and there are ways to predict exactly how it will fall.

有好幾種方法預測它會如何掉落

But there’s something we’ve been leaving out: forces, and why they make things accelerate.

但我們一直沒有提到力 以及他們為何會產生加速度

And for that, we’re going to turn to a physicist you’ve probably heard of: Isaac Newton.

正因如此 今天要來談談你可能早有耳聞的 艾薩克牛頓

With his three laws, published in 1687 in his book Principia, Newton outlined his understanding of motion

牛頓在這本1687年出版的書中 概述了他的想法

-- and a lot of his ideas were totally new.

而許多想法在當時相當先進

Today, more than 300 years later, if you’re trying to describe the effects of forces on

直至300多年後的今日 若你要解釋力的運動

just about any everyday object -- a box on the ground, a reindeer pulling a sleigh, or

幾乎是生活上的所有事 放在地上的箱子 鹿拉動雪橇

an elevator taking you up to your apartment -- then you’re going to want to use Newton’s Laws.

或公寓中的電梯 你都會想用牛頓的運動定律來解釋

And yes. I’ll explain the reindeer thing in a minute.

喔 我一定會解釋剛剛那鹿在幹嘛的

[Theme Music]

牛頓的第一運動定律在談論慣性 基本上就是在說物體的傾向

Newton’s first law is all about inertia, which is basically an object’s tendency

為何它會一直運動

to keep doing what it’s doing.

它是這麼說的 一個物體在運動時必會持續運動

It’s often stated as: “An object in motion will remain in motion, and an object at rest

除非它被其他力給阻止

will remain at rest, unless acted upon by a force.”

這就代表著 如果想改變一物體的運動方式

Which is just another way of saying that, to change the way something moves -- to give

也就是給它加速度 你需要施加有淨外力

it ACCELERATION -- you need a net force.

然而 要如何衡量慣性呢?

So, how do we measure inertia?

最重要的一點是明白它的質量 假設你有兩顆大小相同的球

Well, the most important thing to know is mass. Say you have two balls that are the

一個是充氣後的海灘球 另一個是保齡球

same size, but one is an inflatable beach ball and the other is a bowling ball.

保齡球既會比較難以推動 也比較難以讓它靜止

The bowling ball is going to be harder to move, and harder to stop once it’s moving.

因為質量越大 慣性也越大

It has more inertia because it has more mass.

合理吧? 質量越大代表它有越多東東 會有個傾向讓它持續運動

Makes sense, right? More mass means more STUFF, with a tendency to keep doing what it was

直到你施加了另一力去打斷它

doing before your force came along, and interrupted it.

而這又完美的帶到牛頓第二運動定律: 施加於物體的淨外力=物體質量*加速度

And this idea connects nicely to Newton’s second law: net force is equal to mass x acceleration.

有個公式: F=ma

Or, as an equation, F(net) = ma.

一定要記得我們說的是【淨力】

It’s important to remember that we’re talking about NET force here -- the amount

因為遺留的力可能會抵消所有你剛剛施加的力

of force left over, once you’ve added together all the forces that might cancel each other out.

舉個例子 把個冰球放在完全沒摩擦力的溜冰場

Let's say you have a hockey puck sitting on a perfectly frictionless ice rink. And I know ice isn’t

我知道冰是有摩擦力的啦 但仔細聽我說 如果你用球棍貼緊冰球去推它

perfectly frictionless but stick with it. If you’re pushing the puck along with

力將不會被任何東西抵銷

a stick, that’s a force on it - that isn’t being canceled out by anything else.

因此冰球產生了加速度

So the puck is experiencing acceleration.

但如果冰球沒動起來 或在你推之後才滑出去

But when the puck is just sitting still, or even when it’s sliding across the ice after

那麼所有的力早已相抵了

you’ve pushed it, then all the forces are balanced out.

這就叫平衡

That’s what’s known as equilibrium.

一個受到平衡力的物體是可以繼續運動的 但它的速度不會改變

An object that’s in equilibrium can still be MOVING, like the sliding puck, but its VELOCITY won’t be changing.

但若力沒有達到平衡 好玩的事就發生了

It’s when the forces get UNbalanced, that you start to see the exciting stuff happen.

最常見的淨力就是萬有引力

And probably the most common case of a net force making something move is the gravitational force.

假設你直線上丟了一顆五公斤的球 接下來 你懂的 快離開現場

Say you throw a 5 kg ball straight up in the air -- and then, yknow, get out of the way,

因為它會打得你兩眼冒金星

because that could really hurt if it hits you...

在地球引力拉下那顆球之前 它會以每秒 9.81公尺的加速度落下

But the force of gravity is pulling down on the ball, which is accelerating downward at a rate of 9.81 m/s^2.

