yoeo24 翻译制作
oCourse
Last time we discussed that an acceleration is caused by a push or by a pull.
上次，我们讨论了推和拉作用产生的加速度
yoeo24 翻译制作
oCourse
Today we will express this more quantitatively in three laws
今天，我们将用三条定律更深入地定量分析
which are called Newton's Laws.
这就是牛顿定律
The first law really goes back to the first part of the 17th century.
第一定律可以追溯到17世纪上半叶
It was Galileo who expressed what he called the law of inertia
当时伽利略提出了惯性定律
and I will read you his law.
原文如下
"A body at rest remains at rest
“如无外力作用”
"and a body in motion continues to move at constant velocity
“静止物体将保持静止”
"along a straight line, unless acted upon by an external force."
“运动物体将始终沿直线匀速运动”
And now I will read to you Newton's own words in his famous book, Principia.
下面再读一下牛顿名著《原理》中的原文
Philosophiæ Naturalis Principia Mathematica《自然哲学的数学原理》，牛顿代表作
"Every body perseveres in its state of rest or of uniform motion in a right line
“一切物体总保持匀速直线运动状态或静止状态”
"unless it is compelled to change that state by forces impressed upon it."
“直到有外力迫使它改变这种状态为止"
.
牛顿第一定律（惯性定律）：
一切物体总保持匀速直线运动状态或静止状态 直到有外力迫使它改变这种状态为止
Now, Newton's First Law is clearly against our daily experiences.
显然，牛顿第一定律有违日常经验
Things that move don't move along a straight line
日常生活中，运动物体不会总沿直线运动
and don't continue to move, and the reason is, there's gravity.
也不会保持运动，原因是，存在重力
And there is another reason.
还有一个原因
Even if you remove gravity then there is friction, there's air drag.
哪怕没有重力，还存在摩擦和空气阻力
And so things will always come to a halt.
因此物体终究会停止
But we believe, though, that in the absence of any forces indeed an object,
根据牛顿定律我们相信，如无这些作用力存在
if it had a certain velocity
若自身拥有速度
would continue along a straight line forever and ever and ever.
物体将永远做匀速直线运动
Now, this law, this very fundamental law does not hold in all reference frames.
不过，这条基本定律并不在任意参考系中成立
For instance,
比如
it doesn't hold in a reference frame which itself is being accelerated.
在具有加速度的参考系中它不成立
Imagine that I accelerate myself right here.
假设我现在正在加速
Either I jump on my horse, or I take my bicycle or my motorcycle or my car
我可以是骑马、骑自行车或开车
and you see me being accelerated in this direction.
你们看到我正在加速
And you sit there and you say, "Aha, his velocity is changing.
你们坐着不动，发现我速度在变化
"Therefore, according to the First Law, there must be a force on him."
那么根据第一定律，我身上有力的作用
And you say, "Hey, there, do you feel that force?"
你问我“感受到力了吗”
And I said, "Yeah, I do! I really feel that, I feel someone's pushing me."
我回答“感受到了，似乎有人在推我”
Consistent with the first law.
符合第一定律
Perfect, the First Law works for you.
很好，它对你们适用
Now I'm here. I'm being accelerated in this direction
但对于我，我在往此方向加速
and you all come towards me being accelerated in this direction.
因此在我看来，你们会沿反方向加速
I say, "Aha, the First Law should work so these people should feel a push."
我想“第一定律适用，你们肯定也会感觉到推力”
I say, "Hey, there! Do you feel the push?"
我问“你们，感觉到推力了吗”
And you say, "I feel nothing. There is no push, there is no pull."
你们会回答“没有，推和拉都没有”
Therefore, the First Law doesn't work from my frame of reference
因此，第一定律对于加速中的参考系并不适用
if I'm being accelerated towards you.
因此，第一定律对于加速中的参考系并不适用
So now comes the question, when does the First Law work?
那么问题是：第一定律何时成立
Well, the First Law works when the frame of reference
当参考系为惯性参考系时
is what we call an "inertial" frame of reference.
牛顿第一定律成立
And an inertial frame of reference would then be
惯性参考系也就是
a frame in which there are no accelerations of any kind.
不存在任何形式加速度的参考系
Is that possible?
这可能存在吗
Is 26.100... is this lecture hall an inertial reference frame?
教室是一个惯性参考系吗
For one, the earth rotates about its own axis and 26.100 goes with it.
首先，地球会自转，教室会随之旋转
That gives you a centripetal acceleration.
这就会有向心加速度
Number two, the earth goes around the sun.
其次，地球还绕太阳公转
That gives it a centripetal acceleration
这又有向心加速度
including the earth, including you, including 26.100.
地球如此，你们如此，教室也如此
The sun goes around the Milky Way, and you can go on and on.
太阳还要绕银河系旋转，我们也必然如此
So clearly 26.100 is not an inertial reference frame.
那么显然教室不是惯性参考系
We can try to make an estimate on how large these accelerations are
我们可以简单估计下，上述加速度有多大
that we experience here in 26.100
我们可以简单估计下，上述加速度有多大
and let's start with the one that is due to the earth's rotation.
首先从地球自转开始算起
So here's the earth... rotating with angular velocity omega
这里是地球，旋转角速度为ω
and here is the equator, and the earth has a certain radius.
这里是赤道，半径一定
The radius of the earth... this is the symbol for earth.
地球的半径…地球的符号是⊕
Now, I know that 26.100 is here
教室实际大概在这里
but let's just take the worst case that you're on the equator.
我们就假设在赤道上，这里加速度最大
You're there
我们在赤道上
You go around like this and in order to do that
我们像这样旋转
you need a centripetal acceleration, a c
需要向心加速度ac
which, as we have seen last time, equals omega squared R.
上一讲讲过，它等于ω²R
How large is that one?
它有多大呢
Well, the period of rotation for the earth is 24 hours times 3,600 seconds
地球自转一周需要24小时×3600秒
so omega equals two pi divided by 24 times 3,600
因此ω=2π/(24×3600)
and that would then be in radians per second.
