Physics in Ballet

Energy: It is the ability to do work. The main type of energy is mechanical energy.

Mechanical Energy: The energy due to the position and movement of an object.

Potential Energy: Stored energy. It depends on the weight and the height of the object.

Kinetic Energy: Energy of motion. It depends on the mass and the speed of an object.

Chemical Energy: It is energy stored in the bonds of chemical compounds. The chemical energy can be found in food, and this energy can be converted into mechanical energy and heat by the body.

Gravitational Energy: Gravitational energy is the potential energy a body with mass has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field. 

Elastic Energy: It refers to the energy that is stored when a force is applied to deform an elastic object, and it is stored until the force is removed and the object returns to its original shape.

Applying the terms in jumps:
During a grand jete (Figure 4), the dancer experiences different types of energy such as chemical, kinetic, potential, gravitational, and elastic energy. 

• The first type of energy that is present is chemical energy, which is present all the time, when the dancer jumps, as well as when she is still. The chemical energy in the food the dancer ate is present before the jump, and it is converted into mechanical energy that is used to perform the jump, as well as heat that is emitted from the dancer's body. 
• The next types of energy are kinetic and potential energy. When the jump starts she has a lot of kinetic energy and is gaining potential energy. At the highest point of the jump that energy transforms into less kinetic energy and more potential energy. Finally, as she lands the energy is completely kinetic energy, until she stops. When she lands the energy is transferred to the floor as well. 
• The third one is gravitational energy, which again is present all the time in the dancer's body. When the dancer is standing on the floor, she experiences the pull that the center of the Earth is exerting on her body, which is her normal gravitational energy, but as she performs the jump, and moves higher, further apart from the center of the Earth, the gravitational energy in her body is increasing, which is what pulls her back down.
• The last type of energy is elastic energy. When applying a force to deform, stretch or bend an object you are storing elastic energy that is stored in the object, in this case in the dancer, until it goes back to its original shape. The dancer has elastic energy at the beginning of the jump as she bends her legs to get the impulse she needs to perform the jump. Then, as she jumps she gains elastic energy when she extends her legs (they are stretching), and opens them in the position of the jump. Finally, as she lands she retracts her legs to the original position they had, and that's when she releases all of the stored elastic energy she accumulated.
• As the kinetic, potential, gravitational and elastic energy are all types of mechanical energy the dancer experiences mechanical energy throughout the whole jump.

Figure 4

Momentum: Inertia in motion. It's the product of the mass of an object times its velocity.

Angular Momentum: Rotational inertia times rotational velocity.

Impulse: It's the product of force and time interval. The shorter the time the stronger the force, and viceversa.

Rotational Motion

Rotational Speed: Is the number of rotations per unit of time.

Radial Distance: Distance from the axis of rotation. 

Rotational Inertia: The property of objects to resist change in their rotational state of motion.

Torque: The force needed to change the state of motion of a rotation.

Center of Mass: The average position of all the mass that makes up the object. It's proportional to the center of gravity.

Centripetal Force: Any force directed toward a fixed center.

Applying the terms in:

• Balance: 

For a dancer to have balance, the dancer must keep his or her center of mass/gravity above the base area to have stability. This means that for example, if a dancer was doing an attitude, (Figure 6) her center of gravity/mass must be directly on top of her right foot, which is the base, to give her stability and balance. 

In the same way, the idea of this position is to make an arc with the leg that’s folded (the left leg), and feel like it is trying to get to the opposite shoulder. In this way, as all of the weight of her body is at one single foot the other foot needs to put some of the weight to the opposite side to keep her center of gravity directly on top of her foot and keep her balance.

Her hips, on the other hand, need to keep facing the front and form a perfect square in order to keep her balance, because a misplacement of the hips to either side would cause the center of gravity to shift and cause trouble in the dancer's stability.

Figure 6

• Turns: 

Fouettes: To start the turn the dancers push on the floor, which pushes back with an equal and opposite reaction (Newton’s 3rd Law), to generate torque, which allows dancers to start the turn, but the friction between their foot and the floor cause them to lose speed, and momentum, which is why after every turn dancers pause, face the audience, and their foot flattens as they push on the floor again to generate more torque. In the same way, their arms open and close again which also increases their momentum. Their other leg also extends and folds after every turn to gain more momentum. 

As their legs move from the front to the side it is storing momentum that is then used by dancers when the leg folds back in. This sometimes allows dancers to perform more than one turn out of every leg extension, because the longer the leg is extended, the more momentum it stores, and the more momentum is given to the body when that leg retracts, allowing them to turn more than once out of that single extension.  This happens because of the turn’s angular momentum, which is measured by multiplying the rotational inertia, times the angular velocity. So the rotational inertia is the resistance of a dancers' body to rotational motion, this inertia is increased by increasing the radial distance, or the distance between dancers, or in this case their arms and the legs, form the axis of rotation. This is why after extending the arms and legs they retract them again quickly, decreasing the rotational inertia, so the velocity of the turn must increase in order to maintain the angular momentum. (Figure 7 & 8)

Figure 7

Figure 8

It is important for dancers to keep their center of gravity directly on top of the base, which is their foot, to maintain their balance. For this purpose, the dancers focus on a specific point and turn their heads as quickly as possible to maintain stability. 

In this other turn (Figure 9), the dancer does not retract the leg for the first turns so all of the torque comes from the base foot that flattens and pushes on the floor again after each turn, and from the arms that open and retract. All of this is done to gain more rotational velocity, to maintain the angular momentum, because as the leg is extended throughout most of the turns, the rotational inertia is greater. The dancer retracts the leg slowly and extends it again once to gain more momentum and perform more turns, before continuing his turns a la seconde (with the leg extended).

Figure 9