因此淨力=m*a 但這裡唯一的力是地球引力

So the net force is equal to m a, but the only force acting here is gravity.

這表示 如果我們算得出球落下產生的加速度 就能計算地球引力了

This means that, if we could measure the acceleration of the ball, we’d be able to calculate the force of gravity.

而我們當然算得出加速度 加速度就是9.81 m/s^2 我們叫它g

And we CAN measure the acceleration -- it’s 9.81 m/s^2, the value we’ve been calling small g.

那顆球受到的引力必定是5公斤 然後在乘上g

So the force of gravity on the ball must be 5 kg, which is the mass of the ball… times small g

會等於49.05 公斤除每秒平方再乘上公尺

which comes to 49.05 kilograms times meters per second squared!

我們太常用這公式了 因此簡寫成F(g)=mg

We use this equation for gravity so much that it’s often just written as F(g) = mg.

這就是得知地球引力總量的方式 當然你也可以秤重

That’s how you determine the force of gravity, otherwise known as weight.

而這些有點太繁複 因此我們直接叫它牛頓

Now, those units can be a bit of a mouthful, so we just call them Newtons.

沒錯 為了紀念他 我們用牛頓來形容重量 不是公斤喔

That’s right! We measure weight in Newtons, in honor of Sir Isaac, and NOT kilograms!

公斤是計算質量的

Kilograms are a measure of mass!

但引力並不是唯一作用在物體上的力

But gravity often isn’t the only force acting on the object.

因此當我們在算淨力時 通常會計算比這更多

So when we’re trying to calculate a NET force, we usually have to take into account more than just gravity.

這就是為何這力那麼常被提到 這解釋了牛頓第三運動定律

This is where we get into one of the forces that tends to show up a lot, which is explained by Newton’s third law.

你大概聽過:當兩個物體互相作用時， 彼此施加於對方的力，其大小相等、方向相反

You probably know this law as “for every action, there’s an equal but opposite reaction.”

這表示當你施力於一個物體 它也會作用相同的力給你

Which just means that if you exert a force on an object, it exerts an equal force back on you.

我們叫它反作用力( normal force)

And that’s what we call the normal force.

“反(normal)”在這裡表示"垂直" 因為反作用力永遠是垂直於物體的

“Normal” in this instance means “perpendicular”, and the normal force is always perpendicular

不管你將物體放在何處

to whatever surface your object is resting on.

至少當你在推一個又大又重的東西時 就像桌子

At least, it is when you're pushing on something big, and macroscopic, like a table.

把書放在桌上 反作用力將作用給書 書因此能被放在桌上

If you put a book down on a table, the normal force is pushing -- and therefore pointing -- up.

但若你把它放在傾斜表面 反作用力會與傾斜面垂直

But if you put it on a ramp, then the normal force is pointing perpendicular to the ramp.

反作用力和其他力不太一樣 它會改變大小

Now, the normal force isn’t like most other forces. It’s special, because it changes its magnitude.

假設你把一張鋁箔緊包在碗的上方

Say you have a piece of aluminum foil stretched tightly across the top of a bowl, and then

再把一顆葡萄放上去

you put one lonely grape on top of it.

因為地球引力 葡萄會施加力給鋁箔

Because of gravity, that grape is exerting a little bit of force on the foil, and the

但反作用力又將它推回去 用一樣的力

normal force pushes right back, with the same amount.

若你又再加上一顆葡萄 讓下壓力變成雙倍 反作用力也會以雙倍的力頂回去

But then you add another grape, which doubles the force on the foil -- in that case, the normal force doubles too.

這會持續到你加了太多葡萄讓鋁箔破掉為止

That’ll keep happening until eventually you add enough grapes that they break through the foil.

這就是當反作用力無法承受太多下壓力時發生的

That’s what happens when the normal force can’t match the force pushing against it.

但牛頓有名的第三運動定律到底在說啥?

But, what does Newton’s famous third law really mean, though?

當我用指頭推這個書桌時 我是在給書桌增加壓力

When I push on this desk with my finger right now, I’m applying a force to it.

而同時反作用力也會作用給我 而且只有我能感受到

And it’s applying an equal force right back on my finger -- one that I can actually feel.

但假如這是真的 已經是了啦 為何我們還可以推動東西?

But if that’s true -- and it is -- then why are we able to move things?

我如何撿起這馬克杯? 鹿如何拉動雪橇?

How can I pick up this mug? Or how can a reindeer pull a sleigh?

基本上 物體之所以得以運動 不是只關於作用力與反作用力

Basically, things can move because there’s more going on, than just the action and reaction forces.

比方說 當鹿拉雪橇時 牛頓告訴我們那個雪橇

For example, when a reindeer pulls on a sleigh, Newton’s third law tells us that the sleigh

也會同樣推回鹿 因為作用力等於反作用力

is pulling back on it with equal force.

