Energy and Momentum of Rotating Systems
AP Physics 1 Topic 6 7:54 English narration · English + 中文 subtitles burned in
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Watch a skater spinning with her arms stretched out.
看这位滑冰运动员张开双臂旋转。
She pulls them in — and suddenly she spins much faster.
她把手臂收回来，忽然之间转得快多了。
Nobody pushed her. No motor started.
没有人推她，也没有马达启动。
So where did that extra speed come from?
那么，多出来的转速是从哪里来的？
This is Unit Six: energy and momentum of rotating systems.
这是第六单元：转动系统的能量与动量。
Everything you know about energy and momentum has a spinning twin.
你已经学过的关于能量和动量的一切， 都有一个「旋转版」的孪生兄弟。
Learn the pairs, and half of this unit is done.
记住这些对应关系，这个单元就完成了一半。
Start with energy.
先从能量开始。
A spinning object stores kinetic energy even when its centre never moves: every piece of it is moving, so every piece carries energy.
旋转的物体即使中心完全不动也储存着动能—— 它的每一块都在运动，所以每一块都带有能量。
Add them up and you get one half of the rotational inertia, times the angular velocity squared — one half m v squared, with mass replaced by rotational inertia and speed by angular velocity.
把它们加起来， 得到二分之一的转动惯量乘以角速度的平方——也就是二分之一 m v 平方， 只是质量换成转动惯量，速度换成角速度。
Double the spin rate and you get four times the energy.
转速翻倍，能量就是四倍。
So how does spinning energy get in, or out?
那么转动的能量怎样进出呢？
Through torque.
靠力矩。
Push at the rim and the disk turns through an angle.
在边缘施力，圆盘就转过一个角度。
The work done is the torque multiplied by that angle, in radians, never degrees.
做的功等于力矩乘以这个角度，角度要用弧度，不能用度。
Do it faster and you have more power: torque times angular velocity.
做得更快，功率就更大： 功率等于力矩乘以角速度。
And the net work equals the change in rotational kinetic energy.
而净功等于转动动能的变化。
A torque does work, so let us put a number on it.
力矩会做功，我们来算一个数。
A motor applies a steady torque of eight point zero newton metres, and the wheel turns through ten radians.
一台马达施加八点零牛米的恒定力矩，轮子转过十弧度。
Multiply: that gives eighty joules of work.
相乘：得到八十焦耳的功。
Now look at the graph — torque up the side, angle along the bottom. The work is simply the area of that rectangle.
再看这张图—— 纵轴是力矩，横轴是角度，功就是那个矩形的面积。
Even when the torque changes, the area under the graph is still the work.
即使力矩在变化，图线下的面积仍然就是功。
Momentum has a spinning twin too: angular momentum.
动量也有一个「旋转版」的孪生兄弟：角动量。
For a rigid body it is the rotational inertia times the angular velocity — spin that resists being changed.
对刚体来说， 它等于转动惯量乘以角速度——可以理解为「抗拒被改变的旋转」。
To change it you need a torque, acting for a length of time.
要改变它，你需要一个力矩，并且作用一段时间。
Torque times time is called angular impulse, and it equals the change in angular momentum.
力矩乘以时间叫做角冲量，它等于角动量的变化。
Here is the part students find strange. An object does not have to be spinning to have angular momentum.
下面这一点学生常觉得奇怪：物体不必旋转，也可以有角动量。
A ball moving in a perfectly straight line has angular momentum about any point off to the side.
一个沿直线运动的球，相对于路径旁边的任何一点都有角动量。
Its size is the mass, times the speed, times the perpendicular distance from that point to the line.
它的大小等于质量乘以速率，再乘以该点到运动直线的垂直距离。
Now change the point: move it closer to the line, and the same ball has much less.
现在换一个点：把它移近直线，同一个球的角动量就小得多。
Here is a flywheel already carrying thirty units of angular momentum.
这里有一个飞轮，它已经带有三十个单位的角动量。
A torque of five newton metres acts for four seconds.
一个五牛米的力矩作用了四秒。
Five times four is twenty, so the angular momentum rises to fifty.
五乘四等于二十，于是角动量升到五十。
On a graph of torque against time, the area under the line is the angular impulse.
在力矩对时间的图上，图线下的面积就是角冲量。
On a graph of angular momentum against time, the slope of the line is the net torque.
在角动量对时间的图上，图线的斜率就是合外力矩。
Now the rule that explains our skater.
现在来看解释滑冰运动员的那条规则。
If the net external torque on a system is zero, its total angular momentum cannot change.
如果系统所受的合外力矩为零， 它的总角动量就不会改变。
Rotational inertia times angular velocity before equals the same product after — so if the rotational inertia goes down, the angular velocity must go up.
转动惯量乘以角速度，之前等于之后—— 所以转动惯量变小，角速度就必须变大。
Pulling her arms in moves mass closer to the axis — and the smooth ice gives her almost no torque.
她把手臂收回来， 质量移到更靠近转轴的位置——而光滑的冰几乎不给她任何力矩。
A skater spins at two point zero revolutions per second, rotational inertia four point zero kilogram metres squared.
一位滑冰运动员以每秒二点零转旋转，转动惯量为四点零千克二次方米。
She pulls her arms in and it drops to one point six. Find her new spin rate.
她把手臂收回来，转动惯量降到一点六，求她新的转速。
Pause here and try it.
先暂停，自己试一试。
No external torque, so angular momentum is conserved.
没有外力矩，所以角动量守恒。
Rearrange for the new angular velocity: the old one, times four point zero over one point six. That is two point five times two point zero — five point zero revolutions per second.