单位是弧度每秒
And so you can calculate now what omega squared R earth is
这样就能计算ω²R⊕了
if you know that the radius of the earth is about 6,400 kilometers.
已知地球的半径大约6400千米
Make sure you convert this to meters, of course.
记得将其换算为米
And you will find, then that the centripetal acceleration at the equator
最后算得赤道处的向心加速度
which is the worst case-- it's less here-- is 0.034 meters per second squared.
…这是最大的情况…也小于0.034m/s²
And this is way, way less--
这非常非常小
this is 300 times smaller than the gravitational acceleration
只有地球上重力加速度的1/300
that you experience here on Earth.
只有地球上重力加速度的1/300
And if we take the motion of the earth around the sun
如果再算一下公转的加速度
then it is an additional factor of five times lower.
结果还要小五倍
In other words, these accelerations even though they're real
换句话说，虽然存在加速度
and they can be measured easily with today's high-tech instrumentation--
而且这些加速度也能被精密仪器探测到
they are much, much lower than what we are used to
但它们相比于我们熟悉的重力加速度
which is the gravitational acceleration.
实在微不足道
And therefore, in spite of these accelerations
因此，可以忽略
we will accept this hall as a reasonably good inertial frame of reference
可以将教室看作一个很好的惯性参考系
in which the First Law then should hold.
这里第一定律是成立的
Can Newton's Law be proven?
第一定律能证明吗
The answer is no, because it's impossible to be sure
答案是不能，因为不可能确定
that your reference frame is without any accelerations.
我们所处的参考系有无加速度
Do we believe in this?
它是否可信
Yes, we do.
很可信
We believe in it since it is consistent
我们相信此定律，是因为
within the uncertainty of the measurements
在测量不确定度范围内
with all experiments that have been done.
所有实验均支持该定律
Now we come to the Second Law, Newton's Second Law.
下面来讲牛顿第二定律
I have a spring...
这里有一根弹簧
Forget gravity for now-- you can do this somewhere in outer space.
不考虑重力，假设在外太空
This is the relaxed length of the spring and I extend the spring.
这里是松弛状态下的长度，现在拉伸弹簧
I extend it over a certain amount, a certain distance-- unimportant how much.
我将它拉伸一定长度
And I know that I when I do that that there will be a pull-- non-negotiable.
拉它的时候它对我有一个拉力，毫无疑问
I put a mass, m1, here, and I measure the acceleration
在这里挂个质量m1的物体，然后放开弹簧
that this pull causes on this mass immediately after I release it.
放开瞬间测量拉力产生的加速度
I can measure that. So I measure an acceleration, a1.
测量得到加速度为a1
Now I replace this object by mass m2
之后用质量m2的物体进行同样实验
but the extension is the same, so the pull must be same.
弹簧伸长相同，因此拉力相同
The spring doesn't know what the mass is at the other end, right?
这与挂有多大质量无关
So the pull is the same.
拉力相同
I put m2 there, different mass and I measure the new acceleration, a2.
换成m2之后，测得新加速度为a2
It is now an experimental fact that m1 a1 equals m2 a2.
根据测量结果可以得到m1a1=m2a2
And this product, ma, we call the force.
这个积ma，称为“力”
That is our definition of force.
这就是力的定义
So the same pull on a ten times larger mass
所以如果质量取10倍大
would give a ten times lower acceleration.
则加速度会小10倍
The Second Law I will read to you:
第二定律原文如下
"A force action on a body gives it an acceleration
“物体的加速度由力作用产生”
which is in the direction of the force..."
“方向与力相同”
That's also important--
这一点也很重要
the acceleration is in the direction of the force.
加速度与力方向相同
"And has a magnitude given by ma."
“力的大小为ma”
ma is the magnitude
ma为力的大小
and the direction is the direction of the force.
而加速度方向与力相同
.
牛顿第二定律（F=ma）：
物体的加速度方向跟作用力相同 大小跟作用力成正比，跟物体的质量成反比
→
And so now we will write this in all glorious detail.
下面将它写成公式形式
This is the Second Law by Newton
这就是牛顿第二定律
perhaps the most important law in all of physics
物理中最重要的定律
but certainly in all of 801:
本课程最重要的定律
F equals ma.
F=ma
The units of this force are kilograms times meters per second squared.
力的单位为千克×米/秒²
In honor of the great man, we call that "one newton."
为了纪念伟大的牛顿，我们将此称为“1牛”
Like the First Law, the Second Law only holds in inertial reference frames.
与第一定律类似，第二定律也只在惯性参考系下成立
Can the Second Law be proven?
第二定律能够证明吗
No.
不能
Do we believe in it?
它可信吗
Yes.
可信
Why do we believe in it?
为何可信
Because all experiments and all measurements
因为所有实验和测量
within the uncertainty of the measurements
在测量不确定度范围内
are in agreement with the Second Law.
均符合第二定律
Now you may object and you may say
也许有人会有不同意见
"This is strange, what you've been doing.
“这太奇怪了”
"How can you ever determine a mass if there is no force somewhere?
“没有力那怎么定义质量呢”
"Because if you want to determine the mass maybe you put it on a scale,
“要知道质量都是通过称来称量的”
"and when you put it on a scale to determine the mass
“只有称量才能知道质量”
"you made use of gravitational force
“这就用到重力”
"so isn't that some kind of a circular argument that you're using?"
“这不就成了循环论证了吗”
And your answer is "No."
我认为并非如此
I can be somewhere in outer space where there is no gravity.
比如在外太空无重力环境下
I have two pieces of cheese; they are identical in size.
有两块奶酪，它们大小相同
This is cheese without holes, by the way.
上面没有孔
They are identical in size.
大小相同
The sum of the two has double the mass of one.
两个的质量就是一个的两倍
Mass is determined by how many molecules-- how many atoms I have.
质量由分子数或原子数的多寡决定
I don't need gravity to have a relative scale of masses
确定物体质量相对值无需重力
so I can determine the relative scale of these masses
所以就算在没有重力条件下
without ever using the force.
