Energy and Momentum of Rotating Systems
AP Physics C: Mechanics Topic 6 8:12 English narration · English + 中文 subtitles burned in
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Transcript
Watch a figure skater spin.
看花样滑冰运动员旋转。
Arms stretched out, she turns slowly.
手臂张开时，她转得很慢。
She pulls her arms in — and without pushing off anything, she spins much faster.
她把手臂收回来—— 没有蹬任何东西，却转得快得多。
Nobody touched her, so where did that speed come from?
没有人碰她，那么这份速度是从哪里来的？
One quantity stayed exactly the same the whole time.
整个过程中，有一个量始终没有改变。
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 in a straight line comes back, rewritten for spinning.
你在直线运动中学过的能量和动量， 会全部回来，只是改写成转动的版本。
Kinetic energy gets a rotational twin. So does momentum, and so does impulse.
动能有一个转动版的孪生兄弟， 动量也有，冲量也有。
Learn the translation, and half of this unit is already done.
掌握了这套对应关系，这个单元就已经完成了一半。
Here is the plan.
本节的安排是这样的。
First, the energy of spin.
首先是转动的能量。
Then the work and power a torque delivers.
然后是力矩做的功和它的功率。
Then angular momentum, and the angular impulse that changes it.
接着是角动量，以及改变它的角冲量。
Then the big one: conservation of angular momentum, which this exam asks about every year.
然后是最重要的一个：角动量守恒， 这门考试每年都会考。
Then rolling without slipping, where translation and rotation lock together.
再然后是无滑滚动，在那里平动和转动被锁在一起。
And last, satellites in orbit.
最后是绕轨道运行的卫星。
A spinning body stores energy.
转动的物体储存着能量。
Its rotational kinetic energy is one half of the rotational inertia — the moment of inertia — times the angular speed squared — the twin of one half mass times speed squared.
它的转动动能等于二分之一的转动惯量乘以角速度的平方—— 正是二分之一质量乘以速度平方的孪生兄弟。
Notice something strange: this wheel's centre never moves, yet it is full of energy — every particle inside it is moving.
注意一件奇怪的事： 这个轮子的中心从不移动，它却充满了能量，因为它里面的每一个质点都在运动。
And a body that both travels and spins carries both terms at once.
而一个既平动又转动的物体，同时带有这两项。
Let's use that.
我们来用一用。
A uniform cylinder rolls along the ground. How much of its energy is spin? Pause and try it.
一个均匀圆柱沿着地面滚动，它的能量有多少是转动的？
For a cylinder the rotational inertia is one half mass times radius squared, and rolling means the angular speed is the speed over the radius.
先暂停，自己试一试。 对圆柱来说，转动惯量是二分之一质量乘以半径的平方，而滚动意味着角速度等于速度除以半径。
Put those in and the radius cancels: the total is three quarters mass times speed squared.
把它们代进去，半径完全约掉：总能量是四分之三质量乘以速度的平方。
So one third of the energy is rotational, whatever the mass or size.
所以三分之一的能量是转动的，与质量和大小都无关。
A torque turning something through an angular displacement does work on it.
让物体转过一个角度的力矩，对它做了功。
The work is the integral of the torque over the angle turned — on a graph of torque against angle, it is the area underneath.
功等于力矩对转过角度的积分—— 在力矩对角度的图上，功就是曲线下面的面积。
Power follows the same pattern: torque times angular velocity.
功率遵循同样的规律： 力矩乘以角速度。
Try the numbers on the right.
看看右边的数字。
A steady fifteen newton metres through eight radians does one hundred and twenty joules; at twenty radians per second that is three hundred watts.
恒定的十五牛顿米，转过八弧度， 做功一百二十焦耳；当角速度是二十弧度每秒时，功率是三百瓦。
Now momentum, rewritten for turning.
现在是动量的转动版本。
For one particle, angular momentum is the position vector crossed with the momentum: mass times speed times perpendicular distance.