把新的角速度解出来： 等于原来的角速度乘以四点零除以一点六，也就是二点五乘以二点零—— 每秒五点零转。
Now the twist the exam loves: her kinetic energy rose by the same factor, and her muscles did that work.
现在是考试最爱考的转折： 她的动能上升了同样的倍数，这份功是她的肌肉做的。
Next: rolling.
接下来是滚动。
When a wheel rolls without slipping, the point touching the ground is, for that instant, not moving at all.
轮子做纯滚动时，接触地面的那一点在那一瞬间完全不动。
Follow the marked point on the rim: it pauses every time it reaches the ground.
跟着轮缘上做了记号的那一点看：它每次到达地面都会停顿一下。
That condition ties the two motions together — the speed of the centre equals the radius times the angular velocity.
这个条件把两种运动联系起来——中心的速度等于半径乘以角速度。
A rolling object owns two kinds of kinetic energy at once: one half m v squared for moving forward, and one half rotational inertia times angular velocity squared for spinning.
滚动的物体同时拥有两种动能：向前运动的二分之一 m v 平方， 以及旋转的二分之一转动惯量乘以角速度的平方。
The top of the wheel moves at twice the speed of the centre, while the bottom is momentarily still.
轮子顶部的速度是中心的两倍， 而底部在那一瞬间是静止的。
While it rolls without slipping friction does no work; if it slips, sliding friction turns energy into heat.
只要是纯滚动，摩擦力就不做功； 一旦打滑，滑动摩擦就会把能量变成热。
Race three shapes down one ramp.
让三个形状从同一个斜面下滑。
The potential energy at the top has to split between moving and spinning, and the shape decides how.
顶端的势能必须在平动和转动之间分配， 而分配比例由形状决定。
Take a solid disk: its spinning energy is one quarter m v squared and its moving energy one half — three quarters in total, so exactly one third sits in the spin.
先看实心圆盘： 它的转动能是四分之一 m v 平方，平动能是二分之一，总共四分之三， 所以恰好三分之一在旋转里。
A hoop keeps all its mass at the rim, so a full half goes to spin.
圆环把全部质量放在边缘，整整一半进入旋转。
A solid sphere keeps its mass near the axis: only two sevenths.
实心球的质量靠近转轴，只有七分之二。
So the sphere reaches the bottom first, and the hoop comes last.
所以球最先到底，圆环最后。
The same conservation laws run the sky.
同样的守恒定律也支配着天空。
This is the space station, about four hundred kilometres up — and it is falling, right now.
这是空间站，在地面上方大约四百公里处， 而它此刻正在下落。
Gravity pulls it toward the Earth and it keeps missing, because it moves sideways fast enough.
引力把它拉向地球，但它总是「错过」地球， 因为它横向运动得足够快。
That is all an orbit is: free fall that never lands.
轨道运动就是这么回事：永远落不到地面的自由落体。
For a circular orbit, gravity is the centripetal force — not an extra force, the same one.
对圆轨道来说，引力就是向心力。 不是额外多出来的力，而是同一个力。
Set the gravitational pull equal to the centripetal requirement and the satellite's mass cancels, leaving the orbital speed: the square root of big G, times the planet's mass, divided by the radius.
令引力等于向心力所需的大小，卫星的质量就约掉了。 剩下的就是轨道速度： 大 G 乘以行星质量，再除以半径，然后开平方根。
So the speed never depends on the satellite's mass, and a bigger orbit is slower.
所以速度从不依赖卫星自身的质量，而且轨道越大速度越慢。
For a low Earth orbit, about seven point nine kilometres per second.
对地球低轨道来说，大约每秒七点九公里。
Not every orbit is a circle.
并不是每条轨道都是圆。
On an ellipse, gravity always pulls straight toward the planet, so it never twists the satellite around it. No torque means the angular momentum stays constant.
在椭圆轨道上，引力总是笔直指向行星， 所以它不会让卫星绕着行星「拧」——没有力矩，角动量就保持不变。
Watch the speed change — fast when close, slow when far, sweeping equal areas in equal times.
看它的速率变化：靠近时快，远离时慢，在相等的时间里扫过相等的面积。
The energy trades like a swing, and the total stays the same.
能量的交换就像荡秋千，而总量保持不变。
Escape velocity is the launch speed that just lets an object leave for good.
逃逸速度是刚好能让物体一去不返的发射速度。
The total mechanical energy has to be exactly zero, so the kinetic energy at launch cancels the negative gravitational potential energy.
总机械能必须恰好等于零， 所以发射时的动能要抵消掉为负的引力势能。
Set the sum to zero and solve for the speed.
令两者之和为零，把速度解出来。
That gives the square root of two times the circular orbit speed.
结果正是圆轨道速度的根号二倍。
So from the Earth's surface, about eleven point two kilometres per second — and the object's own mass never appears.
所以从地球表面出发，大约是每秒十一点二公里—— 而物体自身的质量从头到尾都没有出现。
Three habits before you go.
走之前记住三个习惯。
First, when no external torque acts, angular momentum is conserved — but the kinetic energy is not, so never assume both.
第一，没有外力矩时，角动量守恒， 但动能并不守恒，千万不要同时假设两者都守恒。
Second, for anything rolling, split the energy into moving plus spinning; a rolling object goes down a ramp more slowly than a sliding one.
第二，凡是滚动的问题，把能量分成平动加转动； 滚动的物体下斜面总是比滑动的慢。
Third, for a circular orbit, set gravity equal to the centripetal requirement — the satellite's mass always cancels.
第三，对圆轨道， 令引力等于向心力所需的大小——卫星的质量总会约掉。
Get those three, and Unit Six is yours.
掌握这三点，第六单元就是你的了。