也能确定质量相对值
So this is a very legitimate way of checking up on the Second Law.
因此第二定律的描述是合理的
Since all objects in this lecture hall and the earth
由于地球上所有的物体
fall with the constant acceleration, which is g
均有向下的恒定加速度g
we can write down that the gravitational force
因此重力可以写作
would be m times this acceleration, g.
m乘以此加速度g
Normally I write an "a" for it, but I make an exception now
一般加速度写作a
because gravity, I call it "gravitational force."
这里是个例外，因为涉及到重力
And so you see that the gravitational force
公式表明
due to the earth on a particular mass is linearly proportional with the mass.
重力由地球产生，它正比于物体质量
If the mass becomes ten times larger
如果质量增加10倍
then the force due to gravity goes up by a factor of ten.
那么重力也会增加10倍
Suppose I have here this softball in my hands.
假设我手上有一个球
In the reference frame...
在惯性参考系中
26.100 we will accept to be an inertial reference frame.
…教室是一个很好的惯性参考系
It's not being accelerated in our reference frame.
球在参考系中没有加速度
That means the force on it must be zero.
意味着它受合力为零
So here is that ball.
这个球
And we know if it has mass, m-- which in this case is about half a kilogram--
它具有质量m，大概为1/2千克
that there must be a force here, mg
必然存在重力mg
which is about five newtons, or half a kilogram.
大小为5N，1/2公斤
But the net force is zero.
由于合力为0
Therefore it is very clear that I, Walter Lewin, must push up with a force
意味着我略文，需要给它一个推力
from my hand onto the ball, which is about the same...
方向竖直向上，大小与重力相等
which is exactly the same, five newtons.
大小也是5N
Only now is there no acceleration
只有这样加速度才为零
so I can write down that force of Walter Lewin
写出来就是：略文给的力
plus the force of gravity equals zero.
加上重力等于零
Because it's a one-dimensional problem
因为这是一维问题
you could say that the force of Walter Lewin equals minus mg.
可以写作略文给的力=-mg
F equals ma.
F=ma
Notice that there is no statement made on velocity or speed.
注意了，这里没有涉及到速度和速率
As long as you know f and as long as you know m, a is uniquely specified.
当f和m已知，a唯一确定
No information is needed on the speed.
这与速度无关
So that would mean, if we take gravity and an
这意味着，对于重力
object was falling down with five meters per second that the law would hold.
当物体坠落速度为5m/s时，定律成立
If it would fall down with 5,000 meters per second it would also hold.
坠落速度达到5000m/s时，定律依然成立
Will it always hold?
定律永远成立吗
No.
不是
Once your speed approaches the speed of light
一旦速度接近光速
then Newtonian mechanics no longer works.
牛顿力学不再适用
Then you have to use Einstein's theory of special relativity.
这时需要引入爱因斯坦的狭义相对论
So this is only valid as long as we have speeds
因此想让定律成立
that are substantially smaller, say, than the speed of light.
速度必须远小于光速
Now we come to Newton's Third Law:
下面讲牛顿第三定律
"If one object exerts a force on another
“如某物体对另一物体施加力”
the other exerts the same force in opposite direction on the one."
“另一物体会对施力物体产生大小相同、方向相反的力”
I'll read it again.
再读一遍
"If one object exerts a force on another
“如某物体对另一物体施加力”
.
牛顿第三定律（作用力=-反作用力）：
两个物体之间的作用力与反作用力总是 大小相等，方向相反，作用在一条直线上
the other exerts the same force in opposite direction on the one."
“另一物体会对施力物体产生大小相同、方向相反的力”
And I normally summarize that as follows,
一般来说定律可以总结如下
the Third Law as "Action equals minus reaction."
第三定律“作用力=-反作用力”
And the minus sign indicates, then, that it opposes so you sit on your seats
这个负号表示方向相反，比如你们坐在椅子上
and you are pulled down on your seats because of gravity
由于重力，你们压着椅子
and the seats will push back on you with the same force.
椅子则会以同等的力支撑着你们
Action equals minus reaction.
作用力等于负的反作用力
I held the baseball in my hand.
我拿一个球
The baseball pushes on my hand with a certain force.
球对手有一定的力
I push on the baseball with the same force.
手对球也有同样的力
I push against the wall with a certain force.
我推这堵墙有个力
The wall pushes back in the opposite direction with exactly the same force.
同时墙在反方向上也对我有同样的力
The Third Law always holds.
第三定律成立
Whether the objects are moving or accelerated makes no difference.
不管是否运动、是否加速，定律都成立
All moments in time, the force--
任何时候，力
we call it actually the "contact force" between two objects--
两物体间的相互作用力
one on the other is always the same as the other on one
作用力与反作用力总是大小相等
but in the opposite direction.
方向相反
Let us work out a very simple example.
下面讲一个简单的例题
We have an object which has a mass, m1.
某物体质量为m1
We have object number one and m1 is five kilograms.
这是物体一，m1=5kg
And here, attached to it, is an object two and m2 equals 15 kilograms.
连在一起的有物体二，m2=15kg
There is a force and the force is coming in from this direction.
假设有一个力，朝这个方向
This is the force-- and the magnitude of the force is 20 newtons.
这个力…其大小为20N
What is the acceleration of this system?
那么系统的加速度是多少
F equals ma.
F=ma
Clearly the mass is the sum of the two--
显然质量为两者之和
this force acts on both--
力同时作用于两物体
so we get m1 plus m2 times a.
所以是(m1+m2)a
This is 20,
这个是20
this is 20
这个是20
so a equals one meters per second squared in the same direction as f.
因此a=1m/s²，方向与F相同
So the whole system is being accelerated with one meters per second squared.
整个系统以1m/s²进行加速
Now watch me closely.
下面看好了
Now I single out this object--
我把物体分开考虑
here it is... object number two.
这是物体二
Object number one, while this acceleration takes place
加速时，物体一必然对物体二有推力
must be pushing on object number two.
加速时，物体一必然对物体二有推力
Otherwise object number two could never be accelerated.
