：重力，以及轨道向下的压力。
Both of them point downward — toward the centre of the circle.
两个力都指向下方——也就是指向圆心。
So their sum supplies the centripetal force.
所以它们的和提供了向心力。
The slowest possible speed is the one where the track barely touches, so its push is zero.
最慢的速度是轨道刚好不再施力的那一刻，也就是压力为零。
Gravity alone does the whole job, and the speed is the square root of the field strength times the radius.
这时全部由重力来完成这个任务，速度等于重力场强度乘以半径的平方根。
For a radius of two point five metres, the smallest speed is about four point nine metres per second.
对于半径二点五米的圆环，最小速度约为四点九米每秒。 再慢一点，小车还没到最高点就会脱离轨道。
Three habits that save marks.
三个能保住分数的习惯。
First, draw the free-body diagram first, every single time, and tilt the axes for a slope.
第一，每一次都先画受力图，遇到斜面就把坐标轴倾斜过来。
Second, define your system before you write an equation — internal forces cancel, external forces do not.
第二，在写方程之前先定义你的系统——内力相互抵消，外力不会。
Third, never draw a centripetal force as an extra arrow.
第三，绝不要把向心力当作一支额外的箭头画上去。
Identify which real forces point toward the centre, and set their sum equal to the mass times the speed squared over the radius.
要判断哪些真实的力指向圆心， 再让它们的和等于质量乘以速率的平方再除以半径。
Get those three, and this unit is yours.
做到这三点，这个单元就是你的了。