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神奇的数学实验-双色旋转管

Strange Spinning Tubes - Numberphile

我带来了一个小管
I brought a little tube
管的两头分别是黑色和橙色
whose ends have been colored black and orange
现在我要让管子旋转起来
and I’m going to launch it into spin
旋转时它的重心在中间
and it will spin around the middle
正因为如此
and because it is spinning around the middle
所以没有理由你看到的只是黑色
a priori, there is no reason why you would see black better than orange
或只看到橙色
or orange better than black.
好的 我现在开始转动小管
Ok, but now when I launch
瞧我们看到了什么?
what do we see?
我们看到了黑色
We see black.
而且只看到黑色
And every time we seem to be seeing black.
这真有趣
That’s really funny.
我们看着它 却只看到黑色
We seem to be seeing, be seeing black
但是你不喜欢黑色 那么让我来晃动它
but you don’t like black, I’m going to shake it
然后发射它 让它旋转
and launch it
现在我们只看到橙色
and now we’ll see orange instead.
这真是令人感到好奇
That’s really curious.
这无关它旋转的方向是左还是右
It’s not a matter of launching from right to left
因为我换左手让它旋转 看到的还是橙色
because I can launch it with my left hand and it’s still orange.
毕竟你不喜欢橙色 那么
After all, you don’t like orange then
我将管子靠在我的黑色头发上摩擦一下
so I’m going to rub it against my black hair
然后你看到的颜色又不一样了 它现在是黑色的
and that changes the color again, and it’s black.
所以第一个问题就是 为什么有时候你看到了黑色 有时候你又只看到橙色
So the first mystery is why sometimes do we see black and sometimes orange?
第二个问题就是
Mystery number two is
当我发射它的时候
when I launch this
我们能看到几个黑点
how many black spots do we see?
我们现在看到的是四个点
I think we see four.
为什么是四个?似乎并不是因为我发射它的方式导致的
Why four? It’s not as if I’m launching it with very much care.
我都是很随意的发射它
Its initial conditions seem to be random.
那么我们为什么看到四个黑点?
So why do we see four?
当它开始匀速旋转的时候 我们能看到四个点
When it’s stabilized, it’s four.
那么我现在晃动了它
And I have to shake it around.
然后
And…
我们看到了四个橙色的点
four oranges.
为了让你更好的理解其中的奥妙
See, in order to understand it better,
我拿了另外一根管子
I brought another tube.
外表和我这跟一模一样
Similarly marked, black and orange.
现在 我旋转管子 让你看到黑色的点
Now, this time I’m going to show you black.
这是一个漫长的转变
There’s a much longer transient,
换句话说 你得等着直到管子匀速旋转
in other words, you have to wait for it
但是当它稳定下来的时候
but when it settles into steady state
我们只看到三个黑点
we have three.
再旋转一次
Again.
我们还是看到三个黑点
We see three.
真有趣
That’s interesting.
你可能注意到了
You might have noticed…
但我希望并没有
I hope you didn’t, that
当我从黑色这一段发射管子的时候
when I launch from the black end
它出现的是黑色
it shows black.
然后当我按住橙色 然后发射管子
And when I press down the orange end and launch
它是橙色的
it’s orange.
那么为什么会这样呢?
So why is this?
现在我要跟你讲的是零阶理论
I’d like to begin by giving you a zeroth order theory,
之后我们会做完整的理论
and then we’ll do the full theory later on.
现在我们将零阶理论
So the zeroth order theory goes as follows…
你看 我的这个动作
you see, this motion
我按住管子 让它旋转
I’m pressing down, making it spin
可以被看做是组合或者是叠加的两个动作
can be thought of as a combination or superposition of two motions.
首先
First…
整个的旋转是绕着中心点旋转的
the whole thing is spinning like this around the vertical axis.
我们可以称之为旋转
We shall call it revolution.
没错
Yeah.
同时 这个管绕着它的轴旋转
At same time, this thing spins around it’s own axle.
我们称之为旋转运动
And we shall call that motion rotation.
二者两者的结合可以称之为
And the whole thing is a combination of
公转与自转
revolution and rotation.
比如说我们按住黑色的这一头然后发射它 好了
Let’s say for the time being that we launch it by pressing down the black end, okay.
总是
Always.
请关注其他的点
And, please look at the other point
端点另一头这个橙色的点
here at the orange end.
