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.
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?
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?
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.
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.
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.
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.
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.