对一个质点来说，角动量等于位置矢量叉乘动量： 也就是质量乘以速度，再乘以垂直距离。
Here is the surprise — a particle moving in a straight line still has angular momentum about any point off that line.
令人意外的是—— 沿直线运动的质点，对不在这条直线上的任意一点，仍然具有角动量。
So always say which point.
所以一定要说清楚是对哪一点。
For a rigid body spinning about a fixed axis it is just the rotational inertia times the angular speed.
对绕固定轴转动的刚体，它就是转动惯量乘以角速度。
A torque changes angular momentum, exactly as a force changes momentum.
力矩改变角动量，就像力改变动量一样。
The net torque is the rate of change of angular momentum, so on a graph of angular momentum against time it is the slope.
合力矩是角动量的变化率， 所以在角动量对时间的图上，它就是斜率。
Torque multiplied by the time it acts is angular impulse, and it equals the change in angular momentum.
力矩乘以它作用的时间叫做角冲量， 它等于角动量的变化量。
Watch the example: a wheel starts with thirty units; a torque of five newton metres for four seconds adds twenty, ending at fifty.
看这个例子：一个轮子最初的角动量是三十个单位； 五牛顿米的力矩作用四秒，增加二十，最后是五十。
Now the big idea.
现在来看最重要的思想。
With zero net external torque, a system's total angular momentum is conserved.
如果没有外部力矩作用于一个系统，它的总角动量就不会改变。
The skater pulls her arms in, so her mass sits closer to the axis and her rotational inertia drops.
滑冰者把手臂收回来，质量更靠近转轴，转动惯量下降。
The product has to stay the same, so her spin rate rises.
乘积必须保持不变， 所以她的转速上升。
Her muscles only pulled inward, straight at the axis — and a pull at the axis makes no torque.
她的肌肉只是向内拉，方向正对着转轴—— 而指向转轴的拉力，根本不产生力矩。
Here are the numbers.
来看数字。
A skater spins at two turns per second with a rotational inertia of four kilogram metres squared, then pulls in to one point six.
一位滑冰者以每秒两圈旋转，转动惯量是四千克平方米， 然后收到一点六。
Pause and find her new spin rate.
先暂停，求她新的转速。
Before equals after, so four times two equals one point six times the answer: five turns per second.
前后相等，所以四乘以二等于一点六乘以答案：每秒五圈。
Now check her kinetic energy.
再看她的动能。
It rises from three hundred and sixteen joules to seven hundred and ninety — her muscles did that work.
动能从三百一十六焦耳升到七百九十焦耳——这份功是她的肌肉做的。
Conservation survives collisions too.
守恒定律在碰撞中同样成立。
A bullet flies into the tip of a hanging rod and sticks.
一颗子弹射入一根悬挂细杆的末端并嵌进去。
Take the pivot as your point: gravity acts at the centre and the pivot force acts at the pivot, so neither makes a torque about it.
把支点取作你的参考点：重力作用在杆的中心，支点的力作用在支点本身， 所以两者对支点都不产生力矩。
Angular momentum is conserved.
角动量守恒。
But linear momentum is not, because the pivot shoves sideways — and kinetic energy is not either, because the bullet embeds.
但线动量不守恒，因为支点会横向推杆—— 动能也不守恒，因为子弹嵌了进去。
Now the numbers.
来看数字。
A twenty gram bullet at three hundred metres per second hits the tip of a rod, one and a half kilograms and sixty centimetres long.
一颗二十克的子弹以每秒三百米的速度打在一根细杆的末端， 杆重一点五千克，长六十厘米。
Before, only the bullet carries angular momentum: mass times speed times length, three point six.
碰撞前，只有子弹带有角动量： 质量乘以速度再乘以长度，等于三点六。
After, rod and bullet turn together, so add their rotational inertias: zero point one eight seven.
碰撞后，杆和子弹一起转动， 所以把它们的转动惯量相加：零点一八七。
Divide, and the rod swings away at about nineteen radians per second.
相除，杆以大约每秒十九弧度摆开。
Rolling without slipping ties the two motions together.