这个点会比黑色点先转动 就像这样
That point will be advancing like this,
因为旋转
because of the revolution.
但是 你看
But you see…
我按住黑色的一头 转动时会看到黑色 就像这样
the… rotation that I gave the black end is a back spin like this,
这意味着橙色的一端是上旋转
which means the orange end is going in top spin.
那么这个自转也是先于黑色一端
So that’s a rotation which is also going forward.
[Brady]:没错
[Brady]: Yeah.
公转时橙色一头先转 就像这样
So the revolution makes it advance like this
自转 像这样…像…额
but the rotation which is like… like this
也是这样子的
is making go like that also.
然后 这个公转和自转…啊…相互作用
So, the rotation and revolution ADD to each other,
所以这个末端为转的快一点
so this end is going fast.
两者对比的话…
In contrast…
我们来看看我刚才捏住的黑色部分
let’s look at the black end that I pinch down.
和之前一样 公转时这个黑色会先行转动
Well again, the revolution… the revolution will make it advance like this.
但是这一端 自转时是黑色在旋转
But at this end, the rotation is back spin
所以 旋转速度的方向是后退相反的
so the rotational speed is actually pointing backward.
这两个速度力互相抵消
So those two speeds subtract from each other
然后这一端运动就开始变得缓慢
so this end’s going to move slower.
事实上 我们看到的这一端的速度比另一端相对慢
And the fact of the matter is that the end that we see better is the end that is moving slower
那么快的那一端我们就不能很好的看到 因为太快了
and the end that’s moving faster we don’t see so well.
看 黑色的这一端移动的相对慢一点
So, black end is moving slower
橙色的这一端移动的相对快一点
and orange end is moving faster
那么零阶理论就是 我们所看到的黑色的这一端
and the zeroth order theory is that black end is…
就是两者之间旋转相对慢的一端
the slow end is the one end that we see
而橙色的一端就是转动快速的一端 我们看不清
and the orange end, the fast end, we don’t see.
它转的太快 太好了
So far, so good,
但是你看 零阶理论这个解释我认为
but you see, this explanation, zeroth order as I called it
并不能完全的解释清楚 因为
is not totally satisfying, because
首先 为什么我们看到慢的一端而不是快的一端
first of all, why is it that we see the slow end rather than the fast end?
这听起来挺合理的 但是可能人类的的眼睛…
It sounds plausible, but maybe human eyes have a…
你知道的(就像超声波这种)
you know…
例如要捕捉频率
preferred frequency that it wants to capture
可能是某种频率—即使它很快—但是我们依旧能看到 看到管子的快慢也如此
and maybe at some frequency — even if it is fast — we see best, and so on.
可能吧 只是猜测
Maybe.
当然 零阶理论并没有解释为什么我们看到了四个或者三个点
And also, this zeroth order explanation doesn’t do anything to explain why four, why three.
所以我们现在进行完全的分析
So we’ll wade into the now full analysis
它发生在…你捏它的时候
It so happens that when you… pinch it
最初 整个管子都是滑动的 因为我转动了它
Initially, everything is sliding because I’m launching it.
但是很快 这个滑动的转变为旋转的运动
But soon, this sliding motion goes into ROLLING motion.
旋转 就意味着之后的事
Rolling, by the way, means the following thing:
当两个物体反向旋转
when two bodies roll against each other
就意味着接触点 不存在相对速度
it means that at the point of contact, there is no relative velocity
换句话说 这两个物体以完全相同的方式旋转
In other words, these two bodies are moving in exactly the same way.
这就是旋转的意义
That’s what rolling means.
如果它们有相对速度
If they have a relative velocity
如果它们没有用相同的方式旋转
if they are not moving the same way at point of contact
它们就会相对滑动
they slide against each other
这是理论上的
That makes sense.
所以他们相对旋转
So they roll against each other.
就意味着它们在接触点上的相对速度是零
That means that their relative velocity is zero at the point of contact.
但是如果一个物体在固定的地板上
But if an object is rolling on the fixed floor
地板不会移动 那么意味着接触点
Well, the floor is not moving, so that means that the point of contact at
随时都会停止运动
any instant is instantaneously at rest
这个接触点的速度就会变为零
Has zero velocity…
这就是旋转的意义
at the point of contact. That’s what rolling means.
