So let's say I have a spring.

所以，我們'說我有一個彈簧。

Let me draw the ground so that we know what's going on with

讓我來畫地，這樣我們就知道是怎麼回事了。

the spring.

春天的時候。

So let me see, this is the floor.

讓我看看，這就是地板。

That's the floor, and I have a spring.

那是地板，我有一個彈簧。

It's along the floor.

它沿著地板。

I'll use a thicker one, just to show it's a spring.

我'會用粗一點的，只是為了顯示它'是個彈簧。

Let's say the spring looks something like this.

比方說，彈簧的樣子是這樣的。

Whoops, I'm still using the line tool.

哎呀，我'還在用線條工具。

So the spring looks like this.

所以春天是這樣的。

This is my spring, my amazingly drawn spring.

這就是我的春天，我那畫風驚人的春天。

Let's say at this end it's attached to a wall.

比方說，在這一端它'連著一堵牆。

That's a wall.

那是一堵牆。

And so this is a spring when I don't have any force acting on

所以這是一個彈簧，當我沒有任何力量作用在。

it, this is just the natural state of the spring.

它，這只是春天的自然狀態。

And we could call this, where it just naturally rests, this

我們可以把它叫做，它只是自然地休息，這個

tip of the spring.

彈簧的尖端。

And let's say that when I were to apply a force of 5 Newtons

比如說當我施加5牛頓的力時

into the spring, it looks something like this.

進入春天，它看起來像這樣。

Redraw everything.

重畫一切。

So when I apply a force of 5 Newtons-- I'll draw the wall

所以當我施加5牛頓的力時... 我就會畫出牆來。

in magenta now.

在洋紅色的現在。

When I apply a force of 5 Newtons, the

當我施加一個5牛頓的力時,我就會發現...

spring looks like this.

春天的樣子是這樣的。

It compresses, right?

它可以壓縮，對嗎？

We're all familiar with this.

我們'都很熟悉。

We sit on a bed every day or a sofa.

我們每天坐在床上或者沙發上。

So let's say it compresses to here.

所以我們'說它壓縮到這裡。

If this was the normal resting-- so this is where the

如果這是正常的休息... ...所以這是在哪裡的。

spring was when I applied no force, but when I applied 5

當我沒有施加任何力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是，但當我施加5％的力量時，彈簧是

Newtons in that direction, let's say that this distance

牛頓在這個方向上，比方說這個距離。

right here is 10 meters.

這裡是10米。

And so a typical question that you'll see, and we'll explain

所以一個典型的問題，你'會看到，我們'會解釋。

how to do it, is a spring compresses or elongates when

怎麼做，是彈簧在壓縮或伸長的時候。

you apply a certain force by some distance.

你在一定距離內施加一定的力。

How much will it compress when you apply a different force?

當你施加不同的力時，它將壓縮多少？

So my question is how much will it compress when I apply

所以我的問題是，當我應用 "我 "時，它會壓縮多少？

a 10-Newton force?

一個10牛頓的力？

So your intuition that it'll compress more is correct, but

所以，你的直覺，它會壓縮更多是正確的，但。

is it linear to how much I compress it?

它與我壓縮的程度是線性的嗎？

Is it a square of how much I compress it?

是我壓縮多少的平方？

How does it relate?

它的關係如何？

I think you probably could guess.

我想你大概能猜到。

It's actually worth an experiment.

其實這'值得一試。

Or you could just keep watching the video.

或者你可以繼續看視頻。

So let's say I apply a 10-Newton force.

所以，讓我們'說我施加一個10牛頓的力。

What will the spring look like?

春天會是什麼樣子？

Well, it'll be more compressed.

好吧，它會更加壓縮。

Drop my force to 10 Newtons.

把我的力降到10牛頓。

And if this was the natural place where the spring would

如果這裡是天然的泉源地的話

rest, what is this distance?

休息，這個距離是多少？

Well, it turns out that it is linear.

好吧，事實證明，它是線性的。

What do I mean by linear?

什麼叫線性？

Well, it means that the more the force-- it's equally

意思是力量越大...

proportional to how much the spring will compress.

與彈簧的壓縮程度成正比。

And it actually works the other way.

而實際上是另一種方式。

If you applied 5 Newtons in this direction, to the right,

如果你向這個方向施加5牛頓，向右。

you would have gone 10 meters in this direction.

你會在這個方向走10米。

So it goes whether you're elongating the spring or

所以，無論你是在拉長彈簧還是在拉大彈簧。

compressing the spring within some reasonable tolerance.

將彈簧壓縮在合理的公差範圍內。

We've all had this experience.

我們'都有這樣的經歷。

If you compress something too much or you stretch it too

如果你把東西壓得太緊或者你把它拉得太長的話

much, it doesn't really go back to where it was before.

多，它並沒有真正回到以前的地方。

But within some reasonable tolerance, it's proportional.

但在一些合理的容忍度內，它'是成正比的。

So what does that mean?

那麼這意味著什麼呢？

That means that the restoring force of the spring is minus

這意味著彈簧的恢復力為減去

some number, times the displacement of the spring.