无滑滚动把两种运动锁在一起。
In one turn the wheel lays down one circumference, so the speed of the centre is the radius times the angular speed.
转过一圈，轮子恰好铺开一个周长， 所以中心的速度等于半径乘以角速度。
Watch the marked point on the rim: it pauses each time it touches the ground, because the contact point is at rest.
看轮缘上那个标记点： 每次它碰到地面时都会停顿，因为接触点是静止的。
Static friction does no work here, so energy conservation is safe.
所以这里的静摩擦不做功， 能量守恒可以放心使用。
If the wheel slips, that link breaks and sliding friction drains energy away.
如果轮子打滑，这个联系就断了，滑动摩擦会不断消耗能量。
Release a hoop, a disk and a sphere together from the top of a ramp.
把一个圆环、一个圆盘和一个球，从斜面顶端同时释放。
They finish in that order every time: sphere first, hoop last.
每一次它们到达终点的顺序都一样：球第一，圆环最后。
The acceleration is gravity along the slope, divided by one plus the shape factor.
加速度等于沿斜面方向的重力分量，除以一加上形状因子。
Mass and radius cancel completely, so only the shape decides.
质量和半径完全约掉，所以只有形状说了算。
The hoop keeps its mass at the rim, so more of its energy goes into spin.
圆环把质量都放在边缘， 因此更多能量进入了转动。
On a thirty degree ramp the sphere reaches three point five metres per second squared; a sliding block, four point nine.
在三十度的斜面上，球的加速度是三点五米每二次方秒； 而滑动的物块是四点九。
Now put a satellite in orbit.
现在把一颗卫星送上轨道。
Gravity supplies the centripetal force.
引力提供向心力。
Set the two equal, and the orbital speed depends only on the central mass and the radius.
把两者相等， 轨道速度就只取决于中心天体的质量和轨道半径。
Add the gravitational potential energy, minus G M m over r, and the total mechanical energy of a circular orbit is exactly half of it. A bigger orbit means a slower speed and a longer period: Kepler's third law.
轨道越大，速度越慢，周期越长： 这就是开普勒第三定律。
For a circular orbit the total energy is half the potential energy, and it is negative — that is what keeps the satellite bound.
对圆轨道来说，总能量等于势能的一半，而且是负的—— 正是这一点让卫星被束缚住。
Reach zero and it escapes, which needs the square root of two times the circular speed.
总能量达到零，卫星就逃逸了， 这需要圆轨道速度的根号二倍。
Elliptical orbits look harder, but two things stay fixed.
椭圆轨道看起来更难，但有两样东西保持不变。
Gravity always points at the focus, so it makes no torque about it — angular momentum is conserved.
引力始终指向焦点， 所以它对焦点不产生力矩——角动量守恒。
And with nothing to rub against, energy is conserved too.
而且太空中没有摩擦，总能量也守恒。
Conserved angular momentum means the planet sweeps equal areas in equal times: Kepler's second law.
角动量守恒意味着行星在相等时间内扫过相等的面积：这就是开普勒第二定律。
So it races through the near part and crawls through the far part.
所以它在轨道近端飞快掠过，在远端缓慢爬行。
Three habits that save marks.
三个能保住分数的习惯。
First, always name the axis or the point before you write angular momentum down — and if the axis is not through the centre of mass, shift it with the parallel-axis theorem: I equals I centre of mass, plus M d squared.
第一，在写下角动量之前，一定要先说明是对哪条轴或哪一点—— 如果这条轴不过质心，就用平行轴定理平移：I 等于过质心的 I 加上 M d 平方。
Second, when a body rolls, never forget the second energy term — half the marks live in the spin.
第二，物体滚动时，千万不要漏掉第二项能量——一半的分数就在转动里。
Third, check conservation one law at a time: angular momentum can be conserved while linear momentum and kinetic energy are not.
第三，守恒定律要一条一条地检查：角动量可以守恒，而线动量和动能并不守恒。