否则物体二不可能产生加速
I call that force f12 the force that one exerts on two.
将这个推力记作F12，表示一对二的力
I know that number two has an acceleration of one.
我们知道二号物体加速度为1
That's a given already.
这个已经求出了
So here comes f equals ma. f12 equals m2 times a.
还是F=ma，此时是F12=m2a
We know a is one, we know m2 is 15
已知a=1，m2=15
so we see that the magnitude of the force 12 is 15 newtons.
因此力12的大小为15N
This force is 15.
这个力为15
Now I'm going to isolate number one out.
再单独考虑一号
Here is number one.
这是一号
Number one experiences this force, f, which was the 20
一号受到一个力F，大小为20
and it must experience a contact force from number two.
它必然还受到来自二号的接触力
Somehow, number two must be pushing on number one
二号必然也在推一号
if one is pushing on number two.
力的作用是互相的
And I call that force "f21."
这个力记作F21
I know that number one is being accelerated
已知一号正在加速
and I know the magnitude is one meter per second squared.
已知加速度大小为1m/s²
That's non-negotiable,
毫无疑问
and so we have that f, this one, plus f21 must be m1 times a.
因此两矢量F和F21相加必然等于m1a
This is one, this is five, this is 20
这是1、这是5、这是20
and so this one, you can already see, is minus 15.
因此这个显然等于-15
F21 is in this direction and the magnitude is exactly the same as f12.
F21朝这个方向，大小正好等于F12
So you see?
了解？
One is pushing on two with 15 newtons in this direction.
一号对二号的推力为15N，这个方向
Two is pushing back on one with 15 newtons
二号回敬一号推力15N
and the whole system is being accelerated with one meter per second squared.
然后整个系统以1m/s²加速前进
Now, in these two examples--
我举了两个例子
the one whereby I had the baseball on my hand--
一个是手上的球
you saw that it was consistent with the Third Law.
它符合第三定律
In this example, you also see that it's consistent with the Third Law.
现在大家又看到一个符合第三定律的例子
The contact force from one on the other
物体之间的接触力
is the same as from the other on one but in opposite signs.
作用力与反作用力大小相等、方向相反
Is this a proof?
这能算证明吗
No.
不能
Can the Third Law be proven?
第三定律能证明吗
No.
不能
Do we believe in it?
它是否可信
Yes.
可信
Why do we believe in it?
为何可信
Because all measurements, all experiments
因为所有测量和实验
within the uncertainties are consistent with the Third Law.
在不确定度范围内都符合第三定律
Action equals minus reaction.
作用力等于负的反作用力
It is something that you experience every day.
其实你整天都和它打交道
I remember I had a garden hose on the lawn and I would open the faucet
比如花园草坪上的水管，打开龙头
and the garden hose would start to snake backwards. Why?
水管就会向后蛇行，为何
Water squirts out.
水喷出
The garden hose pushes onto the water in this direction.
水管朝水有此方向的推力
The water pushes back onto the garden hose and it snakes back.
水会反推水管，这就造成蛇行
Action equals minus reaction.
作用力等于负的反作用力
You take a balloon.
再比如气球
You take a balloon and you blow up the balloon and you let the air out.
吹好气球，然后把气放出
The balloon pushes onto the air.
气球推动气体
The air must push onto the balloon.
气体必然反推气球
And therefore, when you let it go
此时放开气球
the balloon will go in this direction which is the basic idea behind the rocket.
气球会向此方向飞出，这就是火箭的基本思路
I love to play with balloons, don't you?
我喜欢玩气球，你们呢
So, if I do it like this, and I let it go the air will come out in this direction
如果我放手，气会沿此方向放出
and so then it means the balloon is pushing on the air in this direction.
表明气球在往此方向推气体
the air must be pushing on the balloon in this direction. There it goes.
因此气体必然反方向推动气球，试试
It didn't make it to the moon but you saw the idea of a rocket.
没飞上月球，但这就是火箭的原理
Action equals minus reaction.
作用力等于负的反作用力
If you fire a gun, the gun exerts a force on the bullet
再比如射击，枪对子弹施加力
the bullet exerts an equal force on the gun which is called the recoil.
子弹对枪会有同样大小的反作用，即后坐力
You feel that in your hands and your shoulder.
手和肩膀感觉会很大
I have here a marvelous device
我这里有个奇妙的装置
which is a beautiful example of "action equals minus reaction."
“作用力等于负的反作用力”的美妙例子
I show you from above what it looks like. You'll see more details later.
先在黑板上比划下，等下再看实物
This rotates about this axis rather freely--
它的叶子会绕轴旋转
the axis is vertical--
轴在竖直方向
and we have here a reservoir of water, which we will heat up.
中间的球体内有水，待会我会加热
It turns into steam
水变成蒸汽
and these are hollow tubes and the steam will squirt out.
两片叶子是空管子，蒸汽从中喷出
And so when the steam squirts out in this direction
蒸汽喷出是这个方向
the tube exerts a force on the steam in this direction
叶子给蒸汽一个此方向的力
so the steam exerts an equal force in the opposite direction
同时会受到蒸汽反方向同大小的力
and so the thing will start to rotate like this.
因此它会开始旋转
And I would like to demonstrate that.
证明给你们看
You can see it now there.
装置就在那里
With a little bit of luck, there you see it.
下面看好了
So we're going to heat it.
我来给它加热
Walking.
说说走路
When you walk, you push against the floor.
走路时，我们对地面有推力
The floor pushes back at you and if the floor wouldn't push back at you
地面会反推我们，如若不然
you couldn't even walk, you couldn't go forwards.
我们无法行走、无法前进
If you walk on ice, very slippery-- you can't go anywhere,
比如走在冰上，很滑…很难前进
because you can't push on the ice so the ice won't push back on you.
因为这时很难推冰，冰也不会推你
That's another example where you see action equals minus reaction.
又是一个例子，作用力等于负的反作用力
This engine is called "Hero's engine."