那我们假设 这个管子像这样旋转起来 像…
So let’s say it starts rolling like this and it goes around, around like…
好了
Okay.
现在 你可以看到
Now, as you can see…
它倾斜了一点 但是忽视这个倾斜
It’s tilted a little, but let’s ignore this tilt for the moment.
这个倾斜角度非常 非常的小 它继续保持转动 绕着
It’s very, very small and it just keeps going around and around
这个接触点 有点像是在地板上画圈
The point of contact then is describing a largish circle on the floor.
这个图…你可以看到有横截面的圆管在大圈上
On that… large circle you can see there’s a cross sectional circle of the tube
这个的话…你可以想象成一个硬币绕着一个圈滚动
which is… you can imagine a coin rolling on the circle like this going around, around
所以这是一个小圈在大圈上滚动旋转
Ok, so there is a small circle rolling on the large circle.
额…
Well…
那么这个小圈在大圈上滚动了几次?
How many times does this small circle roll on the large circle?
当然 这与小圆的和大圆的周长有关
Well it has to do with, of course, the perimeter of the small circle to the perimeter of the large circle.
但是这等同于小圆和大圆之间的直径比
But that’s equal to the ratio of the diameter of the small circle to the diameter of the large circle
大圆直径大约是管子的长度
which is approximately the length of this end
所以 这管子的横纵比实际就是大小圆直径比
So, it’s really the aspect ratio of how long it is to how wide it is.
这就是结果—4:1
And that’s — it turns out — 4 to 1.
顺便说一下 我觉得这个看起来真古怪
By the way, it looks a bit curious, I think.
很多人看到这个管子可能会说
Most people looking at this tube would say that
用外部的管长和圈内的直径比不止4:1
it’s longer and thinner than 4 to 1.
但是事实上它的比就是这样因为…举个例子
But actually it is, because, for example…
这是两个相同宽度的管子
These two have the same width
如果我用一根管子的长度测量这个小圈的单位
and if I measure this in units of the small one
一个…两个…三个…四个
that’s one… two… three… four
所以这个比是正确的
So it is really four to one.
我不到为什么人们的眼睛会认为这个比是不对的
I don’t know why human eyes think of it as longer and thinner than 4 to 1
但这个确实是4:1
but it is 4 to 1.
然后这个小的是3:1
And that’s 3 to 1.
这个就是我们证实的数据
And that is number that we were seeing.
4点 4点 3点
4 points, 4 dots, 3 dots.
但是这里还有更漂亮的
But there is something even more beautiful going on
我来展示给你看
so let me go into this…
到目前为止我们都是忽略这个微小的倾斜
We have been so far neglecting this slight tilt
但是事实上 这个微小的倾斜
but actually, there is a slight tilt
当我发射管子
When I launch this…
倾斜就发生在我下压它的时候…
it so happens that the part that I push down starts…
突然的翘起头
immediately rises it’s head
开始进行运动
and goes into this motion.
为什么他的一个头会翘起?嗯
Why does it rise it’s head? Well…
这很简单 但是
it’s easy, but…
是个很尴尬的解释
an awkward explanation.
这个和陀螺效应有关
It has to do with the gyroscopic effect.
我们暂且先不提这个
So let me, er, slide over that then.
我们来讲讲这个上升…额…它的一头翘起然后就开始…像这样
Let’s accept that rise… it raises it’s head and starts going like this.
当管子开始旋转的时候
While this thing is rolling, rolling…
它本身和地板有着轻微的倾斜角
with a slight tilt against the floor
且保持相对稳定状态
in steady state
想象一下 管子上方有一个透明的天花板
In your imagination, please place a transparent ceiling from above
所以天花板抵住管子翘起的一头
So as the touch this tilted tubes from above…
这个运动发生在两个平面中间
this motion is happening between two planes
上方是假象的天花板然后下方是真实的地板
the imaginary ceiling above and then the real floor below
嗯…你可以看见管子在天花板地板之间的运动
Well… you can see that there is an up and down symmetry between the floor and ceiling
是对称的 它四处旋转…像这样
of this motion, it’s just going around and around like… like that
所以…特别的
So… in particular…
如果这个底部的小圈…在地上旋转
if this thing… thing is rolling on the floor… at the bottom,
那么另一个顶部的小圈就在天花板上旋转
at top its rolling under the ceiling.
在天花板或者地板上
On the ceiling, or under the ceiling rather,
这又像是小圈在大圈上绕圈
this thing is describing a larger circle.
