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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.