回到这里，机器叫作“希罗机器”
希罗Hero of Alexandria：古希腊著名学者；希罗机器又叫作汽转球(aeolipile)
Hero, according to the Greek legend was a priestess of Aphrodite.
根据希腊传说，希罗是阿芙洛狄忒的女祭司
阿芙洛狄忒：希腊神话中的美神，相当于罗马神话的维纳斯
Let's first look at it.
先看这里
She was a priestess of Aphrodite and her lover, Leander
希罗是一名女祭司，其情人叫利安德
这里讲的故事名叫：《希罗与利安德》
would swim across the Hellespont every night to be with her.
利安德每晚横渡赫勒斯滂海峡与她幽会
赫勒斯滂海峡即达达尼尔海峡
And then one night the poor guy drowned and Hero threw herself into the sea.
直到某夜溺死于海峡，之后希罗投海殉情
Very romantic thing to do
非常浪漫
but, of course, also not a very smart thing to do.
不过，这样做很不聪明
On the other hand, it must have been a smart lady
从另一个角度看，如果这个希罗
if she invented, really, this engine.
真的是蒸汽机的发明者，她无疑是聪明的
Yesterday, I looked at the Web, "ask.com."
昨天，我上了一个网站“ask.com”
It's wonderful-- you can ask any question.
很不错，可以问任何问题
You can say, "How old am I?"
可以问“我多大了”
Now, you may not get the right answer but you can ask any question.
答案不一定准确，但肯定会回答你
And I typed in, "Hero's engine."
我输入“希罗机器”
And out popped a very nice high- tech version of Hero's engine.
结果出来了一个非常高科技的版本
A soda can-- you pop four holes in the soda can at the bottom.
一个汽水罐，在底部戳四个孔
So here's your soda can.
这是汽水罐
You pop four holes in here, but when you put a nail in there
戳四个洞，每次戳的时候
you bend every time the nail to the same side so the holes are slanted.
注意往同一个方向弯一下，制造倾斜
You put it in water you lift it out of water and you have a Hero's engine.
把它放到水里拿出来，就得到希罗机器了
And I made it for you-- it took me only five minutes.
我做了一个，只花了5分钟
I went to one of MIT's machines, got myself a soda
我在MIT找了个售贩机，买了听可乐
put the holes in it, and here it is.
打上孔，就是它了
It's in the water there.
浸在水里
When I lift it out, you will see the water squirts.
把它拉起来时，会看到水泄出来
There it goes.
就是这样
High-tech version of Hero's engine.
高科技版的希罗机器
Also makes a bit of a mess, but okay.
把这里搞的一团糟
All right.
没事
Try to make one-- it's fun and it's very quick.
试着去做个，很好玩、很简单
It doesn't take much time at all.
很快就能做好
There are some bizarre consequences of these laws.
牛顿这些定律有些怪异的结论
Imagine that an object is falling towards the earth.
假设物体朝地球坠落
An apple is falling towards the earth from a height, say, of, hmm, I'd say 100 m.
苹果从一定高度坠落，比方说100米
And let's calculate how long it takes for this apple to hit the earth
可以计算出需要多久落到地面
which should for you be trivial, of course.
对你们来说小菜一碟
So here's the earth...
这里是地球
and the mass of the earth is about 6 times 10 to the 24 kilograms.
地球质量大约6×10²⁴kg
And here at a distance, h--
这里是高度h
for which we will take 100 meters--
取100m
is this apple, m, which, say, has a mass of half a kilogram.
这里是苹果质量m约为1/2kg
There's a force from the earth onto the apple and this is that force.
苹果上存在地球的引力，这个方向
And the magnitude of that force is mg and that is 5 newton.
力的大小mg大约是5N
I make g ten and just round it off a little.
g取10，四舍五入下
Now, how long does it take this object to hit the earth?
那么，苹果落到地面需要多久
So, we know that 1/2 gt squared equals h.
我们知道，1/2gt²=h
It doesn't start with any initial speed, so that is 100.
初始速度为零，因此这是100
G is 10, this is 5, so t squared is 20.
g=10，这是5，因此t²=20
So t is about 4 and a half seconds.
求得t大概4.5秒
So after 4 and a half seconds, it hits the earth-- so far, so good.
4.5秒后苹果坠落地球，这没问题
But now, according to the Third Law the earth must experience
但根据第三定律，地球也会受到
exactly the same force as the apple does but in opposite direction.
与苹果大小相同、方向相反的力
So therefore the earth will experience this same force, f--
地球受到F同样大小的力
5 newton, in this direction.
5N，反方向
What is the earth going to do?
地球会如何
Well, the earth is going to fall towards the apple-- f equals ma.
地球会坠向苹果，F=ma
So the force on the earth is the mass of the earth
因此地球上的力等于地球的质量
times the acceleration of the earth.
乘以地球的加速度
The force, we know, is 5.
力为5N
We know the mass, 6 times 10 to the 24
质量为6×10²⁴kg
so the acceleration will be 5 divided by 6 times 10 to the 24
算出加速度为5/(6×10²⁴)
which is about 8 times 10 to the minus 25 meters per second squared.
得到8×10⁻²⁵m/s²
How long will the earth fall?
地球下坠需要多久
Well, the earth will fall roughly 4 and a half seconds before they collide.
在与苹果碰撞前，也大约需要4.5秒
How far does the earth move in the 4 and a half seconds?
这4.5秒中地球移动多少距离
Well, it moves one-half a earth t squared.
移动距离为1/2a⊕t²
That's the distance that it moves.
这就是移动距离
We know a and we know t squared, which is 20.
a已知，t²是20
1/2 times 20 is 10 so that means this distance becomes that number times 10.
1/2×20=10，因此距离为这个数×10
It's about 8 times 10 to the minus 24 meters.
大约8×10⁻²⁴米
The earth moves 8 times 10 to the minus 24 meters.
地球移动了8×10⁻²⁴米
That, of course, is impossible to measure.
这显然无法测量
But just imagine what a wonderful concept this is!