某個數字，乘以彈簧的位移。

So what does this mean?

那麼這意味著什麼呢？

So in this example right here, what was the displacement of

所以，在這個例子中，這裡的位移是什麼？

the spring?

泉水？

Well, if we take positive x to the right and negative x to

好吧，如果我們取正x向右，負x向右

the left, the displacement of the spring was what?

左邊，彈簧的位移是多少？

The displacement, in this example right here, x is equal

在這個例子中，位移，就在這裡，x等於

to minus 10, right?

到負10，對嗎？

Because I went 10 to the left.

因為我向左走了10步。

And so it says that the restorative force is going to

所以說恢復力要

be equal to minus K times how much it's

等於負K乘以多少錢'的。

distorted times minus 10.

失真次數減10。

So the minuses cancel out, so it equals 10K.

所以負數抵消了，所以等於1萬。

What's the restorative force in this example?

這個例子中的恢復力是什麼'？

Well, you might say, it's 5 Newtons, just because that's

好吧，你可能會說，這是5牛頓，只是因為這是我的。

the only force I've drawn here, and you would be to some

我在這裡畫的唯一的力量，你將是一些

degree correct.

度正確。

And actually, since we're doing positive and negative,

而實際上，既然我們'做的是正反兩方面。

and this 5 Newton is to the left, so to the negative

而這個5牛頓是在左邊，所以是負數

x-direction, actually, I should call this minus 5

x方向，其實應該叫這個減去5

Newtons and I should call this minus 10 Newtons, because

牛頓，我應該叫它減10牛頓，因為...

obviously, these are vectors and we're going to the left.

很明顯，這些都是矢量，我們'向左走。

I picked the convention that to the left means negative.

我選取了慣例，向左表示負。

So what's the restorative force?

那麼什麼是恢復力呢'？

Well, in this example-- and we assume that K is a positive

好吧，在這個例子中 -- 我們假設K是一個正值的。

number for our purposes.

為我們的目的，數。

In this example, the restorative force is a

在這個例子中，恢復力是一種。

positive number.

正數。

So what is the restorative force?

那麼什麼是恢復力呢？

So that's actually the force, the counteracting force, of

所以，這其實就是力量，反作用力，的。

the spring.

春天的時候。

That's what this formula gives us.

這'就是這個公式給我們的。

So if this spring is stationary when I apply this

所以，如果當我應用這個彈簧時，這個彈簧是靜止的。

5-Newton force, that means that there must be another

5-牛頓力，這意味著一定還有另一個...

equal and opposite force that's

等反力

positive 5 Newtons, right?

正5牛頓，對吧？

If there weren't, the spring would keep compressing.

如果沒有'，彈簧會一直壓縮。

And if the force was more than 5 Newtons, the spring would go

如果力超過5牛頓，彈簧就會走不動了

back this way.

往這邊走

So the fact that I know that when I apply a 5-Newton force

所以我知道當我施加一個5牛頓的力的時候。

to the left, or a negative 5-Newton force, the spring is

向左，或負5牛頓力，彈簧為

no longer moving, it means that there must be-- or no

不再移動，這意味著必須有--或者說沒有。

longer accelerating, actually, it means that there must be an

不再加速，其實，這意味著，一定有一個。

equal and opposite force to the right, and that's the

右邊的力相等且相反，那就是

restorative force.

恢復力。

Another way to think about it is if I were to let-- well, I

另一種思考方式是，如果我讓... ... 好吧，我

won't go in there now.

不會去那裡了。

So in this case, the restorative force is 5

所以在這種情況下，恢復力是5。

Newtons, so we can solve for K.

牛頓，所以我們可以求解K。

We could say 5 is equal to 10K.

我們可以說5等於10K。

Divide both sides by 10.

兩邊除以10。

You get K is equal to 1/2.

你得到K等於1/2。

So now we can use that information to figure out what

所以，現在我們可以利用這些資訊來計算出什麼是

is the displacement when I apply a

是指當我應用

negative 10-Newton force?

負10牛頓力？

When I push the spring in with 10 Newtons in

當我用10牛頓的力量把彈簧推入時

the leftward direction?

向左的方向？

So first of all, what's the restorative force here?

那麼首先，這裡的恢復力是什麼'？

Well, if the spring is no longer accelerating in either

好吧，如果彈簧不再加速，無論在哪種情況下，都會有一定的加速作用。

direction, or the tip of the spring is no longer

的方向，或者彈簧的尖端不再是

accelerating in either direction, we know that the

在任何一個方向上加速，我們知道。

restorative force must be counterbalancing this force

復原力必須與這種力量相抗衡。

that I'm compressing with, right?

我正在壓縮的，對不對？

The force that the spring wants to expand back with is

彈簧想要回彈的力是...。

10 Newtons, positive 10 Newtons, right?

10牛頓，正10牛頓，對吧？

And we know the spring constant, this K for this

我們知道彈簧常數，這個K對這個

spring, for this material, whatever it might be, is 1/2.

