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In the third act of Swan Lake,
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the Black Swan pulls off a seemingly
endless series of turns,
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bobbing up and down on one pointed foot
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and spinning around, and around,
and around 32 times.
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It's one of the toughest sequences
in ballet,
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and for those thirty seconds or so,
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she's like a human top
in perpetual motion.
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Those spectacular turns
are called fouettés,
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which means "whipped" in French,
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describing the dancer's incredible
ability to whip around without stopping.
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But while we're marveling at the fouetté,
can we unravel its physics?
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The dancer starts the fouetté by pushing
off with her foot to generate torque.
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But the hard part
is maintaining the rotation.
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As she turns,
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friction between her pointe shoe
and the floor,
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and somewhat between her body and the air,
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reduces her momentum.
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So how does she keep turning?
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Between each turn, the dancer pauses
for a split second and faces the audience.
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Her supporting foot flattens,
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and then twists as it rises
back onto pointe,
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pushing against the floor to generate
a tiny amount of new torque.
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At the same time, her arms sweep open
to help her keep her balance.
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The turns are most effective if her center
of gravity stays constant,
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and a skilled dancer will be able to keep
her turning axis vertical.
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The extended arms
and torque-generating foot
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both help drive the fouetté.
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But the real secret and the reason
you hardly notice the pause
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is that her other leg never stops moving.
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During her momentary pause,
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the dancer's elevated leg straightens
and moves from the front to the side,
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before it folds back into her knee.
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By staying in motion, that leg is storing
some of the momentum of the turn.
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When the leg comes back in
towards the body,
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that stored momentum gets transferred
back to the dancer's body,
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propelling her around as she rises
back onto pointe.
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As the ballerina extends and retracts
her leg with each turn,
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momentum travels back and forth
between leg and body,
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keeping her in motion.
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A really good ballerina can get more
than one turn out of every leg extension
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in one of two ways.
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First, she can extend her leg sooner.
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The longer the leg is extended,
the more momentum it stores,
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and the more momentum it can return
to the body when it's pulled back in,
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More angular momentum means
she can make more turns
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before needing to replenish
what was lost to friction.
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The other option is for the dancer
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to bring her arms
or leg in closer to her body
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once she returns to pointe.
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Why does this work?
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Like every other turn in ballet,
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the fouetté is governed
by angular momentum,
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which is equal to the dancer's angular
velocity times her rotational inertia.
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And except for what's lost to friction,
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that angular momentum has to stay
constant while the dancer is on pointe.
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That's called conservation
of angular momentum.
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Now, rotational inertia can be thought of
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as a body's resistance
to rotational motion.
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It increases when more mass is distributed
further from the axis of rotation,
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and decreases when the mass is distributed
closer to the axis of rotation.
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So as she brings her arms closer
to her body,
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her rotational inertia shrinks.
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In order to conserve angular momentum,
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her angular velocity,
the speed of her turn,
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has to increase,
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allowing the same amount
of stored momentum
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to carry her through multiple turns.
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You've probably seen ice skaters
do the same thing,
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spinning faster and faster
by drawing in their arms and legs.
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In Tchaikovsky's ballet, the black swan
is a sorceress,
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and her 32 captivating fouettés do seem
almost supernatural.
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But it's not magic that
makes them possible.
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It's physics.