WEBVTT

00:00:06.779 --> 00:00:09.094
In the third act of "Swan Lake,"

00:00:09.094 --> 00:00:14.021
the Black Swan pulls off a seemingly
endless series of turns,

00:00:14.021 --> 00:00:16.885
bobbing up and down on one pointed foot

00:00:16.885 --> 00:00:22.872
and spinning around, and around,
and around 32 times.

00:00:22.872 --> 00:00:25.444
It's one of the toughest sequences
in ballet,

00:00:25.444 --> 00:00:27.217
and for those thirty seconds or so,

00:00:27.217 --> 00:00:31.142
she's like a human top 
in perpetual motion.

00:00:31.142 --> 00:00:34.390
Those spectacular turns
are called fouettés,

00:00:34.390 --> 00:00:36.103
which means "whipped" in French,

00:00:36.103 --> 00:00:40.348
describing the dancer's incredible
ability to whip around without stopping.

00:00:40.348 --> 00:00:44.437
But while we're marveling at the fouetté,
can we unravel its physics?

00:00:44.437 --> 00:00:48.881
The dancer starts the fouetté by pushing
off with her foot to generate torque.

00:00:48.881 --> 00:00:52.164
But the hard part 
is maintaining the rotation.

00:00:52.164 --> 00:00:53.034
As she turns,

00:00:53.034 --> 00:00:55.541
friction between her pointe shoe 
and the floor,

00:00:55.541 --> 00:00:58.469
and somewhat between her body and the air,

00:00:58.469 --> 00:01:00.181
reduces her momentum.

00:01:00.181 --> 00:01:02.177
So how does she keep turning?

00:01:02.177 --> 00:01:07.425
Between each turn, the dancer pauses
for a split second and faces the audience.

00:01:07.425 --> 00:01:08.966
Her supporting foot flattens,

00:01:08.966 --> 00:01:12.375
and then twists as it rises 
back onto pointe,

00:01:12.375 --> 00:01:16.788
pushing against the floor to generate
a tiny amount of new torque.

00:01:16.788 --> 00:01:21.357
At the same time, her arms sweep open
to help her keep her balance.

00:01:21.357 --> 00:01:25.508
The turns are most effective if her center
of gravity stays constant,

00:01:25.508 --> 00:01:30.308
and a skilled dancer will be able to keep
her turning axis vertical.

00:01:30.308 --> 00:01:33.038
The extended arms 
and torque-generating foot

00:01:33.038 --> 00:01:35.359
both help drive the fouetté.

00:01:35.359 --> 00:01:38.701
But the real secret and the reason
you hardly notice the pause

00:01:38.701 --> 00:01:41.921
is that her other leg never stops moving.

00:01:41.921 --> 00:01:43.727
During her momentary pause,

00:01:43.727 --> 00:01:48.263
the dancer's elevated leg straightens
and moves from the front to the side,

00:01:48.263 --> 00:01:50.848
before it folds back into her knee.

00:01:50.848 --> 00:01:56.282
By staying in motion, that leg is storing
some of the momentum of the turn.

00:01:56.282 --> 00:01:58.602
When the leg comes back in 
towards the body,

00:01:58.602 --> 00:02:02.265
that stored momentum gets transferred
back to the dancer's body,

00:02:02.265 --> 00:02:06.260
propelling her around as she rises
back onto pointe.

00:02:06.260 --> 00:02:09.638
As the ballerina extends and retracts
her leg with each turn,

00:02:09.638 --> 00:02:14.183
momentum travels back and forth
between leg and body,

00:02:14.183 --> 00:02:16.333
keeping her in motion.

00:02:16.333 --> 00:02:20.517
A really good ballerina can get more
than one turn out of every leg extension

00:02:20.517 --> 00:02:22.258
in one of two ways.

00:02:22.258 --> 00:02:24.571
First, she can extend her leg sooner.

00:02:24.571 --> 00:02:28.306
The longer the leg is extended,
the more momentum it stores,

00:02:28.306 --> 00:02:32.555
and the more momentum it can return
to the body when it's pulled back in.

00:02:32.555 --> 00:02:35.229
More angular momentum means
she can make more turns

00:02:35.229 --> 00:02:38.907
before needing to replenish
what was lost to friction.

00:02:38.907 --> 00:02:40.603
The other option is for the dancer

00:02:40.603 --> 00:02:44.054
to bring her arms 
or leg in closer to her body

00:02:44.054 --> 00:02:45.819
once she returns to pointe.

00:02:45.819 --> 00:02:47.040
Why does this work?

00:02:47.040 --> 00:02:48.719
Like every other turn in ballet,

00:02:48.719 --> 00:02:51.427
the fouetté is governed 
by angular momentum,

00:02:51.427 --> 00:02:56.506
which is equal to the dancer's angular
velocity times her rotational inertia.

00:02:56.506 --> 00:02:58.662
And except for what's lost to friction,

00:02:58.662 --> 00:03:03.464
that angular momentum has to stay
constant while the dancer is on pointe.

00:03:03.464 --> 00:03:06.850
That's called conservation
of angular momentum.

00:03:06.850 --> 00:03:09.383
Now, rotational inertia can be thought of

00:03:09.383 --> 00:03:12.799
as a body's resistance 
to rotational motion.

00:03:12.799 --> 00:03:17.788
It increases when more mass is distributed
further from the axis of rotation,

00:03:17.788 --> 00:03:22.548
and decreases when the mass is distributed
closer to the axis of rotation.

00:03:22.548 --> 00:03:25.029
So as she brings her arms closer
to her body,

00:03:25.029 --> 00:03:28.088
her rotational inertia shrinks.

00:03:28.088 --> 00:03:29.988
In order to conserve angular momentum,


00:03:29.988 --> 00:03:32.689
her angular velocity,
the speed of her turn,

00:03:32.689 --> 00:03:33.975
has to increase,

00:03:33.975 --> 00:03:36.292
allowing the same amount 
of stored momentum

00:03:36.292 --> 00:03:39.294
to carry her through multiple turns.

00:03:39.294 --> 00:03:42.152
You've probably seen ice skaters
do the same thing,

00:03:42.152 --> 00:03:46.014
spinning faster and faster
by drawing in their arms and legs.

00:03:46.014 --> 00:03:49.813
In Tchaikovsky's ballet, the Black Swan
is a sorceress,

00:03:49.813 --> 00:03:55.036
and her 32 captivating fouettés do seem
almost supernatural.

00:03:55.036 --> 00:03:57.505
But it's not magic that 
makes them possible.

00:03:57.505 --> 00:03:58.723
It's physics.