The Physics of Ballet: Turning
When asked where I’ve seen physics in the world around me, I immediately thought of the many years I have spent dancing, specifically studying ballet and pointe. While there are many different types of turns in ballet there is one, the fouetté turn, that is considered especially challenging (I can confirm). Unsurprisingly, it involves many physics concepts from this semester such as force, torque, angular velocity, moment of inertia, and conservation of angular momentum.
To break down the turn there is the following diagram.
The beginning of this video shows a principal dancer from the Pacific Northwest Ballet Company performing a famous variation from Swan Lake where the role of black swan typically completes 32 fouetté turns.
https://www.youtube.com/watch?v=wxT5gnXs4Ug
In order to start the rotational motion, there is torque applied by the two feet. The back foot pushes off of the floor completely while the front foot pushes up to the tip of the pointe shoe. Thus the lever arm is the distance between the two feet and the force is perpendicular out to the side. The axis of rotation is going through the top of the dancer’s head down into the floor. It is important that when starting, the pelvis is tucked under, in line with the upper body which is positioned upright. This is so that when the weight transfers onto the standing leg, the body is in line with the vertical axis of rotation. If the dancer is not upright and has more of her weight distributed forward and she allows her upper body to be at an angle away from the axis of rotation (towards the ground), she is creating an x-component to the force of gravity. In an ideal situation where the dancer is staying in place when she turns and her body is in line with the axis of rotation, gravity is acting in the y direction (equal and opposite to the normal force) with no forces acting in the x direction. An added x-component of the gravitational force will make the turn more challenging because she will have to exert a force equal and opposite to keep herself from falling in whatever direction she’s leaning. This remains true at all times while turning, it is best to maintain a center of gravity closely in line with the axis of rotation.
*Disregarding the force of friction and tension, in the moment where the leg is not extended
Similar to the examples we have seen with ice skating, the motion of the arms and the leg that extends pertain to the moment of inertia. In this system, angular momentum is considered to be conserved despite the force of friction between the tip of the pointe shoe and the floor. In order to keep turning, the dancer will lower her foot flat and bend her standing leg before straightening and raising back up onto the tip of her pointe shoe. At the same time as this is happening, her other leg is extending out straight away from her knee and being pulled to the side. Additionally her arms are mimicking the same motion before returning to a circular form. This increases her moment of inertia by lengthening the radius from the axis of rotation which in turn slows down her angular velocity due to conservation of angular momentum.
I0ω0 = Ifωf
When she then pulls her leg and both arms back in closer to her body, the radius decreases so the moment of inertia decreases. This causes her angular velocity to then increase.
References:
https://www.ted.com/talks/arleen_sugano_the_physics_of_the_hardest_move_in_ballet?language=en#t-78798
http://iceskatingresources.org/physicsballet.pdf
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