以及
And this…
有点像立起的硬币挂在天花板上旋转
sort of coin like cross-sectional circle is hanging from the ceiling circle
就这样转啊转
going around, around in a circle.
滚动使得使得管子的一点与
Rolling means, as we explained, that in fact instantaneously whatever…
天花板的一点接触的那一刹
point is a point of contact to the… with the ceiling
是瞬间静止的 因为是从下方撞到天花板的
is instantaneously at rest, because it’s now touching the ceiling from below.
没错
Yeah.
之前我们有说过黑色的一端转的慢
So… earlier we said something to the effect that well the black end is the slow end then the, erm…
然后橙色的那一端转的快
orange end is the fast end.
但是慢没关系 反正那一瞬是静止的
But never mind slow, it’s actually instantaneously at rest
管子的速度变为了零
the velocity becomes ZERO
在每次它立起的时候
Every time it comes up
它发出paff声[声音拟词]
it goes paff, paff, paff, paff, paff!
然后停下
Stops for us
这就是我们为什么能看的这么清晰的原因
And that’s why we see them so clearly
那么它每一圈停顿几次?
and how many times did it stop per circuit?
实际上是四个 因为那是一个比例
Well, four exactly because that’s the ratio…
这个比例问题我们讲到过
aspect ratio would we talk about
低频嗡嗡作响 慢速旋转
[low frequency humming — slowed down rotation]
每圈都停顿在了相似的四个地方
[Brady]: Stopping four times in every possible position too, though?
不仅仅是黑色部分停止了
[Brady]: It’s not just stopping with the black part up,
它停住了…
[Brady]: it’s stop…
啊…这个是对的 但是你看…
Ahh… that’s right, but you see… the…
在其它部分我们看到了这个白色是静止的
in other positions we are seeing the white bit stop
事实上在黑色中间 你可以看到很多很多的白色部分相对我们而言是停止的
so between the black bits, in fact you see lots and lots of white bits that are stopping for us
这就是为什么我们看到白色的圈和四个黑色的点的原因
and that’s why we see in fact the white circle with four black dots
确实
Indeed.
所以每个点 只要它停止上升
So every point, whenever it comes upward is stopped.
下边的点也是
Similarly, every point… down below
会在与地板有接触时停下来
is stopping whenever it comes into point of contact
所以如果你也做了这个实验你可能会喜欢在
so if you did this experiment, maybe you would like to do it together…
(不像我现在做实验的实心地板而是)玻璃地板
between not just a floor like this, but it’s a glass floor
或者透明地板上做实验 然后从桌下进行拍摄观察
and transparent floor and we filmed from, ah… below
然后我们会看到橙色的点
then we would see the orange bits…
[低沉而有回响的声音]
boom, boom, boom, boom
[声音拟词]
paff, paff, paff, paff
然后其它全部都是白色的
and then all the rest would be white.
这里有一个美丽的曲线 俗称为摆线
There is a very beautiful curve called a cycloid.
这个摆线粗略地讲就是这条曲线
So a cycloid is roughly speaking this curve.
看看这个有黑色东西随着
If you look at the trajectory of this black thing as the…
这个类似硬币立起的圈滚动的运动轨迹
the cross-sectional circle rolls
现在我要放大这个尺寸
I’m going to exaggerate the size
这个…黑点差不多就是这样运动的 这样子就叫做摆线
It… describe a… describes a curve like this and that’s called a cycloid.
这个黑点的运动轨迹…有着尖点
The point about the cycloid… it has cusps.
你看当它接触地面
You see whenever it touches the floor
固定支撑时候
the stationary support
它…就有了尖点
it… it has a cusps.
这个尖点原来是部分垂直的
and this cusp turns out to be locally vertical
你知道它没有出现这样的角…
you know it doesn’t make an angle like this…
但实际上它像这样…
but actually it’s like this
因为这个点弯曲了然后
because the point is curving in and then
垂直弹跳的出去又进来
kind of bouncing vertically, and going out… and going out.
(呼吸声)
(breath)
所以…就有了以下有趣的结果
So… this has the following very very interesting consequence as well.
如果你将这个动作倒退回去一会儿
If you go back for a moment to this motion…
橙色的点就会碰触到地板 然后就像一个摆线
Orange one, which is touching the floor, is then describing a cycloid.