但鹿仍然可以推動雪橇 是因為牠有立足點

But the reindeer can still move the sleigh forward, because it’s standing on the ground.

當牠向前一步 表示牠在用腳後推地板 而地板也推牠向前

When it takes a step, it’s pushing backward on the ground with its foot -- & the ground is pushing it forward.

因此 雪橇在向後運動時 鹿在拉牠前進

Meanwhile, the reindeer is also pulling on the sleigh, while the sleigh is pulling right back.

而地板推鹿向前的力大於

But the force from the GROUND PUSHING the reindeer forward is STRONGER than the

雪橇向後的力 因此動物向前的同時也拉動了雪橇

force from the sleigh pulling it back. So the animal accelerates forward, and so does the sleigh.

因此要記得 沒有物理學就沒有聖誕節!

So, one takeaway here is that: there would be no Christmas without physics!

現在我們都大概了解力了 那麼來想想

But, now that we have an idea of some of the forces we might encounter, let’s describe

當一個箱子放在地上時究竟發生了甚麼事?

what’s happening when a box is sitting on the ground.

你要先----記得遇到類似問題時一定要先

The first thing to do -- which is the first thing you should ALWAYS do when you’re solving

畫一個力圖分析

a problem like this -- is draw what’s known as a free body diagram.

基本上就是畫一條粗線在物體下 再畫一個點在物體在中央

Basically, you draw a rough outline of the object, put a dot in the middle, and then

再畫標籤箭頭 代表所有的力

draw and label arrows, to represent all the forces.

還要記得區分何者為正向 在這邊我們會選擇向上的力

We also have to decide which direction is positive -- in this case, we’ll choose up to be positive.

箱子的分析力圖其實很簡單 有個箭頭代表向下的力

For our box, the free body diagram is pretty simple. There’s an arrow pointing down,

也就是地球引力 而向上的力代表著

representing the force of gravity, and an arrow pointing up, representing the force

代表著桌子給它的支撐力

of the ground pushing back on the box.

而箱子是靜止的 由此可知它並沒有在做加速度運動

Since the box is staying still, we know that it’s not accelerating, which tells us that

也就是它所受到的力是相等的 合力=0

those forces are equal, so the net force is equal to 0.

但如果你連一條繩在箱子上方 並把它掛在天花板上呢?

But what if you attach a rope to the top of the box, then connect it to the ceiling so

讓箱子懸浮在空氣中?

the box is suspended in the air?

你的淨力依舊等於0 因為箱子沒有受到任何加速度

Your net force is still 0, because there’s no acceleration on the box. And gravity is

依舊像之前受同樣引力

still pulling down in the same way it was before.

但現在 抵銷向上的力來自連接箱子的繩子 我們稱之為張力

But now, the counteracting upward force comes from the rope acting on the box, in what we call the tension force.

為了讓例子簡單一點 我們假設繩子是沒有質量並且牢固的

To make our examples simpler, we almost always assume that ropes have no mass and are completely

不論你如何向下拉 它都會依同樣的力往上

unbreakable -- no matter how much you pull on them, they’ll pull right back.

這代表著張力是不固定的 假設箱子重5公斤

Which means that the tension force isn’t fixed. If the box weighs 5 Newtons, then the

繩子的張力將會是5牛頓 但若箱子再加了5牛頓

tension in the rope is also 5 Newtons. But if we add another 5 Newtons of weight, the

繩子ˇ的張力將也變成10牛頓

tension in the rope will become 10 Newtons.

有點像反作用力 還記得在鋁箔上的葡萄嗎?

Kind of like how the normal force changes, with the grapes on the foil. But in this case,

但在這裡影響的是拉力而非推力

it's in response to a pulling force instead of a push.

重點是 無論如何你都可以將所有的力相加

The key is that no matter what, you can add the forces together to give you a particular

去得出淨力 即使淨力不永遠是0

net force -- even though that net force might NOT always be 0. Like, in an elevator.

假設你在電梯( elevator )裡 就是英式英文的lift

So let’s say you’re in an elevator -- or as I call them, a lift.

電梯再加上你的質量是1000公斤 其運動受到

The total mass of the lift, including you, is 1000 kg. And its movement is controlled

平衡力影響 上頭連接了滑輪

by a counterweight, attached to a pulley.

計劃就是在滑輪另一端施重850公斤 讓電梯得以移動

The plan is to set up a counterweight of 850kg, and then let the lift go. Once you let go,

電梯將會加速向下 因為已經超過了它的最大平衡力

the lift is going to start accelerating downward - because it’s HEAVIER than the counterweight.