但这个想法非常奇妙
When this ball falls back to me
扔出的球坠向我
the earth and you and I and MIT are falling towards the ball.
同时地球、你、我、学校都在坠向球
Every time that the ball comes down we're falling towards the ball.
每次扔出球，这都会发生
Imagine the power I have over you and over the earth!
想想我对地球的影响多大
But you may want to think about this--
而且还可以想象
if I throw the ball up, going to be away from the earth
扔球，球远离地球的同时
I'll bet you anything that the earth will also go away from the ball.
地球同样远离球
So as I do this, casually playing--
我随便玩着球
believe me, man, what a glorious feeling it is--
相信我，这感觉很好
earth is going down, earth is coming towards the ball.
地球往下，然后又坠向球
The earth is going down and I'm part of the earth
我作为地球的一部分
and I'm shaking this earth up and down by simply playing with this ball.
只通过丢球，就能上下撼动整个地球
That is the consequence of Newton's Third Law
这就是牛顿第三定律的结果
even though the amount by which the earth moves
当然，撼动地球的大小
is, of course, too small to be measured.
小得微不足道
I now want to work out with you a rather detailed example
下面，讲一个更详尽的例子
of something in which we combine what we have learned today--
将今天所学的东西串联起来
a down-to-earth problem--
一个地球上的问题
the kind of a problem that you might see on an exam or on an assignment.
这种题很可能出现在作业或考试中
We hang an object on two strings
用两条细绳挂起物体
and one string makes an angle of 60 degrees with the vertical
一条细绳与竖直呈60°
and the other makes an angle of 45 degrees with the vertical.
一条细绳与竖直呈45°
So this is the one that makes an angle...
这是第一条细绳，角度是…
oh, 60 degrees with the horizon, 30 degrees with the vertical
噢，应该是与水平呈60°，与竖直呈30°
and this one, 45 degrees.
而这一条是45°
Let's assume that the strings have negligible mass.
假设细绳的质量可以忽略
So they are attached here to the ceiling and I hang here an object, m.
它们上方连在屋顶，下方挂有物体，质量m
Well, if there's an object m for sure there will be a force mg, gravitational force.
有质量m，就必然存在重力mg
This object is hanging there, it's not being accelerated
物体静止悬挂，没有加速度
so the net acceleration must be zero.
合加速度为零
And so one string must be pulling in this direction
因此一条细绳在此方向必然存在拉力
and the other string must be pulling in this direction
另一条细绳在这个方向也必然存在拉力
so that the net force on the system is zero.
这样系统的合力为零
Let's call this pull, for now, "T1."
这个拉力记作T1
We'll call that the tension in the string
T表示tension(张力)
and we call the tension in this string "T2."
这个张力则记作T2
And the question now is how large is T1 and how large is T2?
问题是T1和T2分别多大
There are various ways you can do this.
计算方法很多
One way that always works-- pretty safe--
一种方法很保险，也总有效
you call this the x direction.
这个方向记作x方向
You may choose which direction you call "plus."
以此方向为正方向
I call this plus, I call this negative.
这边为正、这边为负
And you could call this the y direction
然后这个为y方向
and you may call this plus and this negative.
向上为正方向、向下为负
I know, from Newton's Second Law-- F equals ma--
根据牛顿第二定律F=ma
that there is no acceleration, so this must be zero
合加速度为零，因此公式为零
so the sum of all forces on that mass must be zero.
物体上所有力的和必为零
These three forces must eat each other up, so to speak.
这三个力必须相互抵消
Well, if that's the case, then the sum of all forces in the x direction
因此，x方向所有力之和也必然为零
must also be zero because there's no acceleration in the x direction
因为x方向也没有加速度
and the sum of all forces in the y direction must be zero.
同理y方向上的合力也为零
And so I am going to decompose them-- something we have done before.
这就要用到矢量分解，我们学过的
I am going to decompose the forces into an x and into a y direction.
将这些力分解到x和y方向
So here comes the x component of T1
这里是T1的x分量
and its magnitude is T1 times the cosine of 60 degrees.
其大小为T1cos60°
Now I want to know what this one is.
那么这一个呢
This one is T1 times the sine of 60 degrees.
它等于T1sin60°
This projection, T2, cosine 45 degrees
这个投影为T2cos45°
and the y component, T2 times the sine of 45 degrees.
y分量为T2sin45°
So we go into the x direction.
先看x方向
In the x direction I have T1 cosine 60 degrees
x方向有T1cos60°
minus T2 cosine 45 degrees equals zero-- that's one equation.
减去T2cos45°等于0，一个方程
The cosine of 60 degrees is one-half
cos60°等于1/2
and the cosine of 45 degrees is one-half square root two.
而cos45°为二分之根号二
Now I go to the y direction.
下面看y方向
This is plus, this is minus, so we get one component here
上为正，下为负，这个方向上有
which is T1 times the sine of 60 degrees
T1sin60°
plus T2 times the sine of 45 degrees minus mg.
加上T2sin45°减去mg
It's in the opposite direction-- must be zero.
mg方向不同…结果为零
That's my second equation.
这是第二个方程
The sine of 60 degrees equals one-half the square root three
sin60°等于二分之根号三
and the sine of 45 degrees is the same as the cosine one-half square root two.
sin45°与cos45°一样，为二分之根号二
Notice I have two equations with two unknowns.
这里得到两方程两未知数
If you tell me what m is I should be able to solve for T1 and for T2.
只要知道m是多少，就能求出T1和T2
In fact, if we add them up
两式相加会很简单
it's going to be very easy because we lose this
中间这个消掉了
because we have both one-half square root two.
上下都是二分之根号二
And so you see immediately here that one-half times T1
得到1/2T1+(根号3)/2T1=mg
plus one-half square root three times T1 equals mg
得到1/2T1+(根号3)/2T1=mg
and so you find that the tension 1 equals two mg
结果T1=2mg/(1+根号3)
divided by one plus the square root of three.
结果T1=2mg/(1+根号3)
I can go back now to this equation--
代回到这个方程
T1 times one-half equals T2 times one-half square root of two.
