In the third act of "Swan Lake," the Black Swan pulls off a seemingly endless series of turns, bobbing up and down on one pointed foot and spinning around, and around, and around 32 times. It's one of the toughest sequences in ballet, and for those thirty seconds or so, she's like a human top in perpetual motion.
在「天鵝湖」第三幕, 黑天鵝成功完成 那似乎無止境的連續旋轉, 以一個腳尖上下移動, 不停不停地旋轉了 32 次。 是芭蕾最難的一串動作之一, 在那大約三十秒裡, 她像個陀螺人一樣不停歇地轉動。
Those spectacular turns are called fouettés, which means "whipped" in French, describing the dancer's incredible ability to whip around without stopping. But while we're marveling at the fouetté, can we unravel its physics?
那驚人的旋轉叫做 單足趾尖旋轉/fouettés , 法文是 快速攪拌/whipped , 描述舞者無間斷旋轉的非凡能力。 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. But the hard part is maintaining the rotation. As she turns, friction between her pointe shoe and the floor, and somewhat between her body and the air, reduces her momentum. So how does she keep turning?
舞者開始做 fouetté 是用腳頂推地面製造扭轉力。 但困難的部份是保持旋轉。 當她旋轉時, 摩擦力在尖頭鞋和地板之間, 以及有些在身體及空氣間, 降低了她的動量。 所以她是如何保持旋轉?
Between each turn, the dancer pauses for a split second and faces the audience. Her supporting foot flattens, and then twists as it rises back onto pointe, pushing against the floor to generate a tiny amount of new torque. At the same time, her arms sweep open to help her keep her balance. The turns are most effective if her center of gravity stays constant, and a skilled dancer will be able to keep her turning axis vertical.
每個旋轉之間, 舞者會有一瞬間暫停且面對觀眾, 她支撐的腳會放平, 當又踮起腳尖時會再次旋轉, 因為她頂推地板 再次產生少許的新扭轉力。 同時,她的手臂迅速展開以保持平衡。 如果她的重心保持穩定, 則旋轉最具效率, 一位技術純熟的舞者 能保持旋轉軸垂直。
The extended arms and torque-generating foot both help drive the fouetté. But the real secret and the reason you hardly notice the pause is that her other leg never stops moving. During her momentary pause, the dancer's elevated leg straightens and moves from the front to the side, before it folds back into her knee. By staying in motion, that leg is storing some of the momentum of the turn. When the leg comes back in towards the body, that stored momentum gets transferred back to the dancer's body, propelling her around as she rises back onto pointe. As the ballerina extends and retracts her leg with each turn, momentum travels back and forth between leg and body, keeping her in motion.
展開的手臂和產生扭轉力的腳 共同協力推動 單足趾尖旋轉。 但真正秘訣以及 你幾乎不覺得停頓的原因 是因為她的另一條腿永遠在動。 在動量暫停之際, 舞者那隻抬起的腿打直, 並且從前面移到旁邊, 然後向後折回到膝蓋。 藉著持續動作, 那下肢儲存了一些旋轉的動量, 當下肢縮回靠近身體時, 儲存的動量轉移到舞者身上, 當又踮起腳尖時,動量推動她旋轉。 當芭蕾舞者每次旋轉, 伸直與縮回她的下肢, 動量來回於下肢和身體之間, 讓她繼續轉動。
A really good ballerina can get more than one turn out of every leg extension in one of two ways. First, she can extend her leg sooner. The longer the leg is extended, the more momentum it stores, and the more momentum it can return to the body when it's pulled back in. More angular momentum means she can make more turns before needing to replenish what was lost to friction.
一位非常好的芭蕾舞者 能每次下肢伸直時 旋轉多於一圈。 就是用下列兩個方法之一。 第一,她可提早伸直下肢。 下肢伸直的時間越久, 所儲存的動量就越多, 當下肢縮回時, 就有更多動量回到身體, 更多的角動量意味著 她可轉更多圈 ── 在因摩擦力損失動量,需再補充之前。
The other option is for the dancer to bring her arms or leg in closer to her body once she returns to pointe. Why does this work? Like every other turn in ballet, the fouetté is governed by angular momentum, which is equal to the dancer's angular velocity times her rotational inertia. And except for what's lost to friction, that angular momentum has to stay constant while the dancer is on pointe. That's called conservation of angular momentum. Now, rotational inertia can be thought of as a body's resistance to rotational motion. It increases when more mass is distributed further from the axis of rotation, and decreases when the mass is distributed closer to the axis of rotation. So as she brings her arms closer to her body, her rotational inertia shrinks. In order to conserve angular momentum, her angular velocity, the speed of her turn, has to increase, allowing the same amount of stored momentum to carry her through multiple turns. You've probably seen ice skaters do the same thing, spinning faster and faster by drawing in their arms and legs.
另一個方法是 當舞者又踮起腳尖時, 馬上將手臂和下肢縮得更靠近身體。 這為什麼管用呢? 就像芭蕾其他每個旋轉一樣, 單足趾尖旋轉 受角動量 (L) 的影響, 舞者的 角動量 (L) = 角速度 (ω) * 轉動慣量 (I) 除了摩擦力流失的動量外, 當舞者踮腳尖時, 角動量 (L) 是維持恆定的, 這稱為「角動量守恆定律」。 現在,轉動慣量(I) 可想成是身體對旋轉的阻力, 當越多質量離旋轉軸越遠時, 轉動慣量(I)會增加; 當質量離旋轉軸越近時, 轉動慣量(I)會減少。 所以當她把手臂縮更靠近身體時, 轉動慣量(I)會變小, 為了保持角動量 (L), 角速度(ω)── 即她的旋轉速度必須增加。 用相同的儲存動量, 就能讓她旋轉許多圈。 你可能看過溜冰者作過相同動作, 旋轉得越來越快── 藉著縮回他們的手臂和下肢。
In Tchaikovsky's ballet, the Black Swan is a sorceress, and her 32 captivating fouettés do seem almost supernatural. But it's not magic that makes them possible. It's physics.
在柴可夫斯基的芭蕾舞劇, 黑天鵝是位女巫, 她迷人的 32 單足趾尖旋轉 真是神乎其神, 但並不是魔術讓旋轉成為可能, 是物理!