此時我們希望平衡力不會讓它過快下降

And the hope is that the counterweight will keep it from accelerating TOO much.

要怎麼知道怎樣算安全? 電梯應該以何種速度下降?

But how will we know if it’s safe? How quickly is the lift going to accelerate downward?

首先要先畫分析力圖 讓向上力成為正方向

To find out, first let’s draw a free body diagram for the lift, making UP the positive direction.

將電梯向下拉的引力=電梯質量*g(9.81)

The force of gravity on the lift is pulling it down, and it’s equal to the mass of the lift x small g --

9810牛頓就是在負方向

9810 Newtons of force, in the negative direction.

而繩子拉它向上的張力在正方向

And the force of tension is pulling the lift UP, in the positive direction.

也就是說 淨力= 張力-電梯質量*g

Which means that for the lift, the net force is equal to the tension force, minus the mass of the lift x small g.

現在 因為牛頓第一運動定律說 淨力等於質量乘上加速度 我們就可以假設

Now! Since Newton’s first law tells us that F(net) = ma, we can set all of that to be

所有的力等於電梯質量 也就是電梯質量乘上加速度

equal to the lift’s mass, times some downward acceleration, -a. That’s what we’re trying to solve for.

所以 就來這樣解解看我們的平衡力吧

So, let’s do the same thing for the counterweight.

引力以8338.5牛頓向下拉

Gravity is pulling it down with 8338.5 N of force in the negative direction.

而張力又在向上拉 因此淨力等於張力(拉力)

And again, the force of tension is pulling it up, so that the net force is equal to the

再減掉平衡力*g

tension force, minus the mass of the counterweight times small g.

再一次地 因為牛頓第二定律 我們明白所有的力都等於

And again, because of Newton’s second law, we know that all of that is equal to the mass

平衡力的質量乘上加速度(a) 這次的a是正方向

of the counterweight, times that same acceleration, “a” -- which is positive this time since

也就是拉電梯向上的方向

the counterweight is moving upward.

因此 整理一下後會有兩個公式 兩個都充滿未知數

So! Putting that all together, we end up with two equations -- and two unknowns.

我們不知道張力是多少 也不知道加速度是多少

We don’t have a value for the tension force, and we don’t have a value for acceleration.

但當我們要解題時 會用代數去做

But what we’re trying to solve for is the acceleration. So we use algebra to do that.

當你有了兩個這樣的方程式 可以選擇將兩個式子相加或相減

When you have a system of equations like this, you can add or subtract all the terms on each

讓它們變成一個式子

side of the equals sign, to turn them into one equation.

打個比方 你知道1+2=3 也知道4+2=6

For example, if you know that 1 + 2 = 3 and that 4 + 2 = 6, you can subtract the first

兩個式子相減會得到3=3

equation from the second to get 3 = 3.

由此可知 我們的情況是要兩個式子相減

And in our case, with the lift, subtracting the first equation from the second gets rid

讓等號後等於張力

of the term that represents the tension force.

現在就只要算出加速度了 這代表著 我們要將式子的等號後全變a

We now just have to solve for acceleration -- meaning, we need to rearrange the equation to set everything equal to “a.”

寫出a和重量的不同處來結束式子

We end up with an equation that really just says that “a” is equal to the difference

或是a和淨力的不同處再除全部質量

between the weights -- or the net force on the system -- divided by the total mass.

事實上 這就是複雜版的F(net) = ma.

Essentially, this is just a fancier version of F(net) = ma.

在此我們可以算出a 也就是a= 0.795 m/s

And we can solve that for “a”, which turns out to be 0.795 m/s^2.

這個加速度量實在微乎其微啊

Which is not that much acceleration at all!

所以只要你不降得太深 是很安全的

So, as long as you aren’t dropping too far down, you should be fine.

即便降落處有點凹凸不平

Even if the landing is a little bumpy.

在這一集 你學到了牛頓三大運動定律:慣性如何運作,

In this episode, you learned about Newton’s three laws of motion: how inertia works, that

淨力=質量*加速度 , 平衡的定義是甚麼

net force is equal to mass x acceleration, how physicists define equilibrium, and all

以及反作用力和張力(拉力)

about the “normal” force and the “tension” force.

Crash Course Physics和'PBS數位工作室聯合製作

Crash Course Physics is produced in association with PBS Digital Studios. You can head over

你可以直接去他們的頻道看看像 BrainCraft的酷影片 當聰明人超棒的

to their channel to check out amazing shows like: BrainCraft, It’s OK To Be Smart, and PBS Idea Channel.

這集Crash Course在 Cheryl C. Kinney工作室拍攝

This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

多虧了這些人的幫忙 還有我們的動畫團隊是Thought Cafe

with the help of these amazing people and our Graphics Team is Thought Cafe.