T1/2=(根号2)T2/2
I lose my half and so T2 equals T1 divided by the square root of two.
消掉1/2，得到T2=T1/(根号2)
So the bottom line is, you tell me what m is
只要我们知道m是多少
I'll tell you what T1 is and I'll tell you what T2 is.
求出T1和T2没有问题
Suppose we take a mass of four kilograms--
假设质量为4kg
m equals four kilograms, so mg is about 40 if we make g ten for simplicity.
m=4kg，mg大约40N，g取10
Then T1, if you put in the numbers, is about 29.3 and T2... 29.3 newtons
那么T1，代入数值，大约为29.3N
and T2 is about 20.7 newtons, I believe.
T2大约20.7N
It's very difficult to rig this up as an experiment but I've tried that.
这很难设计为实验，不过我可以试一下
I'll show you in a minute.
马上给你们演示
I want you to know that there is another method
先讲例题的另一种解法
which is perhaps even more elegant
这个方法更加巧妙
and which you may consider
这个方法中
in which there is no decomposition in the two directions.
不用将矢量分解为两个方向
Here is mg-- that's a given.
这是mg，已知量
And we know that the other directions are also given--
同时其它方向也已经给出
this angle of 30 degrees here and this angle of 45 degrees.
这个角度为30°，这个是45°
If these two forces must cancel out this one why don't I flip this one over?
如果这两个力必须与重力抵消，为何不把重力转过来呢
Here it comes. I flip it over. There it is.
就是这样，转过来，成了这样
T1 and T2 now, together, must add up to this one.
T1和T2合起来，必然等于这个
Then the problem is solved, then the net force is zero.
此时总的合力就是0了
Well, that's easy-- I do this.
这很简单
And now I have constructed a complete fair construction of T1 and of T2.
这就构造了一个很好的T1、T2图形
No physics anymore now, it's all over.
题目的物理部分到此为止
You know this angle here, 45 degrees, so this is 45 degrees.
这个角度已知，为45°
This is 30, this is 30.
这个是30°
You know all the angles and you know this magnitude is mg
所有角度均已知，长度mg也已知
so it's a high school problem.
这是高中问题
You have a triangle with all the angles and one side;
三角形所有角和一个边已知
you can calculate the other sides
其它边都能求出
and you should find exactly the same answer, of course.
算出的答案肯定与前面相同
We made an attempt to rig it up.
我尝试设计了一个实验
How do we measure tension?
那么如何测量张力
Well, we put in these lines, scales, tension meters
这些方向上要用到张力计
and that is problematic, believe me.
这里问题很大，相信我
We put in here a tension meter, we put in here a tension meter
这里放一个张力计，这里也放一个
and the bottom one, we hang on a string with a tension meter
下面这里也设置一个张力计
and then here we put four kilograms.
下面挂4kg物体
These scales are not masses.
刻度不是重力刻度
That's already problematic.
这已经很成问题了
The scales are not very accurate
而张力计又不很准确
so we may not even come close to these numbers.
所以不能指望得到很精确数值
For sure, if I put four kilograms here
不过如果这里挂4kg
then I would like this one to read 40 newtons
我肯定希望张力计读数为40
or somewhere in that neighborhood
或和40相差不远
depending on how accurate my meters are.
这取决于张力计的精度
These are springs, and the springs extend
张力计上有弹簧，可以拉伸
and when the springs extend, you see a handle... a hand go.
弹簧拉伸，指针就会偏转
You can clearly see how that works
原理其实很简单
because if there is a force on that bottom scale in this direction,
如果张力计底部有这样一个力
which is mg, and it's not being accelerated
大小mg，系统又没有加速
then the string must pull upwards and so... in order to make the net force zero.
因此细绳上必然有向上的拉力，使合力为零
And if you have a pull down here and you have a pull up here
这里往下拉，这里产生向上的拉力
and you have in here a spring
中间是根弹簧
then you see you have a way of measuring that force.
这就能测出力的大小了
We often do that-- we measure with springs the tension in strings.
用弹簧测张力是常用的方法
For whatever it's worth, I will show you what we rigged up.
不管实验有没价值，反正我是设计好了
Now a measurement without knowledge of uncertainties is meaningless--
“不考虑不确定度的测量，毫无意义”
I told you that.
我说过的
So maybe this is meaningless, what I am going to do now.
也许这个实验确实没什么意义
Let me do something meaningless for once.
没意义也做一次吧
And remember, when I show it, you can always close your eyes
我做我的，你们可以闭上眼睛
so that you haven't seen it.
就当没看见
So we have here something that approaches this 60 degrees
这个悬挂角度大概是60°
and this approaches the 45 degrees
这个大概是45°
and we're going to hang four kilograms at the bottom.
下面挂4kg的质量
There it is, and here it is.
就是这样了
All right, this one-- it's not too far from 40.
下面这个读数和40相差不大
It's not an embarrassment.
不算很糗
This one is not too far from 20.7.
这个离20.7也不远
This one is a bit on the low side. Maybe I can push it up a little.
这一个读数小了点，往上推一下
I think that's close to 30; it's not bad.
接近30，不坏
So you see, it's very difficult to get these angles right but it's not too far off.
这些角度很难弄准，但结果也不算太离谱
So let's remove this again because this will block your view.
做完了，取下来，免得挡着视线
These scales were calibrated in newtons, as you could see.
张力计的刻度是牛顿，你们看到了
Now we come to something very delicate.
下面来一个很精妙的实验
Now I need your alertness and I need your help.
需要各位打起精神，并协助完成
I have a block-- you see it there-- and that block weighs two kilograms.
那里有一个方块，重2千克
A red block. So here it is. It's red.
红色方块，画出来，红色
And I have two strings.
有两条细绳
It's hanging from a black string here and a black string there.
一条挂在上方，一条在底下，均为黑色
Ignore that red string, that is just a safety.
红色细绳不管，那是保险
But it's a very thin thread here and here.
细绳非常之细，这里和这里
And they are as close as we can make them the same.