彈簧，對於這種材料，不管它是什麼，是1/2。

So we know the restorative force is equal to 1/2 times

所以我們知道恢復力等於1/2倍

the distance, right?

的距離，對嗎？

And the formula is minus K, right?

而公式是減K，對嗎？

And then, what is the restorative

然後，什麼是恢復性的

force in this example?

在這個例子中，力?

Well I said it's 10 Newtons, so we know that 10 Newtons is

我說了是10牛頓，所以我們知道10牛頓是... ...

equal to minus 1/2x.

等於負1/2x。

And so what is x?

那麼x是什麼呢？

Well, multiply both sides by minus 1/2, and

好吧，把兩邊都乘以減去1/2，然後...

you get minus 20.

你得到負20。

I'm sorry, multiply both sides by minus 2, you get minus 20

對不起，兩邊都乘以負2，就可以得到負20。

is equal to x.

等於x。

So x goes to the left 20 units.

所以x向左走20個組織、部門。

So that's all that it's telling us.

所以這就是它告訴我們的全部內容。

And this law is called Hooke's Law, and it's named after--

這個定律叫做胡克定律 它的名字是以...

I'll read it-- a physicist in the 17th century, a British

我讀一下--17世紀的物理學家，英國人。

physicist. And he figured out that the amount of force

物理學家。他想出了力的大小

necessary to keep a spring compressed is proportional to

保持彈簧被壓縮的必要條件與以下因素成正比

how much you've compressed it.

你把它壓縮了多少。

And that's all that this formula says.

這就是這個公式的全部內容。

And that negative number, remember, this formula gives

而這個負數，記住，這個公式給出了：

us the restorative force.

我們的恢復力。

So it says that the force is always in the opposite

所以說，力的作用總是在反面的

direction of how much you displace it.

的方向，你把它移位多少。

So, for example, if you were to displace this spring in

所以，舉例來說，如果你要把這個彈簧移位在

this direction, if you were to apply a force and x were a

在這個方向上，如果你要施加一個力，而x是一個。

positive and you were to go in that direction, the force-- no

而你要往那個方向走，那力量... ...

wait, sorry.

等等，對不起。

This is where the spring rests.

這就是彈簧的作用。

If you were to apply some force and take the spring out

如果你施加一些力，把彈簧取出來。

to here, this negative number tells us that the spring will

到這裡，這個負數告訴我們彈簧將

essentially try to pull back with the restorative force in

基本上是想用恢復性的力量來拉回在

the other direction.

另一個方向。

Let's do one more problem and I think this

讓我們再做一個問題，我覺得這個

will be clear to you.

你會明白的。

So let's say I have a spring, and all of these problems kind

是以，讓我們說我有一個彈簧，和所有這些問題種。

of go along.

的隨。

So let's say when I apply a force of 2 Newtons, so this is

所以我們說，當我施加2牛頓的力時，所以這就是。

what I apply when I apply a force of 2 Newtons.

當我應用2牛頓的力時，我應用什麼。

Well, let's say it this way.

好吧，讓我們'這樣說吧。

Let's say when I stretch the spring.

比方說，當我拉伸彈簧的時候。

Let's say this is the spring, and when I apply a force of 2

比方說這是彈簧，當我施加2的力時

Newtons to the right, the spring gets stretched 1 meter.

牛頓向右，彈簧被拉伸1米。

So first of all, let's figure out what K is.

所以首先，我們要弄清楚什麼是K。

So if the spring is stretched by 1 meter, out here, its

所以，如果彈簧被拉伸了1米，在這裡，它就會被拉伸。

restorative force will be 2 Newtons back this way, right?

復原力這樣一來就會有2個牛頓回來，對不對？

So its restorative force, this 2 Newtons, will equal minus K

所以它的恢復力，這個2牛頓，將等於減去K。

times how much I displaced it.

倍，我把它移位了多少。

Well I, displaced it by 1 meter, so then we multiply

好吧，我，把它移位1米，所以我們乘以。

both sides by negative 1, and we get K is equal to minus 2.

兩邊都乘以負1，我們得到K等於負2。

So then we can use Hooke's Law to note the equation for

所以，我們可以利用胡克定律來記下以下公式

this-- to figure out the restorative force for this

這... ... 弄清楚恢復力為這一點。

particular spring, and it would be minus 2x.

特定的彈簧，它將是負2倍。

And then I said, well, how much force would I have to

然後我說，好吧，我得用多大的力氣才行

apply to distort the spring by 2 meters?

適用於扭曲彈簧2米？

Well, it's 2 times 2, it would be 4.

嗯，是2乘以2，應該是4。

4 Newtons to displace it by 2 meters, and, of course, the

4牛頓，使其位移2米，當然還有

restorative force will then be in the opposite direction, and

恢復力就會向相反的方向發展，而。

that's where we get the negative number.

這就是我們得到負數的地方。

Anyway, I've run out of time.

總之，我'已經沒有時間了。

I'll see you in the next video.

我'下一個視頻裡見。