the Black Swan pulls off a seemingly endless series of turns,

黑天鵝成功完成 那似乎無止境的連續旋轉，

bobbing up and down on one pointed foot

以一個腳尖上下移動，

and spinning around, and around, and around 32 times.

不停不停地旋轉了 32 次。

It's one of the toughest sequences in ballet,

是芭蕾最難的一串動作之一，

and for those thirty seconds or so,

在那大約三十秒裡，

she's like a human top in perpetual motion.

她像個陀螺人一樣不停歇地轉動。

Those spectacular turns are called fouettés,

那驚人的旋轉叫做 單足趾尖旋轉／fouettés ，

which means "whipped" in French,

法文是 快速攪拌／whipped ，

describing the dancer's incredible ability to whip around without stopping.

描述舞者無間斷旋轉的非凡能力。

But while we're marveling at the fouetté, can we unravel its physics?

但當我們對 fouetté 讚嘆之餘， 可否用物理解釋它？

The dancer starts the fouetté by pushing off with her foot to generate torque.

舞者開始做 fouetté 是用腳頂推地面製造扭轉力。

But the hard part is maintaining the rotation.

但困難的部份是保持旋轉。

As she turns,

當她旋轉時，

friction between her pointe shoe and the floor,

摩擦力在尖頭鞋和地板之間，

and somewhat between her body and the air,

以及有些在身體及空氣間，

reduces her momentum.

降低了她的動量。

So how does she keep turning?

所以她是如何保持旋轉？

Between each turn, the dancer pauses for a split second and faces the audience.

每個旋轉之間， 舞者會有一瞬間暫停且面對觀眾，

Her supporting foot flattens,

她支撐的腳會放平，

and then twists as it rises back onto pointe,

當又踮起腳尖時會再次旋轉，

pushing against the floor to generate a tiny amount of new torque.

因為她頂推地板 再次產生少許的新扭轉力。

At the same time, her arms sweep open to help her keep her balance.

同時，她的手臂迅速展開以保持平衡。

The turns are most effective if her center of gravity stays constant,

如果她的重心保持穩定， 則旋轉最具效率，

and a skilled dancer will be able to keep her turning axis vertical.

一位技術純熟的舞者 能保持旋轉軸垂直。

The extended arms and torque-generating foot

展開的手臂和產生扭轉力的腳

both help drive the fouetté.

共同協力推動 單足趾尖旋轉。

But the real secret and the reason you hardly notice the pause

但真正秘訣以及 你幾乎不覺得停頓的原因

is that her other leg never stops moving.

是因為她的另一條腿永遠在動。

During her momentary pause,

在動量暫停之際，

the dancer's elevated leg straightens and moves from the front to the side,

舞者那隻抬起的腿打直， 並且從前面移到旁邊，

before it folds back into her knee.

然後向後折回到膝蓋。

By staying in motion, that leg is storing some of the momentum of the turn.

藉著持續動作， 那下肢儲存了一些旋轉的動量，

When the leg comes back in towards the body,

當下肢縮回靠近身體時，

that stored momentum gets transferred back to the dancer's body,

儲存的動量轉移到舞者身上，

propelling her around as she rises back onto pointe.

當又踮起腳尖時，動量推動她旋轉。

As the ballerina extends and retracts her leg with each turn,

當芭蕾舞者每次旋轉， 伸直與縮回她的下肢，

momentum travels back and forth between leg and body,

動量來回於下肢和身體之間，

keeping her in motion.

讓她繼續轉動。

A really good ballerina can get more than one turn out of every leg extension

一位非常好的芭蕾舞者 能每次下肢伸直時 旋轉多於一圈。

in one of two ways.

就是用下列兩個方法之一。

First, she can extend her leg sooner.

第一，她可提早伸直下肢。

The longer the leg is extended, the more momentum it stores,

下肢伸直的時間越久， 所儲存的動量就越多，

and the more momentum it can return to the body when it's pulled back in.

當下肢縮回時， 就有更多動量回到身體，

More angular momentum means she can make more turns

更多的角動量意味著 她可轉更多圈 ──

before needing to replenish what was lost to friction.

在因摩擦力損失動量，需再補充之前。

The other option is for the dancer

另一個方法是

to bring her arms or leg in closer to her body

當舞者又踮起腳尖時， 馬上將手臂和下肢縮得更靠近身體。

once she returns to pointe.

這為什麼管用呢？

Why does this work?

就像芭蕾其他每個旋轉一樣，

Like every other turn in ballet,

單足趾尖旋轉 受角動量 (L) 的影響，

the fouetté is governed by angular momentum,

舞者的 角動量 (L) = 角速度 (ω) * 轉動慣量 (I)

which is equal to the dancer's angular velocity times her rotational inertia.

除了摩擦力流失的動量外，

And except for what's lost to friction,

當舞者踮腳尖時， 角動量 (L) 是維持恆定的，

that angular momentum has to stay constant while the dancer is on pointe.

這稱為「角動量守恆定律」。

That's called conservation of angular momentum.

現在，轉動慣量（I） 可想成是身體對旋轉的阻力，

Now, rotational inertia can be thought of

當越多質量離旋轉軸越遠時， 轉動慣量（I）會增加；

as a body's resistance to rotational motion.

當質量離旋轉軸越近時， 轉動慣量（I）會減少。

It increases when more mass is distributed further from the axis of rotation,

所以當她把手臂縮更靠近身體時，

and decreases when the mass is distributed closer to the axis of rotation.

轉動慣量（I）會變小，

So as she brings her arms closer to her body,

為了保持角動量 (L)，

her rotational inertia shrinks.

角速度（ω）── 即她的旋轉速度必須增加。

In order to conserve angular momentum,

用相同的儲存動量，

her angular velocity, the speed of her turn,

就能讓她旋轉許多圈。

has to increase,

你可能看過溜冰者作過相同動作，

allowing the same amount of stored momentum

旋轉得越來越快── 藉著縮回他們的手臂和下肢。

to carry her through multiple turns.

在柴可夫斯基的芭蕾舞劇， 黑天鵝是位女巫，

You've probably seen ice skaters do the same thing,

她迷人的 32 單足趾尖旋轉 真是神乎其神，

spinning faster and faster by drawing in their arms and legs.

但並不是魔術讓旋轉成為可能，

In Tchaikovsky's ballet, the Black Swan is a sorceress,

是物理！

and her 32 captivating fouettés do seem almost supernatural.

翻譯：Ann Chen

But it's not magic that makes them possible.

It's physics.