我尽可能地让它们完全一样
They come from the same batch.
它们是一个批次的产品
This one has a mass of two kilograms and this string has no mass.
方块质量2kg，而细绳没有质量
This is two kilograms.
这个是2kg
So what will be the tension in the upper string which is string number one?
上面的称为细绳一号，那么一号的张力是多少
This is string number two.
下面是细绳二
Well, this string must be able to carry this two kilograms
由于一号需要承载2kg重量
so the tension has to be 20 newtons.
所以其张力为20N
So you will find here the tension-- call it T1-- which is about 20 newtons.
这里张力，称为T1，大约20N
So it's pulling up on this object.
向上拉拽物体
It's also pulling down from the ceiling, by the way.
同时向下拉拽天花板
Think about it, it's pulling from the ceiling.
想想，向下拉拽天花板
The tension is here, 20 newtons.
张力是20牛顿
We could put in here one of these scales
可以在这里放一个张力计
and you would see approximately 20 newtons.
读数肯定大约在20N
What is the tension here? Well, the tension here is very close to zero.
那么这里张力是多少，显然很接近于0
There's nothing hanging on it and the string has no weight
细绳没挂东西，下面没有重力
so there's no tension there-- you can see that.
所以它没有张力，你们可以看到
Now I am going to pull on here
然后我给它一个拉力
and I'm going to increase the tension on the bottom one
我会持续增加二号绳上的张力
until one of the two breaks.
直到两绳有一根断裂
So this tension goes up and up
这个张力不断增大、增大
and therefore, since this object is not being accelerated--
又因为物体没有加速
we're going to get a force down now on this object--
物体受到向下的力
this tension must increase, right?
这个张力必然增加
You see that?
了解？
If I have a force on this one... so there's a force here, and there is mg
如果这里有一个这样的力和mg
then, of course, this string must now be mg plus this force.
那么当然细绳上力为mg加这个力
So the tension will go up here and the tension will go up here.
这样两条绳上的张力都会增加
The strings are as identical as they can be.
而细绳又基本完全相同
Which of the strings will break first? What do you think?
哪条绳会先断，你们怎么想
Excuse me?
什么
I can't hear you.
我听不到
The one on top.
上面一条？
Who is in favor of the one on top?
同意上面一根的举手
Who says no, the bottom one?
谁同意下面一根先断
Who says they won't break at all?
有没认为都不会断的
Okay, let's take a look at it.
好，我们试试
The one on top-- that's the most likely, right?
上面一条理论上应该先断
Three, two, one, zero.
3 2 1 0
The bottom one broke.
下面那根断了
My goodness.
天啊
Newton's Second Law is at stake.
第二定律岌岌可危
Newton's Third Law is at stake.
第三定律岌岌可危
The whole world is at stake!
整个世界岌岌可危
Something is not working.
哪里出问题了
I increased tension here, this one didn't break.
这里张力增加，但这根绳没有断
This one's stronger, perhaps. No, I don't cheat on you; I'm not a magician.
这根也许更扎实…不会呀，我没使诈，我不是魔术师
I want to teach you physics.
我是教物理的
Did we overlook something?
我们是不是漏掉了什么
You know, I'll give you a second chance.
好吧，再来一次
We'll do it again.
再试试
Let's have another vote.
再举一次手
So I'll give you a chance to change your minds.
你们可以改变主意
It's nothing wrong in life, changing your mind.
生活中改变主意并非坏事
It's one of the greatest things that you can do.
很多时候这是必须的
What do you think will happen now?
现在大家觉得会怎样
Who is in favor still of the top one? Seeing is believing.
谁认为上面的先断，眼见为实
You still insist on the top one?
你仍然坚持上面先断？
Who is now in favor of the bottom one?
谁认为下面的先断
Ah, many of you got converted, right?
啊，很多人叛变了
Okay, there we go.
好的，开始了
Three, two, one, zero.
3 2 1 0
The top one broke.
上面一根断了
So some of you were right.
还是有人对了
Now I'm getting so confused.
现在我很困惑
I can't believe it anymore.
难以置信
First we argued that the top one should break
先前我们认为上面的会先断
but it didn't-- the bottom one broke.
结果下面的却先断了
Then we had another vote and then the top one broke.
之后我们改变想法，上面的却先断了
Is someone pulling our leg?
这不是耍我们吗
I suggest we do it one more time.
我建议再来一次
I suggest we do it one more time
我建议再来一次
and whatever's going to happen, that's the winner.
这次结果决定胜利者的归属
If the top one breaks, that's the winner.
如果上面先断，它就赢了
If the bottom one breaks, well, then, we have to accept that.
如果下面先断，我们也只好接受
But I want you to vote again.
再来一次举手表决吧
I want you to vote again on this decisive measurement
表决这个决定性的实验
whether the top one will break first or the bottom one?
是上面还是下面先断
Who is in favor of the top one?
谁认为是上面
Many of you are scared, right? You're not voting anymore!
我打赌很多人都不敢再投票了
I can tell, you're not voting.
肯定是这样
Who is in favor of the bottom one?
谁认为是下面
Only ten people are voting.
只有10个人举了手
Let's do this in an undemocratic way.
那就别怪我不民主了
You may decide-- what's your name?
让你独裁吧，你叫什么名字
Alicia?
艾丽夏？
Georgia, close enough.
乔吉娅…差不多啦
You may decide whether the top one or the bottom one will break.
由你决定，上面还是下面先断
Isn't that great? Doesn't it give you a fantastic amount of power?
这种无上的权力很棒吧
GEORGIA: The bottom one.
乔吉娅：下面一根
The bottom one.
下面一根
yoeo24 翻译制作
oCourse
The bottom one. You ready?
下面这根，准备
Three, two, one, zero.
3 2 1 0
The bottom one broke.
下面一根断了
You were right.
你是对的
You will pass this course.
这门课你过了
yoeo24 翻译制作
oCourse
Thank you, and see you Wednesday.
谢谢，周三见
By the way, think about this, think about this.
别忘了，想想这是怎么回事