The Physics of Ballet (the nutcracker snow scene)

Andrew Rogowski Comment: Very well put together. Great job, you deserve an A!

The Pirouette

To perform a pirouette, the dancer needs to first find her center of balance. To accomplish this her net force must equal zero, meaning that the gravity pulling down on her and the force pushing up on her supporting leg on the ground must also equal zero. To distribute her weight evenly, she must plie (bend her knees) into a sort of lung position, and avoid leaning towards the right or left because if her center of gravity falls past her supporting leg then she will fall over. The feet are in fourth position; "turned out" so that her toes are pointing completely towards the right or left.


All of this about balance applies to the pirouette as well as the arabesque that the snow queen performs (pictured below) as well as anytime she is on pointe.
external image 1st%20arabesque.JPG

Next (once she is balanced), the dancer musit position her arms to prepare for the turn. To do an "outside" turn (which is used in ballet when a dancer has her leg bent and facing outwards) one arm must be in first position and the other must be fully extended to the side. This is where she gets her momentum.


Next, the real physics comes in. A torque, or rotational force, is exerted on the floor which creates the angular acceleration that is needed for rotational movement. To successfully complete the rotation, the dancer must push her shoulders back, keep her core tight and tuck her bottom under to create a straight line down the center of the body. This straight line serves as the axis of rotation.

Now in this video, it is easier for the dancer to have a defined axis of rotation because the snow queen uses her partner's arm as the tip of her axis. Normally, dancers have to find their center themselves and try to maintain the perfect posture to keep the line of their head, their back and their supporting leg absolutely straight, which is near impossible.
T(torque)=F(Force) x L (distance of the lever arm)

The lever arm connects the axis to the point of force application


In the example above, the dancer's angular velocity decreases and the rotational intertia increases because her leg is extended. The oppostite is true for the snow queen in the video because, unlike other versions of this ballet, in this video her leg is is passe (bent and connected to the other leg) and not extended. So basically, what we can learn from this is that the larger the velocity, the faster the dancer moves because once she pulls her leg in and it is no longer extended, she rotates quicker.
In the Nutcracker clip this is what her turn actually looks like (in passe, the faster version):
external image Pirouette.jpg

The Grande Jete (AKA leap or jump)

Leaps, like turns, can be done a variety of different ways in dance and even within the style of ballet. In the video, the snow queen does a grande jete assisted by her partner with his hands around her waist. But for the sake of simplicity I'll ignore that added force (because usually dancers do leaps on their own, and she still probably did most of the work and the assistsance was for visual effect)
To achieve height in a jump, the dancer must plie (bend) her knees like she did before her turn as described earlier, and the farther she bends the more height she will achieve (farther down the force is exerted, the farther it will go upwards). This leap requires a forward distance too, and so she will follow the path of the trajectory, which follows a parabolic curve (trajectory=path that a moving object follows through space at a certain speed). The velocity decreases as the dancer reaches the height of the parabolic curve and then the velocity increases as the dancer approaches the ground again