如果你喜欢这些圈上的裂片站在…额…地板上
If you like those lobes standing on the, er, circle on the floor.
同样的 这个黑点也像
Similarly, by the same token, this black dot is describing
一个颠倒的摆线序列
a sequence of upside down cycloids
部分裂片是挂在天花板上的…就像这样
whose lobes are hanging from the ceiling… like this.
每次黑色的点出现 那个点实际上就是你看到它摆线中的尖点
And each time the black dot comes up you are at the cusp of this cycloid….
额 颠倒摆线
upside down cycloid.
但是你看这个尖点…是垂直的尖端
But you see this cusp is… vertical cusp
我们看向上方的运动
and we are looking at the motion from above
你看 这意味着每当黑色的点碰触天花板
You see, so that means that whenever the black dot comes into contact with the ceiling
都会瞬间静止
and goes into instantaneous rest.
它不仅会瞬间静止 而且静止前后会垂直移动
Not only does it stop instantaneously, but in that vicinity it’s moving only vertically.
我们从上方向下看
and we are looking at… from above
我们只看到了水平运动
so we see only the horizontal motion.
从上方观察垂直运动是看不到的 对吧?
And for the observer from above vertical motion is invisible, right?
我的意思是它看起来像静止的
I mean it looks like stationary.
换句话说 靠近这个停顿点
See, in other words, near this stationary point
这个摆线的尖端
the cusp of the cycloid
也就是当黑色的点接触上方的天花板的时候
when the black dot comes into contact with the ceiling from above
它不仅…瞬间静止 而且它的速度很夸张
it not only comes… becomes stationary at that moment but it’s velocity is really exaggeratedly
的减慢 对于从上方观察运动的观察者来说
shrunk with respect to the observer from above
所以你不仅看到了零速度 而且
so not only do you get zero velocity, but
在转动的那一刻
around that… moment
从观察者的角度看他的速度是减缓的
the… its velocity actually shrunk because of the angle of the observation.
这就是为什么我们能非常明显的看到这个效果的原因…额
That’s why we see this effect so clearly so it has the, erm,
视频中是夸大版的停止运动效果示范
the effect of exaggerating this stop motion.
现在 我…额…开始思考这个
Now, when I, erm, started thinking about this
我不知道发生了什么 没错
I didn’t know, of course, what was happening
然后 我去东京的五金店买了一个长的 额 塑料管
and so I went to a hardware store in Tokyo and I bought a long, erm, plastic tube
然后切断它…管子横纵比为整数为准
and then cut them in aspect ratios… integers
然后开始这样玩 并算出一切
and then started playing with this and figured out everything.
顺便说一下 如果你想在家里做以上这个
By the way, if you want to do this at, er, at home and so on
我建议你使用空心管 因为你看
I recommend that you use hollow tubes because, you see
空心管意味着它大多数的质量集中在边缘
a hollow tube means that most of the mass is concentrated on the rim
同时 因为它是空心的 所以它并不会太重
and at the same time because it’s hollow, it’s not so heavy
所以像这样…你可以很轻松的转动它
So making it like… has the advantage that you can launch it easily.
大多数的质量集中在边缘
but concentrating the mass on the rim
的优势就是
and has the advantage that
它转动的惯性大于相同质量的(实心管子)
what is called the moment of inertia is large for the same mass
所以一旦空心管子开始转动…额…它就会保持稳定的运动
so once it gets going, it gets going… erm, it keeps going in a stable fashion.
我做了一些管子 它们的横纵比是4:1和3:1的以及等等
I made those things in… aspect ratios of integers: 4 to 1, 3 to 1, and so on.
我也在家做了5:1的管子 但是我后悔做了那个管子 因为
I also have at home, 5 to 1, but I regret doing this because
我很想知道管子的横纵比不是整数的时候会发生什么(管子不够用了)
you see I really would like to know what happens when the aspect ratio is not an integer.
顶端的清单改换成
And instead of listing the vertices
图中这样逐渐切换的
consecutively like this in order
你看到的这些点在跳动
what you see is that you see those dots skipping
每一秒闪动的白点就像这样 就像…
and going every second vertex like this, like a…
[金属物体在木质表面旋转声(啪嗒)]
[metal object spinning on a wooden surface]
一开始你看清楚白点
In the beginning it’s a mess
有一个…[音频消失]
There’s a…

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